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Graduate Studies The Vault: Electronic Theses and Dissertations

2015-09-29

Least Squares and Moments Estimation for Gumbel

AR(1) Model

Zhang, Qicheng

Zhang, Q. (2015). Least Squares and Moments Estimation for Gumbel AR(1) Model (Unpublished

master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/26743

http://hdl.handle.net/11023/2549

master thesis

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UNIVERSITY OF CALGARY

Least Squares and Moments Estimation for Gumbel AR(1) Model

by

Qicheng Zhang

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS AND STATISTICS

CALGARY, ALBERTA

September, 2015

c© Qicheng Zhang 2015

Abstract

The Generalized Extreme Value (GEV) distribution in the Extreme Value Theory (EVT)

has been widely applied to analyze various extreme values. The Gumbel, one of the three

families of GEV, is a light-tailed distribution family. Since the independence among extremes

rarely holds in application, dependent models are more practical. This thesis focuses on a

temporally dependent model, specifically the Gumbel autoregressive model of order 1. We

proposed a special method of moments estimation for the location and the scale parameters

based on a link between Exponential distribution and Gumbel distribution. The existence

and uniqueness of the estimates have been proven. Along with the least squares method

for the autoregressive coefficient, we investigated the performance of those estimators and

compared them with the combination of the Yule-Walker estimator and the ordinary method

of moments estimators, which was used in an environmental study conducted by Toulemonde

et al.(2010). The average run length (ARL) of the model is studied through simulation. Two

examples of real data are used to illustrate our methodology.

i

Acknowledgements

I would like to express my sincere gratitude to all people who helped me or inspired me to

complete this thesis and my MSc program.

Firstly I want to express my great appreciation to my supervisor Dr. Gemai Chen. He

offered me a precious chance to be his student. He also gave me helpful guidance in living,

studying and dealing with various affairs. I benefited so much from his extensive knowledge

and spirit in both academic and applied aspects.

I would like to thank my committee members, Dr. Xuewen Lu and Dr. Tak Fung for

taking their precious time to read my thesis and to give valuable advices.

Furthermore, I would like to express my appreciation to Dr. Chao Qiu who helped me

with my studies and research.

I also would like to appreciate the financial support from the Department of Mathematics

and Statistics at University of Calgary.

Moreover, I would like to thank my parents for their selfless support and love. Without

them I would not have accomplished what I have today.

ii

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction to the Autoregressive Gumbel Model . . . . . . . . . . . . . . . 21.2 Review of Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Asymptotic Probability Models of Extreme Values . . . . . . . . . . . . . . . 5

2.1.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Extremal Types Theorem . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 The Generalized Extreme Value Distribution . . . . . . . . . . . . . . 72.1.4 Return Level and Return period . . . . . . . . . . . . . . . . . . . . . 82.1.5 Asymptotic Models for Minima . . . . . . . . . . . . . . . . . . . . . 9

2.2 Gumbel Autoregressive (1) Model . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 The Ordinary Method of Moments and the Yule-Walker Estimation . 122.2.3 A New Method of Moments and a Least Squares Estimation . . . . . 132.2.4 Return Level and Run Length of Gumbel AR(1) Model . . . . . . . . 16

3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.0.5 Parameters Estimations . . . . . . . . . . . . . . . . . . . . . . . . . 183.0.6 Return Level and Average Run Length . . . . . . . . . . . . . . . . . 26

4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A Proofs of Important Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.1 The parameterization of the positive α-stable variable . . . . . . . . . . . . . 39A.2 The proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.3 The Existence and Uniqueness of the New Method of Moments Estimate . . 41B Useful R code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.1 Generating Gumbel Autoregressive(1) Sequence . . . . . . . . . . . . . . . . 50Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

iii

List of Tables

3.1 Simulated MSEs (n=20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Simulated Biases (n=20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Simulated MSEs (n=50) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Simulated Biases (n=50) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Simulated MSEs (n=200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6 Simulated Biases (n=200) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.7 Average Run Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1 Example: Annual Maximum Water Flow (m3/s)(Bow River, Calgary, 1911-2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Example: Daily Maximum Wind Speed (km/h)(Richmond, VA, USA 1979) . 32

iv

List of Figures and Illustrations

2.1 PDFs and CDFs of standard GEV distributions . . . . . . . . . . . . . . . . 82.2 PDFs and CDFs of Gumbel distributions . . . . . . . . . . . . . . . . . . . . 10

3.1 Simulated MSEs and Biases (n = 20) . . . . . . . . . . . . . . . . . . . . . . 203.2 Simulated MSEs and Biases (n = 50) . . . . . . . . . . . . . . . . . . . . . . 223.3 Simulated MSEs and Biases (n = 200) . . . . . . . . . . . . . . . . . . . . . 243.4 Average Run Length of Gumbel AR(1) Model . . . . . . . . . . . . . . . . . 27

4.1 Plot of annual maxima of water flow . . . . . . . . . . . . . . . . . . . . . . 294.2 Consecutive plot of annual maxima of water flow . . . . . . . . . . . . . . . 304.3 ACF and PACF of annual maxima of water flow . . . . . . . . . . . . . . . . 314.4 QQ plot and PP plot of annual maxima of water flow . . . . . . . . . . . . . 314.5 Plot of daily maxima of wind speed . . . . . . . . . . . . . . . . . . . . . . . 334.6 Consecutive plot of daily maxima of wind speed . . . . . . . . . . . . . . . . 344.7 ACF and PACF of daily maxima of wind speed . . . . . . . . . . . . . . . . 354.8 QQ plot and PP plot of daily maxima of wind speed . . . . . . . . . . . . . 35

v

Chapter 1

Introduction

How tough should the skyscrapers be designed to resist wind power? The 2013 flood in

Calgary caused huge damage to residential buildings and 5 people’s death. The extreme high

temperature in India caused by heat wave killed more than 2,300 people in June 2015.(Inani,

2015) Is there any way to reduce loss in lives and properties before those diasters? These

issues have a common feature for these issues. Once a specific quantity exceeded a red line,

serious situation emerges. We usually do not care how those quantities behave, as long as

they are below the red lines. If the wind speed is greater than the level that a skyscraper’s

capability can endure, then it becomes a dangerous building. If the water level exceeded

the river levee, then we have flood. If the temperature exceeded the maximum that human

bodies can handle, then people’s lives would be threatened. Therefore, to seek solutions to

those questions, it is necessary to focus on a key quantity, extreme value.

The Extreme Value Theory (EVT) is a special statistical discipline with extensive the-

ories for various extreme values. The study of EVT can be traced back to early 20th

century.(Coles, 2001) It has been applied in many fields including environmental studies,

economics, insurance and engineering. The EVT is built on firm mathematical foundations.

It reached its milestone since Fisher and Tippett (1928) discovered that extreme events have

their special distribution families. The theory was applied to predict flood levels based on

historical data with great success.(Ji, 2013) Subsequently, EVT has undergone major devel-

opments. Today many theories and techniques for dealing with extreme values are based on

the Generalized Extreme Value (GEV) distribution founded by Fisher and Tippett (1928).

In this thesis, we focus on one case of the GEV, the Gumbel distribution.

1

1.1 Introduction to the Autoregressive Gumbel Model

The GEV distributions consist of three families, Gumbel, Weibull and Frechet. It is also

known as the Fisher-Tippett extreme value distribution. These three families are proved to

be the only three possibilities of limiting distributions for maxima, assuming observations are

independently and identically distributed (IID). The name of the three families came from

Emil Gumbel (1891), a German mathematician, Leonard Henry Caleb Tippett (1902-1985),

a English statistician and famous Sir Ronald Aylmer Fisher (1890-1962). The role of GEV

distributions in modelling extremes is similar to the role of Normal (Gausian) distribution

in dealing with averages. It is well known that, with the central limit theorem, normal

distribution has been applied in countless situations regardless of the distribution of the

observations.

The three families of GEV have different characteristics. A distinct one is their support.

The Weibull and Frechet families have left bound and right bound respectively. The Gumbel

family has a support on the entire real line. The Gumbel distribution has been mostly used to

model light-tailed extremes whereas the Frechet has been applied in heavy-tailed extremes.

Methodologies of estimation, prediction and diagnostics have been developed for GEV

IID models. They can be found in most classical texts. The popular maximum likelihood

estimation (MLE) with asymptotic efficiency is available and usually preferred. The IID

setting is convenient for computation and interpretation. However, in practice, the inde-

pendence rarely holds. The MLE built upon IID setting will cause biases in parameters

estimation when it’s applied to the Gumbel autoregressive (1) model. (Ji, 2013) Based on

an additive property of Gumbel and positive stable variables, we can construct an additive

autoregressive (1) model.(Fougeres et al, 2009). The Gumbel autoregressive (1) model, also

noted as Gumbel AR(1), is the model we will be working on in this thesis.

2

1.2 Review of Previous Studies

Many attempts have been made to model dependent extremes. Davis et al.(1989) discussed

the Max-ARMA process and provided some results. Leadbetter (1983) studied extremes in

local dependent stationary sequence. Leadbetter and Rootzen (1988) unified some results for

extremes of stochastic process. Cores (1999) studied various dependent extremes including

spatial regression models for extremes and dependent multivariate extremes. Cores (2001)

also systematically summarized dependent extreme values modelling. Zhang and Smith

(2008) obtained method of determining the order of the max-stable process and estimating

parameters. Fougeres, Nolan and Rootzen (2009) used mixtures of positive stable variables

to model dependent extremes., in which a way of constructing the Gumbel AR(1) model

was proposed. Hughes et al.(2006) used MLE of linear ARMA model to analyzed maximum

and minimum temperatures in the Antarctic Peninsula. Toulemonde et al.(2010) obtained

the asymptotic variance-covariance matrix for the Yule-Walker estimator and the moments

estimators of the Gumbel AR(1), applying the method to analyze maxima of methane and

nitrous oxide.

1.3 Thesis Plan

As Ji (2013) pointed out, if a dependent extreme value sequence is analyzed under indepen-

dent assumption, biases of maximum likelihood estimators (MLEs) would emerge especially

when the sample size n is small and the dependence is strong. If we take the dependence into

account, how should we estimate the parameters? In this thesis, we will study an estimation

method, which combines a least squares estimator (LSE) and a new method of moments

estimators (NMMEs). Our goal is to investigate the performance of the estimation on Gum-

bel AR(1) model parameters and compare with the method provided by Toulemonde, G.,

Guillou, A.(2010).

The thesis is organized as follows. In Chapter 2, we introduce the GEV distribution and

3

our Gumbel AR(1). We explain our estimation method, LSE and NMMEs, as well as the

estimation method used in Toulemonde et al.’s study.(2010). The latter will be presented

for comparison. In Chapter 3, we will run simulations to study performance of these two

estimations. The bias and the mean squared error are investigated. We also study the

average run length (ARL) of the Gumbel AR(1). In Chapter 4, we apply our estimation

method to two examples of real data. Chapter 5 will summarize the comparison among our

method and the one in Toulemonde et al.’s study as well as the ARL. Some limitations of

the research and suggestions for future studies are also discussed.

4

Chapter 2

Methodology

In this Chapter, we are going to introduce the Gumbel AR(1) model and its two estimation

methods, the method proposed by Toulemonde et al.(2010) and a combination of a new

method of moments and a method of least squares. In the first section, the general extreme

values (GEV) distributions and their properties are introduced. After that, the Gumbel AR

(1) model and its generating method are provided. In the last section, the two approaches

of estimation for comparison will be discussed.

2.1 Asymptotic Probability Models of Extreme Values

2.1.1 Model Formulation

We start with the foundation of extreme value theory.

As the concept extreme values indicated, the statistics of interest are the maximum and

minimum of a group of data. Our main focus is on the maximum, denoted as Mn, of a

sequence of random variables X1, X2, . . . , Xn, that is,

Mn = max{X1, X2, . . . , Xn}

where X1, X2, . . . , Xn are independent variables with an identical distribution function F . In

practical setting, the Xi usually come from a process measured successively on regular time-

scale such as hourly measurements of temperature, daily electricity consumption or weekly

sales revenue. Then Mn is the maximum of the process among n time units of observation.

If n time units sum up to one year, then Mn is the annual maximum.

If we know the distribution function of X1, X2, . . . , Xn, then we can obtain the exact

distribution function of Mn, namely,

5

P (Mn ≤ x) = P (X1 ≤ x,X2 ≤ x, . . . , Xn ≤ x)

= P (X1 ≤ x)P (X2 ≤ x) . . . P (Xn ≤ x)

= [F (x)]n. (2.1)

Unfortunately, the distribution function F is unknown in most situations. We may use

some standard techniques to estimate F based on sample data, and then derive our distri-

bution of Mn by equation (2.1). The problem of this approach is, as Cores (2001) pointed

out, is that even a small deviation in the F estimate can cause substantial discrepancies for

F n in practice.

Alternatively, we can focus on approximate distribution of Mn directly. It’s similar to

approximate sampling distribution of sample mean by using the central limit theorem. Let

x+ be the upper end-point of F , then for any x < x+, F n(x)→ 0 as n→∞. The distribution

of Mn degenerates to a point mass at x = x+. This problem can be solved by introducing a

linear rescaling of the variable Mn,

M∗n =

Mn − µnσn

with sequences of constants {µn} and {σn > 0}. As n increases, the location and scale of

M∗n are stabilized by choosing proper {µn} and {σn}. Therefore, we are interested in the

asymptotic distribution of M∗n instead of Mn with suitable {µn} and {σn}.

2.1.2 Extremal Types Theorem

All possibilities of limit distributions of M∗n are provided by the following theorem, the

extremal types theorem (Cores, 2001).

Extremal Types Theorem. If there exist two sequences of constants {µn} and {σn > 0}

such that

6

P [(Mn − µn)/σn ≤ x]→ F (x), as n→∞,

where F is a non-degenerate distribution function, then F belongs to one of the following

families:

I : F (x) = exp

[−exp

(−x− µ

σ

)],−∞ < x <∞;

II : F (x) = exp

[−(−x− µ

σ

)β], x < µ;

III : F (x) = exp

[−(x− µσ

)−β], x > µ;

for parameters µ, σ > 0, and in the cases of families II and III, β > 0.

These three families labelled I, II and III are known as the Gumbel, Weibull and Frechet

families, respectively. All families have a location parameter, µ, and a scale parameter, σ.

The Weibull and Frechet families have a shape parameter β.

For the Gumbel family, the following property can be easily derived and will be used in

estimating the location and the scale parameters.

Lemma 1. If X has a Gumbel(µ, σ) distribution, then exp(−X−µ

σ

)has an Exponential

distribution with mean 1.

2.1.3 The Generalized Extreme Value Distribution

The asymptotic distribution of the sample maxima has three distinct forms. In practice, the

selection among the three families may be problematic. However, there is a unified form:

F (x) = exp

[−(

1− ε(x− µσ

))1/ε], (2.2)

7

where 1− ε(x− µ)/σ > 0.

The probability density functions (PDFs) and cumulative distribution functions (CDFs)

of standardized (µ = 0, σ = 1) GEV distributions are provided in Figure 2.1 , in which we

take ε > 0, ε = 0 and ε < 0 to obtain Weibull, Gumbel and Frechet families, respectively.

Figure 2.1: PDFs and CDFs of standard GEV distributions

2.1.4 Return Level and Return period

For the GEV distribution defined in (2.2), the (1− p)-quartile is given by

xp = µ+σ[1− (−log(1− p))ε]

ε.

The quartile of the Gumbel family is obtained by taking the limit of xp as ε approaches 0,

xp = µ− σlog(−log(1− p)). (2.3)

By definition, the xp is the return level corresponding to return period 1/p. That is, under

the independent and the identical distribution (IID) assumption, the level xp is expected to

8

be exceeded on average once every 1/p units of time. For any particular unit of time, the

maximum has probability p to exceed xp.

2.1.5 Asymptotic Models for Minima

Sometimes it’s necessary to model minimum instead of maximum. For example, in assess-

ment of lifetime of a electronic equipment. The components of the equipment are in series

connection so that any component fails would make the equipment fail. Hence the lifetime

of the equipment is the minimal lifetime of all components. Also in the air traffic control

department, if the minimum height of clouds is too low, it would be dangerous for air-

crafts landing and taking off. To model the minima, we denote Mn = min{X1, X2, . . . , Xn},

where the Xi’s are IID. The limit distribution of Mn can be obtained by applying analogous

arguments to Mn which are applied to Mn.

Let Yi = −Xi for i = 1, 2, . . . , n, the change of sign means that the small values of Xi’s

are corresponding to large values of Yi’s. Specifically, if Mn = min{X1, X2, . . . , Xn} and

Mn = max{X1, X2, . . . , Xn}, then Mn = −Mn. Therefore, for large n,

P (Mn ≤ x) = P (−Mn ≤ x)

= P (Mn ≥ −x)

= 1− P (Mn ≤ −x)

≈ 1− exp

[−(

1− ε(−x− µ

σ

))1/ε]

= 1− exp

[−(

1− ε(µ− xσ

))1/ε]

where 1− ε(µ−xσ

)> 0 and µ = −µ. This is the GEV distribution for minima.

9

2.2 Gumbel Autoregressive (1) Model

2.2.1 Model Formulation

The Gumbel family can be viewed as a limiting distribution of GEV in (2.2) by letting ε

approach to 0. The PDFs and CDFs of Gumbel distributions with different parameters are

displayed in Figure 2.2.

Figure 2.2: PDFs and CDFs of Gumbel distributions

The mathematical foundation of the Gumbel autoregressive model is the following addi-

tive property of the positive stable variable and the Gumbel variable.

Lemma (Additive Property). Let S be a positive α-stable variable defined by its Laplace

transform

E(exp(−uS)) = exp(−uα), u ∈ [0,∞), α ∈ (0, 1). (2.4)

Let X be a Gumbel(0, σ) distributed variable independent from S. Then the sum X+σlogS

is also Gumbel distributed with location parameter 0 and scale parameter σ/α.

10

One proof of this additive property was provided by Fougeres et al. (2009).

Based on the additive property, the Gumbel(0, σ) autoregressive (1) sequence is con-

structed by the following Corollary.

Corollary. Suppose St be IID positive α-stable variables defined by (2.4) for any t ∈ Z+.

Let X1 be a Gumbel(0, σ) variable independent from {St}∞t=1. X2, X3, . . . are defined by the

recursive relationship

Xt = αXt−1 + ασlogSt−1, t = 2, 3, . . . (2.5)

where σ > 0. Then every term of {Xt}∞t=1 has the same marginal, Gumbel(0, σ).

The Gumbel(µ, σ) autoregressive (1) sequence can be obtained by simply adding µ to

each term of the Gumbel(0, σ) autoregressive(1) sequence. Or equivalently, let {Xt}∞t=1 be a

stochastic process defined by the following recursive relationship

Xt = αXt−1 + ασlogSt−1 + (1− α)µ, t = 2, 3, . . . (2.6)

where {St}∞t=1 is a sequence of i.i.d positive α-stable variables. Let X1 be a Gumbel(µ, σ)

variable independent from {St}∞t=1, then each term of {Xt}∞t=1 follows the Gumbel(µ, σ)

distribution.

For simulation study, it’s vital to be able to generate the Gumble AR(1) sequence. The

key to achieve this is to generate the positive α-stable variable. There are two approaches

available. An algorithm of generating random variable based on its Laplace transform was

provided by Ridout (2009). We can apply this method directly to (2.4). Another approach

is using an R package, ‘stabledist’, provided by Wuertz et al. (2015). This package

contains calculation programs of probability density, cumulative probability and quartile of

general stable family. It also has a program rstable() which can be used to generate general

stable variable using the parameterization S(α, β, γ, δ; 1) specified by Nolan(2009). That is,

11

if a random variable S has a stable distribution with parameter (α, β, γ, δ; 1), denoted as

S ∼ S(α, β, γ, δ; 1), then the characteristic function (c.f) of S is given by

φS(t) = exp[−γα|t|α

[1− iβtan

(απ2

)sign(t)

]+ iδt

](2.7)

If 0 < α < 1, γ = (cos(απ/2))1α , β = 1 and δ = 0, then S(α, β, γ, δ; 1) is the positive

α-distribution defined by (2.4). A proof is provided in Appendix.

2.2.2 The Ordinary Method of Moments and the Yule-Walker Estimation

A simple estimation method, the method of moments, was used to estimate the location

parameter µ and the scale parameter σ in the study of chemicals CH4 and N2O conducted

by Toulemonnde et al.(2010). Let X1,X2,. . .,Xn be the first n terms of the series defined by

(2.6), then the first two moments can be easily obtained.

E(X) = µ+ γσ

E

(1

n

n∑t=1

X2t

)= (µ+ γσ)2 +

π2σ2

6

where γ is the Euler’s constant. It can be obtained by taking the derivative of Gamma

function at 1. That is γ = −Γ′(1). (Spiegel, 1968)

The ordinary method of moments estimators (OMMEs) of µ and σ are therefore found

to be

µO = X − γ

π

√√√√ 6

n

n∑t=1

(Xt − X)2 (2.8)

σO =1

π

√√√√ 6

n

n∑t=1

(Xt − X)2 (2.9)

12

One advantage of the OMMEs is computational simplicity. It does not involve iterative

algorithm.

The method of moments can not generate estimator of α since any moment of Xt doesn’t

contain α. Therefore the inference on α should be obtained through other approaches. In

the study by Toulemonnde et al.(2010), a method called Yule-Walker equation was used to

estimate α. The estimator is obtained by solving the Yule-Walker equation, that is,

αY =

n−1∑t=1

(Xt+1 − X)(Xt − X)

n∑t=1

(Xt − X)2(2.10)

2.2.3 A New Method of Moments and a Least Squares Estimation

The least squares method used in this study is obtained by minimizing summation of squared

distances between observations and their expected values conditioning on previous observa-

tions. In other words, the conditional expectation of Xt+1 given Xt will be treated as the

centre of Xt+1. The utility function to be optimized is therefore given by

Q1 =n−1∑t=1

(Xt+1 − E(Xt+1|Xt))2 =

n−1∑t=1

(Xt+1 − αXt − (1− α)(µ+ γσ))2 (2.11)

in which γ is the Euler’s constant.

The expression of the least squares estimator (LSE) of α is provided by the following

Corollary.

Corollary 1. Let Q1 be the function of α, µ and σ defined by equation (2.11), then Q1 is

minimized with respect to α, µ and σ if and only if

α = αL =

n−1∑t=1

(Xt+1 −X2,n)(Xt −X1,n−1)

n−1∑t=1

(Xt −X1,n−1)2(2.12)

13

and

µ+ γσ = µL + γσL =X2,n − αLX1,n−1

1− αL(2.13)

in which µL, σL and αL are values of µ, σ and α at which Q1 is minimized, Xm,n =

(n∑

t=m

Xt)/(n−m+ 1).

The following useful asymptotic property of the estimator α in (2.12) is given by Balakr-

ishna and Shiji (2013).

√n(αL − α)

d−→ N(0, 1− α2) as n→∞ (2.14)

There is a problem of both LSE and Yule-Walker estimator (Y.W.E) of α. They both have

convenient analytic formulas. However both two estimators sometimes generate negative

estimates of α.

The Corollary 1 also has falsified the uniqueness of the least squares estimators of µ and

σ. In fact there are infinitely many least squares estimates of these two parameters satisfying

equation (2.13). Hence µ and σ must be estimated through other approaches.

We are going to estimate µ and σ by another method of moments. By Lemma 1, the

first and the second moments of exp(−Xt−µσ

) are 1 and 2, respectively. we may obtain new

method of moments estimators (NMMEs) of µ and σ by solving the following system:

1

n

n∑t=1

exp

(−Xt − µN

σN

)= 1; (2.15)

1

n

n∑t=1

exp

(−2(Xt − µN)

σN

)= 2. (2.16)

The existence and the uniqueness of the NMMEs of µ and σ are supported by the following

Corollary.

Corollary 2. If X = (X1, X2, . . . , Xn)T is a n-dimensional random vector such that n ≥ 3

and P (Xi = Xj) = 0 for 1 ≤ i < j ≤ n, then the system of equations (2.15) and (2.16) has

a unique solution (µN , σN) ∈ R× R+ with probability 1.

14

To calculate the estimates, the NMME of µ can be expressed in terms of σN and X, that

is,

µ1(X, σN) = −σN log

[1

n

n∑t=1

exp

(−Xt

σN

)]; (2.17)

µ2(X, σN) = − σN2

log

[1

2n

n∑t=1

exp

(−2Xt

σN

)](2.18)

where the subscript i of µ(X, σ) indicates the expression is obtained from ith moment.

(i = 1, 2)

The bivariate estimation now has been reduced to the solution of the following univariate

equation.

µ1(X, ˆσN)− µ2(X, ˆσN) = 0 (2.19)

Based on the existence, the uniqueness and the dimensional reduction, the new method

of moments estimates of µ and σ can be computed through various algorithms.

In solving equation (2.19) for σN , there is a problem for numerical root. If we let σ

approaches zero from the right, then both limits of µ1(X, σN) and µ2(X, σN) are exactly the

same. We may consequently have a blind numerical solution of (2.19) which is close to 0

regardless of the real value of σN .

With a small initial value, the Newton-Raphson method may generate the blind zero σ.

Hence a sufficient large initial value is necessary to obtain the correct numerical solution.

The Newton-Raphson method also has another drawback. It sometimes produces nonsense

values such as a negative σN . We can use reparameterization to solve this problem. For

example, we define θ = logσ such that the θ is mapping the parameter space of σ onto R,

then the algorithm of estimating σ is always working in an appropriate range.

The Bisection method and the Secant method are simple and more stable in each iterative

calculation than the Newton-Raphson. As previously mentioned, the issue of blind zero also

15

emerges with bad choice of initial values for these two methods. Two initial values should be

such that the function µ1(X, σN)− µ2(X, σN) at the two initial values have opposite signs.

Due to the existence and the uniqueness, to find the estimates of µ and σ is equivalent

to find optimal (µN , σN) in R× R+ such that the following function is minimized.

Q2 =

[1

n

n∑t=1

exp

(−Xt − µN

σN

)− 1

]2+

[1

n

n∑t=1

exp

(−2(Xt − µN)

σN

)− 2

]2(2.20)

This method works efficiently without the blind zero issue. It’s also more stable than

the Newton-Raphson method. Some programs/packages of optimization in various softwares

can be used directly to obtain the estimates of µ and σ. In this study, we use the R program

optim().

2.2.4 Return Level and Run Length of Gumbel AR(1) Model

The concept of return period is defined as the reciprocal of the probability of extreme event.

However, if the IID setting is not appropriate, then the expected waiting time to observe

the extreme event related to xp in (2.3) would be meaningless. For this reason, we study the

average run length (ARL) instead of the expected waiting time.

For a Gumbel AR(1) sequence {Xt}∞t=1 with parameters α, µ and σ defined by (2.6), let

L be the waiting time to see the first Xt exceeding the return level xp defined in (2.3). For

example, if the sequence is 0,−1,−0.8, 1.2, 0.4, 2.1, 1.2, . . . and xp = 2, then L = 6. The

mean of L, average run length, is defined by the formula

E(L) =∞∑k=1

k · P(L = k) (2.21)

in which the probability is given by

P(L = k) =

∫ ∞xp

∫ xp

−∞. . .

∫ xp

−∞fX1,X2,...,Xk(x1, x2, . . . , xk)dx1 . . . dxk−1dxk (2.22)

16

where fX1,X2,...,Xk(x1, x2, . . . , xk) is the joint probability density function (p.d.f) of the first

k components of {Xt}∞t=1.

If k = 1, the integration is simply p. If k > 1, the integration involves the p.d.f of the pos-

itive α-stable variable which has no closed form.(Hougaard, 1986) The region of integration

is complicated. Therefore the average run length is difficult to obtained analytically. In this

study, the ARL of the Gumbel autoregressive (1) model will be investigated by simulation.

17

Chapter 3

Simulation Study

In this chapter, simulations are conducted to study performance of two estimations discussed

in the chapter 2 and the average run length of Gumbel autoregressive (1) model.

3.0.5 Parameters Estimations

To compare the two estimations, we choose α ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} and

the sample size n = 20, 50, 200. Without losing generality, the µ and σ were set to be 0 and

1, respectively.

For each combination of α and n, we generated N = 20, 000 samples of Gumbel AR(1)

model defined by (2.6). Based on each sample, we computed the least squares method

estimator and the Yule Walker estimator of α, the original method of moments estimators

and the new method of moments estimators of µ and σ. The mean square error (MSE) and

the bias of each estimator are estimated by the following expressions.

MSE(θ) =1

N

N∑i=1

(θi − θ)2

Bias(θ) =1

N

N∑i=1

(θi − θ)

where θ can be µ, σ or α and θi is the estimate of θ computed from ith sample.

The MSE and the biases are presented in Tables 3.1-3.6 and Figures 3.1-3.3. We denote

the Yule-Walker estimator, the least squares estimator, the ordinary method of moments

estimator and the new method of moments estimator by Y.W.E, LSE, OMME and NMME,

respectively. For each comparison in Tables 3.1-3.6, the number printed in bold type indicates

that the corresponding estimator has a better performance in the simulation.

18

Table 3.1: Simulated MSEs (n=20)

αEstimators of α Estimators of µ Estimators of σ

Y.W.E LSE OMME NMME OMME NMME

0.1 0.04442 0.04987 0.07283 0.07208 0.04977 0.050670.2 0.04804 0.05267 0.08963 0.08832 0.05084 0.052940.3 0.05117 0.05421 0.10999 0.10755 0.05736 0.056940.4 0.05356 0.05487 0.13937 0.13617 0.06401 0.062470.5 0.05755 0.05662 0.17165 0.16698 0.07409 0.071690.6 0.06436 0.06042 0.22052 0.21402 0.09120 0.085110.7 0.07061 0.06255 0.31759 0.30769 0.11537 0.109880.8 0.07708 0.06369 0.46113 0.44502 0.15780 0.150990.9 0.08941 0.06850 0.79175 0.76390 0.25615 0.25042

Table 3.2: Simulated Biases (n=20)

αEstimators of α Estimators of µ Estimators of σ

Y.W.E LSE OMME NMME OMME NMME

0.1 -0.06531 −0.06303 0.02822 −0.00374 −0.05460 -0.096020.2 -0.08806 −0.08059 0.03356 0.00144 −0.06637 -0.104000.3 -0.10685 −0.09485 0.04084 0.00787 −0.07218 -0.114570.4 -0.12593 −0.10942 0.04996 0.01629 −0.08821 -0.125230.5 -0.14446 −0.12127 0.06560 0.02901 −0.10420 -0.142430.6 -0.17177 −0.14240 0.07742 0.03889 −0.13413 -0.170070.7 -0.19530 −0.15816 0.10356 0.06245 −0.18216 -0.212080.8 -0.22035 −0.17275 0.14853 0.10502 −0.25686 -0.277460.9 -0.25223 −0.18875 0.22875 0.18852 -0.40439 −0.40415

19

Figure 3.1: Simulated MSEs and Biases (n = 20)

20

Table 3.3: Simulated MSEs (n=50)

αEstimators of α Estimators of µ Estimators of σ

Y.W.E LSE OMME NMME OMME NMME

0.1 0.01877 0.01948 0.02912 0.02894 0.02077 0.019720.2 0.01916 0.01974 0.03568 0.03543 0.02263 0.021150.3 0.01872 0.01894 0.04370 0.04337 0.02420 0.022560.4 0.01882 0.01866 0.05521 0.05467 0.02774 0.025590.5 0.01780 0.01718 0.06959 0.06899 0.03305 0.029860.6 0.01801 0.01685 0.09399 0.09263 0.04182 0.036450.7 0.01751 0.01554 0.13200 0.12977 0.05403 0.047460.8 0.01721 0.01414 0.20558 0.20041 0.07881 0.067880.9 0.01734 0.01278 0.40819 0.39503 0.13789 0.12489

Table 3.4: Simulated Biases (n=50)

αEstimators of α Estimators of µ Estimators of σ

Y.W.E LSE OMME NMME OMME NMME

0.1 -0.02711 −0.02555 0.01332 −0.00141 −0.02309 -0.042130.2 -0.03541 −0.03184 0.01603 0.00055 −0.02655 -0.046510.3 -0.04369 −0.03818 0.01825 0.00244 −0.02987 -0.050050.4 -0.05286 −0.04549 0.02179 0.00528 −0.03727 -0.057510.5 -0.05630 −0.04669 0.02112 0.00253 −0.04535 -0.067570.6 -0.06700 −0.05485 0.03125 0.00999 −0.05915 -0.082510.7 -0.07652 −0.06162 0.04853 0.02333 −0.08105 -0.105930.8 -0.08782 −0.06904 0.07425 0.04285 −0.12229 -0.147160.9 -0.09960 −0.07527 0.13032 0.09090 −0.22082 -0.23972

21

Figure 3.2: Simulated MSEs and Biases (n = 50)

22

Table 3.5: Simulated MSEs (n=200)

αEstimators of α Estimators of µ Estimators of σ

Y.W.E LSE OMME NMME OMME NMME

0.1 0.00486 0.00490 0.00717 0.00725 0.00543 0.005120.2 0.00475 0.00478 0.00896 0.00900 0.00587 0.005330.3 0.00457 0.00458 0.01093 0.01101 0.00638 0.005800.4 0.00438 0.00435 0.01407 0.01411 0.00751 0.006760.5 0.00397 0.00392 0.01824 0.01811 0.00904 0.008030.6 0.00352 0.00342 0.02422 0.02415 0.01116 0.009790.7 0.00297 0.00283 0.03254 0.03233 0.01532 0.013200.8 0.00240 0.00220 0.05397 0.05338 0.02320 0.019900.9 0.00174 0.00146 0.11071 0.10926 0.04531 0.03794

Table 3.6: Simulated Biases (n=200)

αEstimators of α Estimators of µ Estimators of σ

Y.W.E LSE OMME NMME OMME NMME

0.1 -0.00720 −0.00672 0.00242 −0.00159 −0.00637 -0.011140.2 -0.00943 −0.00847 0.00544 0.00147 −0.00657 -0.011890.3 -0.00992 −0.00842 0.00582 0.00153 −0.00692 -0.013320.4 -0.01318 −0.01124 0.00440 −0.00042 −0.00931 -0.016550.5 -0.01481 −0.01238 0.00813 0.00252 −0.01064 -0.018210.6 -0.01704 −0.01403 0.00763 0.00094 −0.01663 -0.025010.7 -0.01892 −0.01527 0.01185 0.00299 −0.02083 -0.031950.8 -0.02113 −0.01698 0.02162 0.00939 −0.03361 -0.045370.9 -0.02350 −0.01849 0.03624 0.01584 −0.06933 -0.08764

23

Figure 3.3: Simulated MSEs and Biases (n = 200)

24

In estimation of α, the LSE has better MSEs when α is greater than 0.4. The graphs of

MSEs showed that as α increases, LSE gains greater advantage over Y.W.E. In simulations

with n = 50 and n = 20, that advantage is more distinct. It has been showed that LSE and

Y.W.E have the same asymptotic property, which is verified by the simulation with large

sample size, n = 200. The MSE of both estimators increases with increasing α when n = 20.

That pattern in n = 50 is decreasing for LSE and unstable for Y.W.E. When n = 200, the

decreasing pattern is clear. Both curves of the two estimators become smooth. In terms

of bias, the absolute values of biases of both estimators are increasing as α increases. The

larger α is, the better bias LSE has against Y.W.E. The LSE dominates the entire range

of α in respect of bias. As mentioned in Chapter 2, both LSE and Y.W.E may generate

negative estimates almost simultaneously. This problem happens more frequently when α is

small and n is small. LSE also produces estimates larger than 1 when α is close to 1.

When estimating µ, the MSEs of both estimators are increasing exponentially as α in-

creases. The differences between the two estimators are so close that two curves of MSE

cannot be distinguished. The MSE of NMME is slightly smaller compared to that of OMME

when n = 20 and 50. The OMME is better compared to NMME when n = 200 and α ≤ 0.4.

In aspect of bias, when n = 20 and n = 50, the differences between the two estimators are

almost equal to the same constant for all values of α as the parallel curves showed. For

n = 200, this difference become large for large value of α. Almost all biases are positive and

increasing as α increases. The NMME is better compared to OMME in terms of bias in all

cases.

In estimating σ, the MSEs of both NMME and OMME are increasing exponentially as α

increases. Similar trend emerges for bias, the greater α is, the worse biases both estimators

have. The NMME has better MSEs in almost all situations. The exceptions emerged in

the simulations with n = 20 and α ≤ 0.2. The difference of the two estimators becomes

distinguishable with n = 50, n = 200 and large values of α. Although the NMME has overall

25

Table 3.7: Average Run Length

αp

0.1 0.05 0.02 0.01

0.1 10.26695 20.23905 50.69040 100.11500.2 10.85580 21.05420 51.30260 101.83000.3 11.45460 21.89230 52.82155 103.46410.4 12.41395 23.24790 55.74265 106.55200.5 13.57390 25.92380 59.24165 112.51260.6 15.87520 29.57860 66.51410 125.06790.7 19.32105 35.25600 78.58425 143.68400.8 26.35120 47.83820 104.83335 186.15560.9 47.59965 84.04015 180.77800 319.0752

advantage in MSE, it’s interesting that the NMME has worse biases compared to OMME in

almost all cases. Even in the only exception appeared at n = 20 and α = 0.9, the biases of

both estimators are almost the same.

3.0.6 Return Level and Average Run Length

To study the average run length of the Gumbel AR(1) model, we use the return level defined

by (2.3). We choose α ∈ {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} and p ∈ {0.1, 0.05, 0.02, 0.01}.

Again, without losing generality, the location parameter and the scale parameter are fixed

to be 0 and 1, respectively. For each combination of p and α, we conducted N = 20, 000

trials. In each trial, we keep generating new terms of a Gumbel AR(1) sequence using (2.5)

until a observation exceeding the return level is observed. Thus there were N = 20, 000 run

lengths generated for each selection of p and α. We then recorded the ARL of those 20, 000

run lengths for each selected p and α. The ARLs simulated are displayed in Table 3.7 and

Figure 3.4.

26

Figure 3.4: Average Run Length of Gumbel AR(1) Model

Each curve in the Figure (3.4) represents ARLs corresponding to one value of p. It shows

that all ARLs at α = 0.1 are close to the expected values of IID Gumbel sequences. All

ARLs are increasing as α increases, regardless of the value of p. When α ≤ 0.7, all curves

have gentle rates of increase. When α ≥ 0.8, those rates become rapid. From the graph, it

seems that the curve of p = 0.01 has the greatest increasing speed at α = 0.8. However, if

we consider the increases relative to the ARLs at α = 0.1, the relative increase for p = 0.1

from α = 0.1 to 0.9 showed nearly four times increase, whereas those relative increases for

p = 0.05, 0.02, 0.01 are 3.15, 2.57, 2.19, respectively.

27

Chapter 4

Examples

This chapter offers two examples of real data to illustrate how to estimate parameters based

on Gumbel autoregressive(1) modelling. Some methods of assumptions/model checking are

also mentioned.

4.1 Example 1

The first example is the water flow in the Bow River in the city of Calgary. The data was

taken from official website of Wateroffice (2015). It provides the record of speed of water

flow of the Bow river in unit of m3/s from 1911 to 2014. Each observation was the maximum

out of 365/366 numbers in a year.

The water flow is directly related to water quantity. It’s one of the most important

parameter related to water movement. If we know how much water is passing through a

river, we can estimate many useful quantities including water level. If the risk of exceeding

water level can be accurately calculated, that would be useful in insurance risk management

and hopefully in prior rescue and evacuation planning.

Our purpose is to apply the estimation method (LSE and NMME) to this data. The

original data is provided in the Table 4.1. The data in the Figure 4.1, most observations

are arround 500 or below 500. We assume the water flow maxima came from Gumbel

AR(1) model. According to the estimator given by (2.10) and the new method of moments

estimation, our estimates of α, µ and σ are

28

Table 4.1: Example: Annual Maximum Water Flow (m3/s)(Bow River, Calgary, 1911-2014)

467 430 416 405 796 767 436 532 411 510 337 365 714 377 385259 547 558 1150 399 306 1160 612 447 374 408 257 413 309 271178 303 323 214 306 354 374 558 200 419 408 513 388 365 306270 250 306 289 391 249 317 351 456 345 382 256 399 331 362402 334 476 224 249 208 238 141 302 425 276 202 217 189 420197 227 238 525 357 275 219 251 452 280 275 293 342 173 167339 220 207 602 244 331 287 159 185 313 442 1750 318

Figure 4.1: Plot of annual maxima of water flow

(α, µ, σ) = (0.1900, 294.1692, 97.7120).

By (2.14), the 95% confidence interval for α is (0.0004, 0.3796). The interval does not

contain zero, so our model is significant.

The estimated return level with risk p is calculated by following formula.

xp = µ− σlog(−log(1− p))

We take p = 0.01 as an example. The corresponding return level is xp = 743.66. Based

29

on the simulation study in Chapter 3, if p = 0.01, the approximate ARL for the nearest

α = 0.2 is 101.83. Therefore we would expect to observe a maximum water flow exceeding

743.66m3/s once approximately every 101.83 years.

To check dependence, the consecutive plot, the ACF and the PACF plots are provided.

In the consecutive plot, three large observations caused 6 points located far away from the

majority of the data points. The data is dispersive, the dependence is weak. The ACF and

the PACF plots showed that the most significant number of lag is the lag 1 though it does

not exceed the dashed critical line.

To check distribution, the QQ plot and the PP plot are displayed in Figure 4.4. The PP

plot shows that the data fits Gumbel distribution well. However the PP plot is not sensitive

to outliers. As we can see in the QQ plot, three extremely large observations far from the

expected line made the data skewed too much to the right. This problem is discussed in the

conclusion Chapter.

Figure 4.2: Consecutive plot of annual maxima of water flow

30

Figure 4.3: ACF and PACF of annual maxima of water flow

Figure 4.4: QQ plot and PP plot of annual maxima of water flow

31

4.2 Example 2

This example includes the data of wind speed in the Richmond, the capital of Virginia, USA.

The data comprised the daily maximum wind speed (km/h) for the entire year of 1979.

The wind speed plays an important role in building design. The structural material

of a building should be able to resist wind force. As Caulfied et al. (2012) mentioned in

their major qualifying project, “Over the last several decades the intensity and number of

extreme wind storms, like tornadoes and hurricanes, has escalated at an alarming rate.”

Extreme wind speed is also problematic for aircrafts landing and taking off. Airports even

closed for storms. Therefore the extreme wind speed should be well analyzed.

The daily maxima of wind speed for Richmond are displayed in Table 4.2 and Figure 4.5.

The maxima are mostly located around 10-20. We can assume the process is stationary.

Table 4.2: Example: Daily Maximum Wind Speed (km/h)(Richmond, VA, USA 1979)

22 20 20 15 10 9 15 20 13 8 14 12 12 22 12 13 17 27 12 834 21 11 20 21 18 9 22 25 19 12 23 17 5 17 15 9 14 13 2217 10 19 18 7 10 22 20 15 26 10 15 10 10 15 17 16 13 6 1012 11 22 21 13 12 7 11 21 21 16 23 26 20 10 7 18 14 13 1212 22 24 18 16 17 13 18 18 16 16 23 17 26 16 29 20 15 26 1411 15 20 17 25 17 17 18 16 9 13 10 15 9 16 17 17 15 17 1415 13 22 23 16 13 13 14 11 12 5 12 18 14 9 14 19 20 16 117 7 17 20 23 17 23 17 12 9 17 17 12 14 16 12 16 12 13 1114 24 12 14 15 8 12 14 15 16 14 12 11 12 16 14 16 7 11 1620 17 20 11 20 16 9 10 7 10 6 5 13 24 11 14 11 12 18 157 13 7 16 22 15 12 12 11 17 15 8 8 16 9 11 12 14 9 4311 16 21 18 11 17 17 17 8 13 7 10 17 10 15 13 14 12 17 1713 9 12 14 17 15 12 30 20 16 16 13 9 5 12 14 21 16 11 76 17 10 15 17 17 15 13 6 10 9 6 14 10 17 16 17 23 20 1912 17 18 10 17 13 20 17 10 7 5 11 11 14 20 22 15 12 12 1018 7 11 12 17 16 25 16 10 17 16 15 14 17 17 17 14 23 15 1616 9 6 8 9 13 16 15 17 33 11 21 18 14 13 24 13 13 15 1614 21 12 10 13 16 22 23 15 10 28 6 13 16 13 6 13 17 23 2017 13 11 7 12

32

Figure 4.5: Plot of daily maxima of wind speed

The LSE and NMMEs of α, µ and σ are computed to yield

(α, µ, σ) = (0.1925, 12.5523, 4.7097).

The 95% confidence interval for α is (0.0918, 0.2931), where 0 is not included. Thus our

model is significant compared to IID model.

The return level can be estimated using the same formula as in the first example. In this

case, if we consider p = 0.05, 0.1, then the return levels are 26.54101 and 23.15084, respec-

tively. The ARLs are 21.05 and 10.86, respectively, according to our simulation results. We

would expect to have a maximum wind speed exceeding 23.15084 km/h once approximately

every 10.86 days. Approximately every 21.05 days there will be a maximum wind speed

larger than 26.54101 km/h.

For model checking, the consecutive plot, the ACF, the PACF, the QQ and the PP

plot are presented in Figures 4.6, 4.7 and 4.8. Since the all of the speeds were recorded in

integers, in the consecutive plot, the majority of the data shows a neat array pattern. A

33

weak positive association is found between xt and xt+1. The ACF and the PACF clearly

showed a significant lag 1. We would ignore the lag 5, since our model is AR(1). It can be

seen from the QQ plot and the PP plot that the extreme wind speed fits Gumbel distribution

well. The data in both plots are close to the expected straight lines.

Figure 4.6: Consecutive plot of daily maxima of wind speed

34

Figure 4.7: ACF and PACF of daily maxima of wind speed

Figure 4.8: QQ plot and PP plot of daily maxima of wind speed

35

Chapter 5

Conclusion

The extreme value theory (EVT) has attracted growing attention over last several decades.

The distinct role of extreme values in risk identification and assessment has been illustrated

by many real-life examples. The generalized extreme value (GEV) distribution provided

a useful probability model, which can be applied to extreme values without knowing their

distribution. As one of the three families of GEV, the Gumbel family is characterized by

its light tail and support on entire real line. The additive property of the positive α-stable

variable and the Gumbel variable was provided by Fougeres et al. (2009) The property allows

us to construct a remarkable Gumbel autoregressive (AR) model for dependent extremes.

In this thesis, we focused on the Gumbel AR(1) model and proposed a new method

of moments estimation (NMME) for the location parameter µ and the scale parameter σ

obtained by the transformation link between the Gumbel and the Exponential families. For

the coefficient of the autoregressive process, α, we used a least squares method estimation

(LSE). We compared the performances of two estimation methods, the combination of the

Yule-Walker estimator (Y.W.E) and the ordinary method of moments estimator (OMME)

used in an environmental study conducted by Toulemonde et al. (2010) and the combination

of the LSE and the NMME. We also proved the existence and the uniqueness of the NMME

with sample size greater than or equal to 3. We also described a method of generating the

positive α-stable variable based on a R package ‘stabledist’ is also provided.

In estimation comparison, the LSE of α is more precise compared to Y.W.E when α is

moderate or large. The LSE is always less biased compared to Y.W.E. For µ, the mean

squared errors (MSEs) of NMME and OMME are close but there is a slim advantage on

NMME in most cases. The NMME of µ is always less biased compared to that of OMME.

36

When estimating σ, the MSE of NMME is slightly smaller compared to that for OMME in

all cases. The bias is in favor of NMME when α is small or moderate. When α is close to 0.9,

the biases of both estimators are approximately the same. In conclusion, the combination of

LSE and NMME provided overall better estimation compared to the combination of Y.W.E

and OMME. The choice between the two should be determined by prior interest.

Additionally, we have discovered that, the average run length (ARL) regarding to a

certain risk p or return level xp is always larger than 1/p for Gumbel AR(1) model. The

ARL increases exponentially as α increases. In IID model, the ARL is called return period.

The traditional interpretation of the return period, such as “once every 1/p years,” fails in

the AR(1) model. That understanding should also be modified in general for all dependent

models. Even with the ARL, we should also be careful when making prediction. Based

on our Gumbel AR (1), for example, the future observations greatly depend on the most

recent observation. The recursive relationship of successive observations implies that, if the

current observation is larger, then there is a higher risk for the next observation exceeding the

return level. Therefore, the ARL is not enough in the risk assessment. The current/recent

observations should be taken into account when evaluating and reporting risk.

The NMME and the LSE have greater advantages over OMME and Y.W.E. However,

the following limitations are worth mentioning.

1. The LSE sometimes produces nonsense estimates of α outside (0, 1).

2. We are currently not able to obtain either variances or asymptotic variances of

NMMEs. There is no way available to do hypotheses testing/decision making

regarding to the location and scale parameters. Furthermore, we also do not

have (asymptotic) variance-covariance matrix of (αL, µN , σN) to make predic-

tion.

3. Both the NMME and the LSE ignore much information of probability structure

of the Gumbel AR (1) model. All method of moments estimators (MME)

37

only take the moments into account. The MME can be used regardless of the

dependence structure among observations. The maximum likelihood estimator

of α has been shown to be more efficient than the LSE by Balakrishna et

al.(2013). The MME is also less efficient than the MLE in most cases.

4. Gumbel distribution is one of the three extreme value distribution (EVD)

families. As the first example in chapter 4 showed, the data has three extremely

large observations. The Gumbel distribution, which is good in modelling light-

tail data, may not be an appropriate choice for Calgary’s flood prediction.

5. The stationary process assumption is not likely in many situations. The stock

price is more likely to grow up or move down than to stay around a constant.

It’s well known that the temperature and the sea level are rising.

For future research, the following potential directions may be valuable to work on.

1. Propose a more general model for dependent extremes with all three families of

GEV available. Obtain the corresponding estimation and prediction methods.

2. Develop nonstationary process or regression models for dependent extremes,

such as nonstationary Gumbel AR(1) and autoregressie integrated moving

average (ARIMA) models. Develop the related methodology.

38

Appendix A

Proofs of Important Results

A.1 The parameterization of the positive α-stable variable

To obtain the appropriate values of the parameters for the positive α-stable variable, we

need the following useful Lemma by Paolella (2007).

Lemma 2. Suppose X is a nonnegative random variable. If the moment generating function

(MGF) of X, MX(t), exists for t ∈ (−∞, 0]. Then the characteristic function (CF) of X,

φX(t), can be obtained by substituting imaginary number it for t in the MX(t),

φX(t) = MX(it), t ∈ R (A.1)

Now we are going to prove that if S ∼ S(α, β, γ, δ; 1) and S is a positive α-stable variable,

then 0 < α < 1, γ = (cos(απ/2))1α , β = 1 and δ = 0.

Proof. Suppose the MGF, MS(t), of a nonnegative random variable S exists for ∀t ∈ (−∞, 0],

then

MS(t) = E(exp(tS)) = E[exp(−(−t)S))] = LS(−t) (A.2)

where LS(.) is the Laplace transform of S.

By equation (A.1),

φS(t) = MS(it) = LS(−it) (A.3)

According to the definition of the positive α-stable variable, LS(t) = exp(−tα), we can

write the CF of the positive α-stable variable S as

φS(t) = LS(−it) = exp[−(−it)α], t ∈ R (A.4)

Hence

39

φS(t) =exp[−(−it)α] = exp[−(−i · sign(t) · |t|)α]

=exp[−|t|α(−i · sign(t))α] = exp

[−|t|αexp

(−sign(t) · απi

2

)]=exp

[−|t|α

(cos(απ

2

)− isin

(απ2

)sign(t)

)]=exp

[−(

cos1/α(απ

2

))α|t|α(

1− itan(απ

2

)sign(t)

)](A.5)

where sign(t) = t/|t| if t is a nonzero real number and sign(t) = 0 if t = 0.

In the equation (2.7), the CF of the general stable variable, if we let 0 < α < 1, γ =

(cos(απ/2))1α , β = 1 and δ = 0, then the CF is exactly the expression (A.5). �

A.2 The proof of Corollary 1

Proof of Corollary 1.

Q1 =n−1∑t=1

(Xt+1 − αXt − (1− α)(µ+ γσ))2

=n−1∑t=1

(Xt+1 − αXt − (X2,n − αX1,n−1)

+ (X2,n − αX1,n−1)− (1− α)(µ+ γσ))2

=n−1∑t=1

(Xt+1 − αXt − (X2,n − αX1,n−1))2

+ (n− 1)((X2,n − αX1,n−1)− (1− α)(µ+ γσ))2

≥n−1∑t=1

(Xt+1 − αXt − (X2,n − αX1,n−1))2

where Xm,n = (n∑

t=m

Xt)/(n−m + 1) and the equality holds if and only if µ + γσ = (X2,n −

αX1,n−1)/(1− α).

Moreover

40

Q1 ≥n−1∑t=1

(Xt+1 − αXt − (X2,n − αX1,n−1))2

=

[n−1∑t=1

(Xt −X1,n−1)2

]α−n−1∑t=1

(Xt+1 −X2,n)(Xt −X1,n−1)

n−1∑t=1

(Xt −X1,n−1)2

2

+n−1∑t=1

(Xt+1 −X2,n)2 −

[n−1∑t=1

(Xt+1 −X2,n)(Xt −X1,n−1)

]2n−1∑t=1

(Xt −X1,n−1)2

≥n−1∑t=1

(Xt+1 −X2,n)2 −

[n−1∑t=1

(Xt+1 −X2,n)(Xt −X1,n−1)

]2n−1∑t=1

(Xt −X1,n−1)2

The equality holds if and only if µ+γσ = (X2,n−αX1,n−1)/(1−α) and α = (n−1∑t=1

(Xt+1−

X2,n)(Xt −X1,n−1))/(n−1∑t=1

(Xt −X1,n−1)2). �

A.3 The Existence and Uniqueness of the New Method of Moments Esti-

mate

To prove Corollary 2, we need the following two important Lemmas.

Lemma 3. Define g(X, σ) = µ1(X, σ) = −σlog

[1n

n∑t=1

exp(−Xt

σ

)]. Let

(X(1), X(2), . . . , X(n)) be the order statistic of the sample (X1, X2, . . . , Xn) and let k be the

number of minimum values in the sample (i.e. a sample {2, 1, 4, 1, 3} has k = 2), then the

g(X, σ) has the following properties.

(i)

limσ→0+

g(X, σ) = X(1)

(ii)

limσ→∞

g(X, σ) = X

41

(iii)

limσ→∞

∂g(X, σ)

∂σ= 0

(iv)

limσ→0+

∂g(X, σ)

∂σ= log(

n

k)

Lemma 4. Let {yt}nt=1 and {wt}nt=1 be two sequences of real numbers such that y1 ≤ y2 ≤

. . . ≤ yn and 0 < w1 ≤ w2 ≤ . . . ≤ wn, then

n∑t=1

w2t yt

n∑t=1

w2t

≥

n∑t=1

wtyt

n∑t=1

wt

(A.6)

The equality holds if and only if y1 = y2 = . . . = yn or w1 = w2 = . . . = wn.

Proof of Lemma 3 . (i)

g(X, σ)−X(1) = −σlog

[1

n

n∑t=1

exp

(−Xt

σ

)]− σlog

[exp

(X(1)

σ

)]

= −σlog

[1

n

n∑t=1

exp

(X(1) −X(t)

σ

)]

= −σlog

kn

+1

n

∑X(t)>X(1)

exp

(X(1) −X(t)

σ

) (A.7)

Consequently

limσ→0+

[g(X, σ)−X(1)] = 0

(ii) By Taylor expansion

g(X, σ) = −σlog

[1

n

n∑t=1

exp

(−Xt

σ

)]= −log

[1− X

σ+ ◦

(1

σ

)]σTherefore g(X, σ)→ −log[exp(−X)] = X as σ →∞.

42

(iii)

∂g(X, σ)

∂σ= −log

[1

n

n∑t=1

exp

(−Xt

σ

)]−(

1

σ

) n∑t=1

Xtexp(−Xtσ

)

n∑t=1

exp(−Xtσ

)

Notice that

X(1) ≤

n∑t=1

Xtexp(−Xt

σ

)n∑t=1

exp(−Xt

σ

) ≤ X(n)

The partial derivative of g(X, σ) with respect to σ is bounded.

−log

[1

n

n∑t=1

exp

(−Xt

σ

)]−X(n)

σ≤ ∂g(X, σ)

∂σ≤ −log

[1

n

n∑t=1

exp

(−Xt

σ

)]−X(1)

σ

Hence

0 = − limσ→∞

[log

(1

n

n∑t=1

exp

(−Xt

σ

))+X(n)

σ

]

≤ limσ→∞

∂g(X, σ)

∂σ

≤ − limσ→∞

[log

(1

n

n∑t=1

exp

(−Xt

σ

))+X(1)

σ

]= 0

(iv) From (A.7), we can write

∂g(X, σ)

∂σ=∂[g(X, σ)−X(1)]

∂σ

=− log

kn

+1

n

∑X(t)>X(1)

exp(X(1) −X(t)

σ)

−(

1

σ

)∑

X(t)>X(1)

(X(t) −X(1))exp(X(1)−X(t)

σ

)k +

∑X(t)>X(1)

exp(X(1)−X(t)

σ

)

Therefore

43

limσ→0+

∂g(X, σ)

∂σ= lim

σ→0+

∂[g(X, σ)−X(1)]

∂σ= lim

δ→∞

∂[g(X, 1/δ)−X(1)]

∂(1/δ)

=− limδ→∞

log

kn

+1

n

∑X(t)>X(1)

exp((X(1) −X(t))δ)

− lim

δ→∞

δ∑

X(t)>X(1)

(X(t) −X(1))exp((X(1) −X(t))δ)

k +∑

X(t)>X(1)

exp((X(1) −X(t))δ)

=log

(nk

)�

Proof of Lemma 4. We use Mathematical Induction.

If n = 1, then y1 ≤ y1. The inequality (A.6) holds.

If n = 2, we have w22/(w

21 + w2

2) ≥ w2/(w1 + w2) since

0 <w2

1 + w22

w22

=

(w1

w2

)2

+ 1 ≤ w1

w2

+ 1 =w1 + w2

w2

hence

w21y1 + w2

2y2w2

1 + w22

− w1y1 + w2y2w1 + w2

=

(w2

2

w21 + w2

2

− w2

w1 + w2

)(y2 − y1) ≥ 0

The equality holds if and only if w2 = w1 or y2 = y1.

Suppose for n = k (k > 2), we have

k∑t=1

w2t yt

k∑t=1

w2t

≥

k∑t=1

wtyt

k∑t=1

wt

(A.8)

and the equality holds if and only if wk = wk−1 = . . . = w1 = or yk = yk−1 = . . . = y1.

Now for n = k + 1, we have wk+1 ≥ wk ≥ . . . ≥ w1 > 0, yk+1 ≥ yk ≥ . . . ≥ y1 and we

want to prove that

44

k+1∑t=1

w2t yt

k+1∑t=1

w2t

≥

k+1∑t=1

wtyt

k+1∑t=1

wt

(A.9)

and the equality holds if and only if wk+1 = wk = . . . = w1 = or yk+1 = yk = . . . = y1.

Notice that

w2k+1

k+1∑t=1

w2t

=1

k∑t=1

(wtwk+1

)2 ≥ 1k∑t=1

wtwk+1

=wt+1

k+1∑t=1

wt

(A.10)

The left side of the inequality (A.9) can be viewed as a weighted average of (∑k

t=1w2t yt)/(

∑kt=1w

2t )

and yk+1.

k+1∑t=1

w2t yt

k+1∑t=1

w2t

=

(k∑t=1

w2t

)[(k∑t=1

w2t yt

)/

(k∑t=1

w2t

)]+ w2

k+1yk+1(k∑t=1

w2t

)+ w2

k+1

≥

(k∑t=1

wt

)[(k∑t=1

w2t yt

)/

(k∑t=1

w2t

)]+ wk+1yk+1(

k∑t=1

wt

)+ wk+1

because yk+1 ≥ (∑k

t=1w2t yt)/(

∑kt=1w

2t ) and the weight of yk+1 is increased according to

(A.10). And by (A.8), we have

45

(k∑t=1

wt

)[(k∑t=1

w2t yt

)/

(k∑t=1

w2t

)]+ wk+1yk+1(

k∑t=1

wt

)+ wk+1

≥

(k∑t=1

wt

)[(k∑t=1

wtyt

)/

(k∑t=1

wt

)]+ wk+1yk+1(

k∑t=1

wt

)+ wk+1

=

k+1∑t=1

wtyt

k+1∑t=1

wt

And it can be easily proved that the equality holds if and only if wk+1 = wk = . . . = w1 =

or yk+1 = yk = . . . = y1. �

Now we can prove Corollary 2.

Proof of Corollary 2. We first prove the existence.

The equation (2.19) indicates that the location parameter estimate µN is uniquely de-

termined by the scale parameter estimate σN . If the estimate of σ exists, then the estimate

of µ also exists. If we can prove that there is a unique positive σ satisfying the equation

(A.11), then the existence and uniqueness of the new method of moments estimate of (µ, σ)

is automatically justified.

Let g(X, σ) = µ1(X, σ) = −σlog

[1n

n∑t=1

exp(−Xtσ

)

], then we can write µ2(X, σ) = log2

2σ +

g(X, σ2), and the equation (2.19) can be written as

g(X, σ)− g(X,

σ

2

)− log2

2σ = 0 (A.11)

By Lemma 3, we have

limσ→0+

[g(X, σ)− g

(X,

σ

2

)− log2

2σ

]= 0 (A.12)

46

and

limσ→0+

∂

∂σ

[g(X, σ)− g

(X,

σ

2

)− log2

2σ

]=

1

2log( n

2k

)(A.13)

Since n ≥ 3 and P (Xi = Xj) = 0 for 1 ≤ i < j ≤ n, the above limit of derivative is

positive with probability 1. It implies that with probability 1, there exists σ > 0 such that

g(X, σ)− g(X, σ

2

)− log2

2σ > 0.

Also we have

g(X, σ)− g(X,

σ

2

)− log2

2σ → −∞, as σ →∞ (A.14)

Consequently there exists σ such that g(X, σ)− g(X, σ

2

)− log2

2σ < 0.

Therefore, with probability 1, the solution of the system of equations (2.15) and (2.16)

exists.

Now we prove the uniqueness.

Suppose there are two distinct estimates (µ1, σ1) and (µ2, σ2) satisfying the system of

equations (2.15) and (2.16) with probability greater than 0.

As previously mentioned, µj is uniquely determined by σj (j = 1, 2) by (2.19).

If σ1 = σ2, then µ1 = µ2, which is contradictory to our assumption.

If σ1 6= σ2, without losing generality, we assume σ1 < σ2, by the first moment,

1

n

n∑t=1

exp

(−Xt − µ1

σ1

)= 1

1

n

n∑t=1

exp

(−Xt − µ2

σ2

)= 1

Take ratio of these two equations.

exp

(µ1

σ1− µ2

σ2

) n∑t=1

exp(−Xt

σ1

)n∑t=1

exp(−Xt

σ2

) = 1

47

Hence

µ1

σ1− µ2

σ2= log

n∑t=1

exp(−Xt

σ2

)n∑t=1

exp(−Xt

σ1

) (A.15)

Similarly, from the second moment, we have

µ1

σ1− µ2

σ2= log

√√√√√√√n∑t=1

exp(−2Xt

σ2

)n∑t=1

exp(−2Xt

σ1

) (A.16)

From (A.15) and (A.16),

n∑t=1

exp(−Xt

σ2

)n∑t=1

exp(−Xt

σ1

)

2

=

n∑t=1

exp(−2Xt

σ2

)n∑t=1

exp(−2Xt

σ1

) (A.17)

Let at = exp(−Xt/σ2) and λ = σ2/σ1, by (A.17), we can write

(n∑t=1

at

)2

n∑t=1

a2t

=

(n∑t=1

aλt

)2

n∑t=1

a2λt

(A.18)

Now we define f(λ) =(∑n

t=1 aλt

)2/(∑n

t=1 a2λt

), then the equation (A.18) can be written

as f(1) = f(λ).

Let (a(1), a(2), . . . , a(n)) be the order statistic of (a1, a2, . . . , an). The derivative of the

natural logarithm of f(λ) is given by

dlogf(λ)

dλ=

2n∑t=1

aλ(t)loga(t)

n∑t=1

aλ(t)

−2

n∑t=1

a2λ(t)loga(t)

n∑t=1

a2λ(t)

(A.19)

In Lemma 4, let wt = aλ(t), y(t) = loga(t), then we have

dlogf(λ)

dλ≤ 0 (A.20)

48

where the equality holds if and only if X1 = X2 = . . . = Xn.

If n ≥ 3, then with probability 1, the function f(λ) is a strictly decreasing function for

all λ ∈ R+ and 1 is the unique root of the equation (A.18). Then λ = σ2/σ1 = 1, which is

contradictory to our assumption.

Therefore, with probability 1, there is a unique (µN , σN) which satisfies the system of

equations (2.15) and (2.16). �

49

Appendix B

Useful R code

B.1 Generating Gumbel Autoregressive(1) Sequence

There are two approaches to generate Gumbel autoregressive(1) sequence as discussed in

chapter 2. The first method is based on the Laplace transform of the positive-α variable. The

corresponding R code can be found in the master thesis of Ji (2013). The second method is

using a ‘stabledist’ package in R. Based on the package, the R code of generating Gumbel

AR(1) sequence and computing the least squares estimate and the new method of moments

estimates are provided.

#==================================================================================

#Install and Load the package "stabledist"

#==================================================================================

install.packages("stabledist")

library(stabledist)

#==================================================================================

#Generating Gumbel autoregressive(1) sequence with fixed length

#==================================================================================

gumbelar1n<-function(alpha,mu,sigma,n){

#-----------------------------------------------------------

# alpha is the autoregressive coefficient (0<alpha<1)

# mu is the location parameter (mu is a real number)

# sigma is the scale parameter (sigma>0)

50

# n is the length of the sequence (n is a positive integer)

#-----------------------------------------------------------

x<-c()

x[1]<-mu-sigma*log(rexp(1,rate=1))

#-------------------------------------------------------

# The fisrt gumbel(mu,sigma) is obtained by transfering

# a Exponential(1) variable according to the lemma 1

#-------------------------------------------------------

S<-rstable((n-1),alpha,1,gamma = (cos(pi*alpha/2))^(1/alpha),delta = 0,pm = 1)

#----------------------------------------------------

# Generate (n-1) iid positive alpha-stable variables

#----------------------------------------------------

for (t in 1:(n-1)){

x[t+1]<-alpha*x[t]+alpha*sigma*log(S[t])+(1-alpha)*mu

#-----------------------------------------------------

# Applying the recursive relationship of the AR model

#-----------------------------------------------------

}

x

}

#==================================================================================

#Generating a Gumbel autoregressive(1) sequence until a exceeding observation

#==================================================================================

gumbelar1u<-function(alpha,mu,sigma,p){

51

#-----------------------------------------------------------------------

# alpha is the autoregressive coefficient (0<alpha<1)

# mu is the location parameter (mu is a real number)

# sigma is the scale parameter (sigma>0)

# p is the predetermined risk and is used to calculate the return level

#-----------------------------------------------------------------------

RL<-mu-sigma*log(-log(1-p))

#------------------------------------------------

# RL is the return level corresponding to risk p

#------------------------------------------------

x<-c()

x[1]<-mu-sigma*log(rexp(1,rate=1))

t<-1

while (x[t]<RL){

S<-rstable(1,alpha,1,gamma = (cos(pi*alpha/2))^(1/alpha),delta = 0,pm = 1)

x[t+1]<-alpha*x[t]+alpha*sigma*log(S)+(1-alpha)*mu

t<-t+1

#-------------------------------------------------

# Keep generating new terms until a x_t exceeding

# the return level is observed

#-------------------------------------------------

}

x

}

52

#==================================================================================

#Computing the LSE of alpha and the NMMEs of mu & sigma

#==================================================================================

Estimate<-function(x){

#----------------------------------------

# x is an observed gumbel AR(1) sequence

#----------------------------------------

n<-length(x)

m1<-mean(x[1:(n-1)])

m2<-mean(x[2:n])

v1<-c()

v2<-c()

for (t in 1:(n-1)){

v1[t]<-(x[t+1]-m2)*(x[t]-m1)

v2[t]<-(x[t]-m1)^2

}

alpha<-(sum(v1))/(sum(v2))

#-----------------------

# The estimate of alpha

#-----------------------

mu1<-mean(x)

sigma1<-(max(x)-min(x))/(log(log(1/n)/log(1-1/n)))

par1<-c(mu1,sigma1)

#------------------------------------

# The initial values of mu and sigma

#------------------------------------

53

fn<-function(theta){

v3<-c()

v4<-c()

for (i in 1:n){

v3[i]<-exp(-(x[i]-theta[1])/theta[2])

v4[i]<-exp(-2*(x[i]-theta[1])/theta[2])

}

(sum(v3)-n)^2+(sum(v4)-2*n)^2

#---------------------------------------------

# Define the utility function to be optimized

# with respect to mu and sigma

#---------------------------------------------

}

Optpar<-optim(par1,fn)

#-------------------------------

# The estimates of mu and sigma

#-------------------------------

c(Optpar$par[1],alpha,Optpar$par[2])

#-------------------------------------

# return estimates (mu, alpha, sigma)

#-------------------------------------

}

54

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