estimation of the burr xii-exponential distribution parameters · unimodal hazard rate functions....

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Appl. Appl. Math.ISSN: 1932-9466

Applications and Applied

Mathematics:

An International Journal(AAM)

Vol. 13, Issue 1 (June 2018), pp. 47 – 66

Estimation of the Burr XII-exponential Distribution Parameters

1Gholamhossein Yari & 2Zahra Tondpour

School of MathematicsIran University of Science and Technology

Tehran, [email protected], [email protected]

Received: January 24, 2017; Accepted: October 18, 2017

Abstract

The Burr XII distribution is one of the most important distributions in Survival analysis. In thisarticle, we introduce the new wider Burr XII-G family of distributions. A special model in thenew family called Burr XII-exponential distribution that has constant, decreasing and unimodalhazard rate functions is investigated. We discuss the estimation of this distribution parameters bymaximum likelihood, three modifications of maximum likelihood and Bayes methods. In Bayesmethod, we use the uniform, triangular and Burr XII-uniform priors for posterior analysis andobtain Bayes estimations under two different loss functions. We obtain two approximations of theBayes estimations, the first one is by importance sampling and the second is based on Lindley’sapproximation. Monte Carlo simulated data are used to evaluate these methods. Finally, we fit thisdistribution to a set of real data set by estimation procedures.

Keywords: Burr XII distribution; exponential distribution; Maximum likelihood; BayesianAnalysis; Importance Sampling; Lindley’s Approximation; Simulation; SurvivalAnalysis

MSC 2010 No.: 62F10; 65C05; 60K10

1. Introduction

Survival analysis is one of the topices of statistics that has application in various fields includingcomputer sciences, insurance, economics, medicine, engineering, epidemiology and agricultural.Survival analysis is a set of statistical methods for data analysis in which desired variable is lifetimeor time of failure.

47

48 G. Yari and Z. Tondpour

A widely used family of distributions in lifetime data analysis is Burr distributions family. The firstpurpose of this article is to define new family of distributions to describe and fit the data sets withnon-monotonic hazard rates, such as the bath-tub, unimodal and modified unimodal (unimodalfollowed by increasing) hazard rates. For this purpose, we first define a family of continuous distri-butions, namely, Burr XII-G distributions. Many of probability distributions that are modificationsor generalizations of important lifetime distributions (Weibull, logistic, normal, gamma, etc.) havebeen introduced in order to achieve this purpose in recent years. But unfortunatly, weakness manyof them was increasing the number of parameters and following that their estimation and also com-plicated forms of survival and hazard rate functions. Several classes to generate new distributionsby adding one or more parameters have been proposed in statistical literatures. Some well-knowngenerators are beta-G by Eugene et al. (2002), Kumaraswamy-G (Kw-G) by Cordeiro and De Cas-tro (2011), transformer (T-X) by Alzaghal et al. (2013), Weibull-G by Bourguignon et al. (2014)and Logistic-X by Tahir et al. (2016).

The motivation to define Burr XII-G family is to describe and fit the data sets with non-monotonichazard rate functions by distributions with less parameters. In this family of distributions, thispurpose is almost achieved. for example, if G is uniform distribution, obtained distribution is dis-tribution with two parameters and with increasing and unimodal hazard rate functions and if Gis exponential distribution, derived distribution has three parameters and constant, decreasing andunimodal hazard rate functions. If also G is gamma distribution, obtained distribution has threeparameters and increasing, bath-tub and modified unimodal hazard rate functions.

In this article, a special model in the new family called Burr XII-exponential distribution is inves-tigated.

In survival analysis, after definition a sufficiently flexible distribution, finding a suitable methodfor estimation its parameters is a major challenge for statisticians.

The main objective of this article is to develop competitive methods of parameter estimation forBurr XII-exponential distribution parameters and to compare them with each other. We use fivemethods for estimation. the first four methods are based on unconstrained and constrained max-imum likelihood and the last method is based on Bayesian analysis with different loss functionsand prior distributions.

The reminder of the article is organized as follows: in Section 2, Burr XII-G distributions familyand Burr XII-exponential distribution are defined and some properties of Burr XII-exponential dis-tribution such that hazard rate function, entropy, skewness and kurtosis are investigated. Parameterestimation of Burr XII-exponential distribution by maximum likelihood, modifications of maxi-mum likelihood and Bayes methods are discussed in Section 3. In Section 4, a simulation study isprovided to compare methods with each other and in Section 5, we fit this distribution to a set ofreal data and show the fits given by estimation procedures. Section 6 provides conclusions.

2. Burr XII-exponential Distribution

Consider a continuous distribution G with probability density function (pdf) g and support (a, b),a < b, a, b ∈ [−∞,+∞], and the Burr XII cumulative distribution function (cdf)B(t) = 1−(1+tc)−p

AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 49

(for t > 0) with positive parameters c and p. Based on pdf of Burr XII distribution, by replacing xwith G(x)

1−G(x) , we define the cdf of the Burr XII-G family of distributions by:

F (x; c, p, ξ) =

∫ G(x;ξ)

1−G(x;ξ)

0cptc−1(1 + tc)−p−1dt

= B

(G(x; ξ)

1−G(x; ξ)

)= 1−

(1 +

(G(x; ξ)

1−G(x; ξ)

)c)−p, x ∈ (a, b), (1)

where G(x; ξ) is a baseline cdf, which depends on a parameter vector ξ. The family pdf reduces to

f(x; c, p, ξ) = cpg(x; ξ)(G(x; ξ))c−1

(1−G(x; ξ))c+1

(1 +

(G(x; ξ)

1−G(x; ξ)

)c)−p−1

. (2)

Using Newton’s generalized Binomial theorem, pdf of Burr XII-G distribution reduces to

f(x; c, p, ξ) =

∞∑j,k=0

wj,khc(j+1)+k(x; ξ),

where

wj,k = c(−1)jΓ(j + p+ 1)Γ(c(j + 1) + k + 1)

j!k!Γ(c(j + 1) + 1)Γ(p)Γ(c(j + 1) + k),

and

ha(x; ξ) = ag(x; ξ)Ga−1(x; ξ).

Note, ha is pdf of exponentiated-G distribution with power parameter a > 0.

Hence the Burr XII-G pdf can be expressed as an infinite linear combination of exponentiated-Gpdfs and so some mathematical properties of the Burr XII-G model can be obtained directly fromthose properties of exponentiated-G distribution. For example, sth moment and moment generatingfunction for Burr XII-G distribution are, respectively, as

µ(s) = E(Xs) =

∞∑j,k=0

wj,kE(Zsj,k), (3)

and

MX(t) =

∞∑j,k=0

wj,kE(etZj,k),

where Zj,k has exponentiated-G distribution with power parameter c(j + 1) + k.

According to Equation 1, the Burr XII-G family of distributions can be simulated by simulation ofBurr XII distribution, T , and then using X = G−1

(T

1+T

).

The quantile function, Q(u), 0 < u < 1, for the Burr XII-G family of distributions can be computedby using the formula

Q(u) = G−1

(B−1(u)

1 +B−1(u)

).


In this article, we investigate a special model of this family that obtains by setting exponentialdistribution as baseline cdf G. If G is cdf of exponential distribution, then g(x; θ) = θe−θx (θ, x > 0)

and G(x; θ) = 1− e−θx. Therefore the Burr XII-exponential has cdf given by

F (x; c, p, θ) = 1− (1 + (eθx − 1)c)−p , x > 0, c, p, θ > 0,

where c and p are shape parameters and θ is scale parameter. The corresponding pdf is

f(x; c, p, θ) = cpθeθx(eθx − 1)c−1(1 + (eθx − 1)c)−p−1 , x > 0 .

If c = 1, then the Burr XII-exponential distribution reduces to exponential distribution with param-eter pθ. Hence Burr XII-exponential distribution is a generalization of exponential distribution.

Figure 1 illustrates shapes of the probability density and hazard rate functions for the Burr XII-exponential distribution.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

18

x

f(x)

c=1/2,p=5,θ=2c=5,p=5,θ=2c=1,p=5,θ=2

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

x

h(x

)

c=1/2,θ=1/2

c=2,θ=1/2

c=1,θ=1/2

Figure 1. Shapes of probability density and hazard rate functions of the Burr XII-exponential distribution for differentvalues of shape parameters c

As shown in Figure 1, this model is flexible enough to accommodate constant (c = 1), decreasing(c < 1) and unimodal (c > 1) hazard rate functions.

According to Equation 3, the moments of Burr XII-exponential distribution is as

µ(s) = E(Xs)

=

∞∑j,k=0

wj,k(−1)s(c(j + 1) + k)

θs∂s

∂psβ(c(j + 1) + k, p+ 1− c(j + 1)− k) |p=c(j+1)+k .

Skewness and kurtosis of a parametric distribution are often measured by S = µ(3)σ3 and K = µ(4)

σ4 ,respectively. For the Burr XII-exponential distribution, skewness and kurtosis can be approximatedby approximations of µ(3) and µ(4) or alternative measures for skewness and kurtosis, based onquantile functions. The measures of skewness and kurtosis defined, respectively, by Galton (1883)and Moors (1988) as follow are based on quantie functions.

S =Q(6

8)− 2Q(48) +Q(2

8)

Q(68)−Q(2

8), (4)


K =Q(7

8)−Q(58) +Q(3

8)

Q(68)−Q(2

8). (5)

To investigate the effect of the shape parameters c and p on the Burr XII-exponential pdf, Equations4 and 5 are used to obtain Galton’s skewness and Moors’ kurtosis. Figure 2 displays the Galton’sskewness and Moors’ kurtosis for the Burr XII-exponential distribution.

02

46

810

0

5

10−0.5

0

0.5

1

cp

S

02

46

810

0

5

100

50

100

150

200

cp

KFigure 2. Galton’s skewness and Moors’ kurtosis for the Burr XII-exponential distribution

One of the applications of this distribution is in the field of insurance. In the area of agriculturalinsurance, efficient implementation of crop revenue insurance contracts requires accurate measuresof risk for both crop prices and yields. Empirical evidence shows that crop prices tend to be pos-itively skewed with fat tails but in premium rate calculations assume crop prices follow a normalor log-normal distribution. According to Figure 2 the Burr XII-exponential distribution is positiveskewed and has fatter tail than normal distribution so it seems logical that the Burr XII-exponentialdistribution provide a better overall fit for the crop prices when compared to the normal distributionand the log-normal distribution.

3. Methods of estimation

In this section, we discuss maximum likelihood, three modifications of maximum likelihood andBayes methods for estimation of parameters of Burr XII-exponential distribution.

3.1. Maximum Likelihood Estimator (MLE)

Using Equation 2, the log likelihood function for the Burr XII-exponential distributed independentobservations x1, x2, ..., xn can be written as

l(x1, x2, ..., xn, c, p, θ)

= n log(cpθ) +

n∑i=1

θxi + (c− 1)

n∑i=1

log(eθxi − 1)− (p+ 1)

n∑i=1

log(1 + (eθxi − 1)c). (6)

The MLEs can be obtained by maximization of log likelihood function with respect to the param-eters c, p and θ, using optimization methods such as quasi-Newton method.


3.2. Modified Maximum Likelihood Estimator 1 (MMLE 1)

In this case, the log likelihood function in Equation 6 is maximized with respect to the parametersc, p and θ subject to the constriant change-point equality of distribution and sample. Change-pointis the point at which the hazard rate function reaches to a maximum (minimum) and then decreases(increases). Mathematically, change-point is local extremum of hazard rate function. For the BurrXII-exponential distribution , this constriant is

−1 + ceθxch − (eθxch − 1)c = 0, (7)

where xch is sample change-point.

Note that this method can only be used when hazard rate of sample is unimodal.


In this modification, constriant is median equality of distribution and sample. For the Burr XII-exponential distribution, this constriant is as

xme =1

θln((2

1

p − 1)1

c + 1), (8)

where xme is sample median.


Added constriant to maximum likelihood method in this modification is

E(F (x(r))) = F (x(r)), (9)

where E(.) is the usual expectation, x(r) is the rth order statistic in a random sample of size n, andF (x(r)) is value of cdf at x(r). Since E(F (x(r))) = r

n+1 , then constraint is reduced tor

n+ 1= 1− (1 + (eθx(r) − 1)c)−p. (10)

In particular case r = 1, above constraint becomes

x(1) =1

θln

(( n

n+ 1

)− 1

p

− 1

) 1

c

+ 1

. (11)

Maximizing the log likelihood function in Equation 6subject to the above constraint is the basicmethodology of the Cohen and Whitten’s computational procedure.

3.5. The Bayes estimators

Let X = (x1, x2, ..., xn) be a observed sample from Burr XII-exponential distribution. In this casethe likelihood function is

L(x1, x2, ..., xn, c, p, θ) = f(X | c, p, θ)

= (cpθ)ne∑ni=1 θxi

n∏i=1

(eθxi − 1)c−1n∏i=1

(1 + (eθxi − 1)c)−p−1.


Here, we suggest the joint prior for c, p and θ as product of marginals of c, p and θ. Hence

π(c, p, θ) = π(c)π(p)π(θ).

Let Θ = (θ1, θ2, θ3) = (c, p, θ). In this article, considered priors for θi are uniform distribution withpdf

π(θi) =1

θiMLE

, 0 < θi < θiMLE ,

Burr XII-uniform distribution on 95% confidence interval of MLEs with pdf

π(θi) =c1p1

b1 − a1

( θi−a1

b1−a1)c1−1

(1− θi−a1

b1−a1)c1+1

(1 +

(θi−a1

b1−a1

1− θi−a1

b1−a1

)c1)−p1−1

,

where a1 < θi < b1, c1, p1 > 0 and a1 = θiMLE − 1.96√− ∂2

∂θ2il(X;cMLE ,pMLE ,θMLE)

and

b1 = θiMLE + 1.96√− ∂2


, and triangular distribution with pdf

π(θi) =

2(θi−a2)

(b2−a2)(c2−a2) a2 ≤ θi < c2

2(b2−θi)(b2−a2)(b2−c2) c2 ≤ θi ≤ b20 else

where a2 = θiMLE − 1.96√− ∂2


, c2 = θiMLE and

b2 = θiMLE + 1.96√− ∂2


.

The posterior joint density of c, p and θ is

π(c, p, θ | X) =f(X | c, p, θ)π(c, p, θ)∫∞

0

∫∞0

∫∞0 f(X | c, p, θ)π(c, p, θ)dc dp dθ

.

But the integral in the denominator, in many cases, is not simply computable. For this distribution isalso. To solve this problem, we use the proposed method in Ross (2002) called importance samplingthat in which the calculation of mentioned integral is not needed.

Algorithm 3.1.

The algorithm used in Ross’s method is as follows:

• A sample with large size N of prior distribution of parameters, π(c), π(p) and π(θ), is generated.This sample is shown as Θ1 = (c1, p1, θ1), Θ2 = (c2, p2, θ2),..., ΘN = (cN , pN , θN ).

• By using observations, X, f(X | Θi), i = 1, 2, ..., N , is calculated.• For i = 1, 2, ..., N , αi = f(X|Θi)∑N

j=1f(X|Θj)

is calculated.

• Each function E(h(Θ) | X) can be estimated by using∑N

i=1 αih(Θi). �

In this article, we have considered Entropy and Precautionary loss functions for Bayesian analysisof estimation of the parameters of Burr XII-exponential distribution.


If θiELF represent Bayes estimator for parameter θi under Entropy loss function, then θiELF =

(E( 1θi| X))−1 and if θiPLF represent Bayes estimator for θi under Precautionary loss function,

then θi2

PLF = E(θ2i | X). Hence to obtain Bayes estimator by Ross’s method, under Entropy loss

function is enough to set h(Θ) = h(θi) = 1θi

and to obtain Bayes estimator under Precautionary lossfunction, h(Θ) = h(θi) = θ2

i .

Another method to get Bayes estimators is the use of Lindley’s approximation. In general to obtainBayes estimator of a parameter need to calculation E(h(Θ) | X), which is

E(h(Θ) | X) =

∫∞0

∫∞0

∫∞0 h(c, p, θ)f(X | c, p, θ)π(c, p, θ)dc dp dθ∫∞

0

∫∞0

∫∞0 f(X | c, p, θ)π(c, p, θ)dc dp dθ

and cannot be expressed in a simple closed form. Hence, we need to obtain a numerical approxi-mation. Here, we use Lindley’s approximation. Lindley’s approximation leads to

E(h(Θ) | X) ' h(Θ) +1

2

3∑j,k=1

(hjk + ρk)σjk +1

2

3∑j,k,m,n=1

ljkmhnσjkσmn, (12)

where l is the log likelihood function, σjk is (j, k)th element of inverse of the matrix (ljk = ∂2l∂θj∂θk

),j, k = 1, 2, 3, and ρ(Θ) = log(π(Θ)), ρk = ∂ρ

∂θk, ljkm = ∂3l

∂θj∂θk∂θm, hn = ∂h

∂θnand hjk = ∂2h

∂θj∂θk.

Moreover, Equation 12 is evaluated at the MLEs of c, p and θ.

To obtain θi based on Lindley,s approximation with uniform prior, Equation 12 reduces to,

E(h(θi) | X) ' h(θi) +1

2hiiσii +

1

2hiσ1i(l111σ11 + 2l121σ12 + 2l311σ31

+l221σ22 + 2l231σ23 + l331σ33) +1

2hiσ2i(l112σ11 + 2l122σ12

+2l312σ31 + l222σ22 + 2l232σ23 + l332σ33) +1

2hiσ3i(l113σ11

+2l123σ12 + l223σ22 + 2l233σ23 + l333σ33 + 2l133σ13).

But if prior distribution of θi is Burr XII-uniform distribution, then

E(h(θi) | X) ' h(θi) +1

2(hii +

3∑k=1

ρk)σii +1

2hiσ1i(l111σ11 + 2l121σ12

+2l311σ31 + l221σ22 + 2l231σ23 + l331σ33) +1

2hiσ2i(l112σ11

+2l122σ12 + 2l312σ31 + l222σ22 + 2l232σ23 + l332σ33) +1

2hiσ3i

×(l113σ11 + 2l123σ12 + l223σ22 + 2l233σ23 + l333σ33 + 2l133σ13),

where

ρk =(c1b1 − b1 + 2θk − a1c1 − a1)(1 + ( θk−a1

b1−θk )c1) + (−p1 − 1)c1(b1 − a1)( θk−a1

b1−θk )c1

(θk − a1)(b1 − θk)(1 + ( θk−a1

b1−θk )c1).


Burr XII-exponential distribution population cases considered in the sampling experiment

Burr XII-exponentialdistribution population c p θ

case 1 12 1 2

case 2 2 1 2case 3 1 1 2

Note that to obtain θiELF , set h(θi) = 1θi

and to obtain θiPLF , set h(θi) = θ2i .

Also note, since the triangular prior distribution is assumed on 95% confidence interval of MLEwith mode in MLE, therefore in this case, prior distribution does not have derivative at MLE andso there isn’t Lindley’s approximation for E(h(Θ) | X) when prior distribution of parameters istriangular.

4. Application to Monte Carlo-Simulated data

4.1. Simulation from the Burr XII-exponential distribution

The cdf of the Burr XII-exponential distribution has a closed form which makes the simulationfrom this distribution easier. In order to simulation a random sample from this distribution, first1000 random numbers are generated from the uniform distribution with range (0,1), these gener-ated random numbers are then transformed using the inverse cdf by solving the equation F (x) = u,numerically.

4.2. Monte Carlo samples

The performance of the MLE, MMLEs and Bayse methods were assessed by using Monte Carlosimulated data. As seen in Section 2, the behavior of hazard rate function of Burr XII-exponentialdistribution is only dependent on value of parameter c. On the other hand, because our purpose ofstudy Burr XII-exponential distribution is different behaviors of its hazard rate function, thereforefor simplicity, we consider arbitrary values θ = 2 and p = 1 for different values c < 1, c = 1 andc > 1. Three population cases, listed in the following Table, were considered. distributions withconstant (c = 1), decreasing (c < 1) and unimodal (c > 1) hazard rate functions were simulated. Foreach population case, 1000 random samples of size 200 to 600 were generated and then parameterswere estimated.

The required numerical evaluations are implemented using MATLAB (version 2013) and R soft-ware (version 3.3.1).

In MLE method, maximum of likelihood function 6 without any constraints should be found.This optimization can be carried out in many scientific computing packages. The ’Optimizationtoolbox’ of MATLAB (Gilat, 2004) contains an algorithm fminunc for unconstrained nonlinearoptimization. fminunc attempts to find a minimum of a scalar function of several variables, starting


at an initial point.

x = fminunc(fun, x0, options) minimizes with the optimization options specified in options. InMLE method, fun is negative of likelihood function.

A important problem in fminunc algorithm is to choose initial point. We choose initial point forusing MLE method as follows:

The Burr XII-exponential distribution with parameters c = 1, p and θ is exponential distributionwith parameter pθ. Hence, (1, 1, 1

x) is chosen as the initial point because 1x is MLE estimation for

parameter λ of exponential distribution.

In MMLE 1, MMLE 2 and MMLE 3 methods, maximum of likelihood function 6 with a con-straint should be found. The ’Optimization toolbox’ of Matlab (Gilat, 2004) contains an algorithmfmincon for constrained nonlinear optimization.

fmincon finds the minimum of a problem specified by

minx f(x) such that

c(x) ≤ 0

ceq(x) = 0

Ax ≤ bAeqx = beq

lb ≤ x ≤ ub

,

where b and beq are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that returnvectors, and f(x) is a function that returns a scalar. f(x), c(x) and ceq(x) can be nonlinear functions.x, lb and ub can be passed as vectors or matrices.

x = fmincon(fun, x0, A, b, Aeq, beq, lb, ub, nonlcon, options) subjects the minimization to the non-linear inequalities c(x) or equalities ceq(x) defined in nonlcon. fmincon optimizes such thatc(x) ≤ 0 and ceq(x) = 0.

In MMLE 1, MMLE 2 and MMLE 3 methods, fun is negative of likelihood function, ceq(x) isequations 7, 8, and 9 respectively, lb is zero and since there aren,t linear inequalities and equalitiesand upper bound for x, hence set A = b = Aeq = beq = ub = []. Also, similar MLE method (1, 1, 1

x)

is chosen as the initial point.

In Baysian method, Bayes with Entropy loss function and uniform, triangular and Burr XII-uniformpriors, respectively, are denoted by BELFU, BELFT, BELFBU and also Bayes with Precautionaryloss function and uniform, triangular and Burr XII-uniform priors, respectively, are denoted byBPLFU, BPLFT, BPLFBU.

Lindley’s approximation of Bayes estimation with Entropy loss function and uniform and BurrXII-uniform priors, respectively, are denoted by LBELFU and LBELFBU and also Bayes withPrecautionary loss function and uniform and Burr XII-uniform priors, respectively, are denoted byLBPLFU and LBPLFBU.

The performance of the MLE, MMLEs and Bayes methods was evaluated by using the following


performance indices:

• Standard bias: BIAS=E(x)−xx .

• Root mean square error: RMSE=E((x−x)2)0.5

|x| , where x is an estimate of x (parameter), and

• E(x) = 1n

∑ni=1 xi, where n is the number of the Monte Carlo samples (n = 1000 in this study).

The number of samples of 1000 may arguably not be large enough to produce the true values ofBIAS and RMSE, but will suffice to compare the performance of estimation methods.

4.3. BIAS in parameter estimation

The bias of parameters estimated by the methods is summarized in Tables 1, 2 and 3. Of all meth-ods, in the first two cases, MMLE 2 and in the third case, MLE produced the least bias in c pa-rameter estimate. In p parameter estimate, MMLE 3, MMLE 1 and MLE produced the least biasin cases 1, 2 and 3, respectively. In three cases, MLE produced the least bias in the θ parameter.

In comparison between approximations of Bayes estimations, in cases 2 and 3, LBELFBU and incase 1, LBPLFU produced the least bias in c parameter estimate. In p parameter estimate, in cases2 and 3, LBPLFBU and in case 1, LBELFU produced the least bias. In θ parameter estimate, incases 2 and 3, LBPLFU and in case 1, LBPLFBU produced the least bias.

4.4. RMSE in parameter estimation

The values of RMSE of parameters estimated by the methods are given in Tables 4, 5 and 6. Ofall methods, in cases 1 and 3, MLE and in the case 2, MMLE 2 produced the least RMSE inc parameter estimate. In p parameter estimate, in cases 1 and 3, MLE and in case 2 MMLE 1produced the least RMSE. In three cases, MLE produced the least RMSE in the θ parameter.

In comparison between approximations of Bayes estimations, in cases 1 and 2, LBELFBU andin case 3, LBELFU produced the least RMSE in c parameter estimate. In p parameter estimate,in cases 2 and 3, LBPLFBU and in case 1, LBELFBU produced the least RMSE. In θ parameterestimate, in cases 1 and 3, LBELFBU and in case 2, LBPLFU produced the least RMSE.

5. Application to real data

Here, we apply MLE, MMLE 1, MMLE 2, MMLE 3 and Bayse estimators to a real data set. Dataset is corresponding to remission times (in months) of a random sample of 128 bladder cancerpatients. These data were previously studied by Lee and Wang (2003), Lemonte and Cordeiro(2011) and Zea et al. (2012).

The scaled empirical TTT transform (Aarset, 1987) can be used to identify the shape of the hazardrate function. The scaled empirical TTT transform is convex (concave) if the hazard rate function is


Tabl

e1.

BIA

Sof

para

met

eres

timat

es

case

1ca

se2

case

3sa

mpl

esi

zem

etho

dc

pθ

cp

θc

pθ

200

ML

E0.

0073

0.03

700.

0494

0.01

700.

1639

-0.0

088

0.01

200.

1301

0.26

90M

ML

E1

——

—-0

.50

0.27

47—

——

MM

LE

2-0

.003

80.

0366

0.05

070.

0223

0.38

57-0

.007

80.

0341

1.92

950.

9332

MM

LE

3-0

.057

5-0

.006

30.

2924

-0.0

416

1.11

60-0

.098

1-0

.073

40.

0464

18.4

886

400

ML

E0.

0016

0.01

170.

0312

0.01

150.

0596

0.00

280.

0051

0.11

800.

1301

MM

LE

1—

——

-0.5

00.

2734

——

—M

ML

E2

-0.0

028

0.01

780.

0313

0.00

840.

0751

-0.0

069

0.02

221.

4101

0.62

57M

ML

E3

-0.0

570

-0.0

055

0.28

09-0

.048

10.

9171

-0.0

450

-0.0

541

-0.3

171

17.0

702

600

ML

E0.

0015

0.00

880.

0183

0.00

580.

0469

-0.0

029

0.00

470.

0912

0.09

21M

ML

E1

——

—-0

.50

0.27

42—

——

-M

ML

E2

-0.0

004

0.01

140.

0206

0.00

290.

0550

-0.0

052

0.00

931.

2575

0.51

96M

ML

E3

-0.0

478

-0.0

027

0.27

34-0

.052

90.

7819

-0.0

346

-0.0

527

-0.4

960

15.5

969


Tabl

e2.

BIA

Sof

para

met

eres

timat

es

case

1ca

se2

case

3sa

mpl

esi

zem

etho

dc

pθ

cp

θc

pθ

200

BE

LFU

-0.9

989

-0.9

990

-0.9

990

-0.9

989

-0.9

990

-0.9

990

-0.9

991

-0.9

980

-0.9

994

BPL

FU-0

.997

1-0

.999

0-0

.999

0-0

.998

9-0

.999

0-0

.999

0-0

.998

6-0

.997

9-0

.999

6B

EL

FT-0

.999

1-0

.999

1-0

.999

8-0

.999

0-0

.998

7-0

.999

2-0

.999

0-0

.998

7-0

.999

3B

PLFT

-0.9

969

-0.9

991

-0.9

988

-0.9

990

-0.9

987

-0.9

992

-0.9

986

-0.9

987

-0.9

993

BE

LFB

U-0

.996

1-0

.998

4-0

.999

3-0

.999

1-0

.998

7-0

.999

2-0

.999

0-0

.998

4-0

.999

4B

PLFB

U-0

.995

9-0

.998

4-0

.999

3-0

.999

0-0

.998

7-0

.999

2-0

.998

6-0

.998

4-0

.999

440

0B

EL

FU-0

.998

7-0

.999

0-0

.999

0-0

.998

7-0

.998

9-0

.998

9-0

.999

0-0

.998

6-0

.999

3B

PLFU

-0.9

970

-0.9

990

-0.9

989

-0.9

988

-0.9

991

-0.9

989

-0.9

986

-0.9

986

-0.9

993

BE

LFT

-0.9

990

-0.9

990

-0.9

986

-0.9

988

-0.9

986

-0.9

990

-0.9

989

-0.9

986

-0.9

991

BPL

FT-0

.996

7-0

.999

0-0

.998

7-0

.998

8-0

.998

5-0

.999

1-0

.998

5-0

.998

7-0

.999

2B

EL

FBU

-0.9

959

-0.9

982

-0.9

992

-0.9

990

-0.9

985

-0.9

991

-0.9

989

-0.9

982

-0.9

991

BPL

FBU

-0.9

957

-0.9

982

-0.9

992

-0.9

989

-0.9

985

-0.9

992

-0.9

986

-0.9

985

-0.9

995

600

BE

LFU

-0.9

988

-0.9

987

-0.9

988

-0.9

987

-0.9

988

-0.9

988

-0.9

989

-0.9

985

-0.9

992

BPL

FU-0

.996

6-0

.998

7-0

.998

8-0

.998

7-0

.999

0-0

.998

7-0

.998

5-0

.998

5-0

.999

1B

EL

FT-0

.998

7-0

.998

7-0

.998

8-0

.998

6-0

.998

5-0

.998

7-0

.998

6-0

.998

5-0

.998

6B

PLFT

-0.9

996

-0.9

990

-0.9

985

-0.9

987

-0.9

983

-0.9

990

-0.9

983

-0.9

986

-0.9

990

BE

LFB

U-0

.995

7-0

.998

2-0

.999

1-0

.998

9-0

.998

3-0

.999

0-0

.998

7-0

.998

1-0

.999

0B

PLFB

U-0

.995

6-0

.998

0-0

.999

0-0

.998

7-0

.998

5-0

.998

9-0

.998

4-0

.998

1-0

.999

2


Tabl

e3.

BIA

Sof

para

met

eres

timat

es

case

1ca

se2

case

3sa

mpl

esi

zem

etho

dc

pθ

cp

θc

pθ

200

LB

EL

FU0.

2995

0.22

591.

0285

0.97

792.

3457

0.09

800.

460

1.77

640.

8312

LB

PLFU

0.24

410.

2414

1.10

240.

9297

1.78

740.

0200

0.60

731.

0717

0.65

55L

BE

LFB

U-0

.697

9-0

.463

1-0

.937

00.

7457

-0.9

179

-0.7

958

-0.0

799

-0.7

673

-0.9

711

LB

PLFB

U2.

6778

1.80

69-0

.344

73.

1856

-0.0

593

1.13

462.

1538

-0.5

476

-0.9

656

400

LB

EL

FU0.

2118

0.19

870.

9989

0.89

211.

9876

0.05

470.

345

1.12

350.

6931

LB

PLFU

0.19

310.

2019

0.91

670.

6721

1.13

980.

0118

0.41

130.

8876

0.53

35L

BE

LFB

U-0

.566

1-0

.233

4-0

.899

10.

5136

-0.8

876

-0.6

968

-0.0

588

-0.5

613

-0.6

755

LB

PLFB

U1.

5863

1.12

32-0

.112

32.

1536

-0.0

346

0.99

871.

93-0

.356

1-0

.813

160

0L

BE

LFU

0.15

460.

1198

0.78

650.

7945

0.99

430.

0113

0.13

40.

8775

0.46

89L

BPL

FU0.

0947

0.09

840.

0541

0.34

990.

8971

0.00

130.

156

0.56

490.

1289

LB

EL

FBU

-0.4

329

-0.1

879

-0.7

741

0.34

90-0

.658

9-0

.456

8-0

.034

5-0

.458

9-0

.355

6L

BPL

FBU

0.98

140.

8854

-0.0

987

1.90

34-0

.014

60.

6559

0.99

31-0

.216

9-0

.568

9


Tabl

e4.

RM

SEof

para

met

eres

timat

es

case

1ca

se2

case

3sa

mpl

esi

zem

etho

dc

pθ

cp

θc

pθ

200

ML

E0.

0823

0.25

010.

3553

0.09

430.

5998

0.22

050.

0955

0.67

641.

0830

MM

LE

1—

——

0.5

00.

28—

——

MM

LE

20.

0839

0.25

940.

3513

0.09

840.

4135

0.23

040.

1840

4.28

073.

5223

MM

LE

30.

1947

0.29

770.

7580

0.19

902.

5210

1.09

720.

3252

2.23

6335

.487

440

0M

LE

0.05

960.

1653

0.24

400.

0692

0.35

150.

1563

0.05

810.

5296

0.58

02M

ML

E1

——

—0.

50

0.27

59—

——

-M

ML

E2

0.05

920.

1892

0.25

320.

0661

0.35

470.

1514

0.10

863.

1933

2.73

18M

ML

E3

0.18

240.

2359

0.61

350.

1802

1.97

780.

8850

0.26

451.

5269

31.6

1960

0M

LE

0.04

920.

1380

0.19

560.

0540

0.25

760.

1230

0.04

580.

4460

0.48

69M

ML

E1

——

—0.

50

0.27

59—

——

MM

LE

20.

0503

0.14

580.

2015

0.05

150.

2775

0.12

730.

0933

2.79

532.

5917

MM

LE

30.

1730

0.19

470.

5233

0.17

301.

6714

0.77

680.

2418

1.18

6829

.330

5


Tabl

e5.

RM

SEof

para

met

eres

timat

es

case

1ca

se2

case

3sa

mpl

esi

zem

etho

dc

pθ

cp

θc

pθ

200

BE

LFU

0.99

950.

9995

0.99

950.

9995

0.99

950.

9995

0.99

951.

0000

0.99

96B

PLFU

1.00

140.

9995

0.99

950.

9995

0.99

950.

9995

0.99

961.

0001

0.99

96B

EL

FT0.

9995

0.99

950.

9995

0.99

950.

9996

0.99

950.

9995

0.99

960.

9995

BPL

FT1.

0016

0.99

950.

9995

0.99

950.

9996

0.99

950.

9996

0.99

960.

9995

BE

LFB

U1.

0037

0.99

970.

9995

0.99

950.

9996

0.99

950.

9995

0.99

970.

9996

BPL

FBU

1.00

440.

9997

0.99

950.

9995

0.99

960.

9995

0.99

960.

9997

0.99

9640

0B

EL

FU0.

9994

0.99

950.

9994

0.99

940.

9993

0.99

940.

9995

0.99

960.

9995

BPL

FU1.

0000

0.99

930.

9993

0.99

940.

9994

0.99

940.

9996

0.99

960.

9995

BE

LFT

0.99

930.

9993

0.99

930.

9994

0.99

930.

9994

0.99

920.

9993

0.99

92B

PLFT

1.00

000.

9993

0.99

930.

9994

0.99

940.

9995

0.99

950.

9994

0.99

94B

EL

FBU

1.00

150.

9992

0.99

930.

9993

0.99

940.

9993

0.99

940.

9995

0.99

94B

PLFB

U1.

0020

0.99

940.

9991

0.99

910.

9990

0.99

910.

9994

0.99

950.

9994

600

BE

LFU

0.99

930.

9994

0.99

930.

9994

0.99

930.

9992

0.99

940.

9994

0.99

93B

PLFU

0.99

960.

9991

0.99

910.

9993

0.99

940.

9993

0.99

940.

9994

0.99

93B

EL

FT0.

9991

0.99

920.

9991

0.99

920.

9991

0.99

920.

9991

0.99

920.

9991

BPL

FT0.

9999

0.99

920.

9992

0.99

930.

9994

0.99

920.

9992

0.99

930.

9993

BE

LFB

U1.

0000

0.99

900.

9991

0.99

910.

9992

0.99

910.

9992

0.99

930.

9992

BPL

FBU

1.00

000.

9990

0.99

890.

9987

0.99

870.

9990

0.99

920.

9991

0.99

92


Tabl

e6.

RM

SEof

para

met

eres

timat

es

case

1ca

se2

case

3sa

mpl

esi

zem

etho

dc

pθ

cp

θc

pθ

200

LB

EL

FU1.

2231

1.45

672.

8737

2.43

953.

7575

0.85

571.

7509

3.32

082.

3857

LB

PLFU

1.20

151.

4385

2.93

362.

4352

2.23

860.

8013

1.74

262.

8059

2.44

55L

BE

LFB

U0.

7763

0.68

790.

4193

1.51

590.

9203

0.87

941.

9016

1.55

210.

4854

LB

PLFB

U2.

3495

3.84

652.

7497

2.13

050.

6153

1.57

092.

5409

1.34

011.

1595

400

LB

EL

FU0.

9388

0.99

981.

1109

1.54

802.

1178

0.65

310.

8331

2.18

571.

1225

LB

PLFU

0.99

131.

024

1.95

61.

5647

1.54

350.

5231

0.98

751.

9301

1.53

45L

BE

LFB

U0.

5263

0.47

520.

2187

0.98

570.

6347

0.71

580.

8931

0.85

490.

2331

LB

PLFB

U1.

2509

2.51

371.

3345

1.03

150.

4251

0.99

931.

4709

0.89

310.

7805

600

LB

EL

FU0.

7654

0.81

230.

9167

0.99

871.

1564

0.32

760.

6453

0.99

890.

8765

LB

PLFU

0.77

650.

6543

0.75

610.

9909

1.12

040.

3324

0.87

690.

8907

0.87

69L

BE

LFB

U0.

4453

0.11

090.

098

0.86

050.

4608

0.54

090.

6389

0.88

760.

1908

LB

PLFB

U0.

9989

1.09

90.

8760

0.80

960.

1105

0.77

530.

9807

0.68

770.

5603


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

i/n

T(i/n

)

Figure 3. TTT plot for remission times of bladder cancer

decreasing (increasing) and for bathtub (unimodal) hazard rate functions, the scaled empirical TTTtransform is first convex (concave) and then concave (convex). The TTT plot for this data in Figure3 shows an unimodal hazard rate function and Therefore, indicates that the Burr XII-exponentialdistribution with c > 1 is suitable for fitting this data set.

The MLE, MMLE 1, MMLE 2, MMLE 3 and Bayes estimations, the value of Kolomogor-ov-Smirnov (K − S) statistic with its respective p-value are listed in table 7. Note, the K − S statisticquantities a distance between the emprical distribution function of data set and the cdf of thefitted distribution, i.e. K − S =

√n supx | Fn(x) − F (x) |, where Fn(x) = 1

n

∑ni=1 I[−∞,x)(xi) is

the emprical distribution function of data set and F (x) is fitted cdf. So if data set comes fromdistribution F (x), then K − S converges to 0 almost surely in the limit when n goes to infinity.Also, note that when n goes to infinity, distribution of K − S converges to Kolomogrov-Smirnovdistribution with cdf H(t) = 1− 2

∑∞i=1(−1)i−1e−2i2t.

Also, p-value=P (K − S > (K − S)0 | distribution of data set is F (x)), where (K − S)0 is value oftest statistic for data set. The p-value simply indicates the degree to which the data set conform tothe fitted cdf. On the other hand, p-value denotes the threshold value of the significance level in thesense that the null hypothesis (distribution of data set is F (x)) will be accepted for all values of αless than the p-value. For example, p-value for Burr XII-exponential distribution fitted by MMLE1is 0.7077, so the null hypothesis will be accepted at all significance levels less than p-value (i.e.0.5 and 0.7), and rejected at higher levels, including 0.8 and 0.9. The p-value can be useful, inparticular, when the null hypothesis is rejected at all predefined significance levels, and you needto know at which level it could be accepted. For further reading on p-value, refer to Greenland etal. (2016).

From values in Table 7, we conclude that the estimation method MMLE2 is better than the otherestimation methods since K−S = 0.0420 and p-value=0.9705 and has less K−S and more p-value.


Table 7. Estimation of parameters of the Burr XII-exponential distribution for remission times of bladder cancer

method c p θ K-S p-valueMLE 1.3414 0.3416 0.2436 0.0584 0.7528

MMLE 1 1.3126 0.4082 0.21 0.0608 0.7077MMLE 2 1.3508 0.3183 0.2690 0.0420 0.9705MMLE 3 1.0246 0.6695 0.1615 0.0834 0.3178

BELFU 1.3426 0.3417 0.2414 0.0617 0.6910BPLFU 1.5951 0.3430 0.2452 0.0567 0.7827BELFT 1.3424 0.3422 0.2444 0.0561 0.7935BPLFT 1.5960 0.3436 0.2452 0.0574 0.7704

BELFBU 1.3353 0.3313 0.2365 0.0830 0.3226BPLFBU 1.5852 0.3541 0.2351 0.0474 0.9224LBELFU 1.5149 0.3337 0.22 0.0688 0.5571LBPLFU 1.3425 0.3379 0.2420 0.0648 0.6322

LBELFBU 1.4821 0.3125 0.2478 0.0502 0.8873LBPLFBU 1.5750 0.3448 0.2433 0.0511 0.8751

6. Conclusion

In this article, we introduced the new Burr XII-G family of distributions. A special model in thenew family called Burr XII-exponential distribution that has constant, decreasing and unimodalhazard rate functions, was investigated. This distribution would be applied in Survival analysis andinsurance.

In this article, we suggested estimate the parameters of Burr XII-exponential distribution by maxi-mum likelihood, the three modifications of maximum likelihood and Bayesian methods. The threemodifications of maximum likelihood were subject to constraints, respectively, (1) change-pointequality of distribution and sample (MMLE 1), (2) median equality of distribution and sample(MMLE 2) and (3) F (x(r)) equality and E(F (x(r))) (MMLE 3). Also Bayesian method was underEntropy and Precautionary loss functions and uniform, triangular and Burr XII-uniform priors.

We used Monte Carlo simulated data to compare these methods. We concluded, for all samplesizes, of all methods, in cases unimodal and decreasing hazard rates, the MMLE 2 and in caseconstant hazard rate, MLE produced the least bias in c parameter estimate. In p parameter estimate,MMLE 3, MMLE 1 and MLE produced the least bias in cases decreasing, unimodal and constanthazard rates, respectively. In three cases, MLE produced the least bias in the θ parameter.

Acknowledgement:

The authors acknowledge the Department of Mathematics, Iran University of Science and Tech-nology.


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