AS305 Statistical Methods For Insurance

REVIEW OF SURVIVAL MODELS

The distribution of T

A survival model is defined in terms of the function S(t),

S(t) = Pr(T> t), T is the failure time random variable. This function of the random variable T is called the Survival

Distribution Function (SDF). It gives the probability that failure (death) will occur after

time t, which is the same as the probability that the entity, known to exist at time ≥ 0, will survive to at least time t.

S(0)=1 and S(∞) = 0

CDF of T

The CDF gives the probability that the random variable will assume a value less than or equal to t.

F(t) = Pr(T ≤ t). probability that failure (death) will occur not later

than time t. F(t) = 1-S(t) F(0) = 0 and F(∞) = 1.

The Probability Density Function

For the case of a continuous random variable, the Probability Density Function (PDF), f(t), is defined as the derivative of F(t).

Consequently,

and

It is important to recognize that f(t) is the unconditional density of failure at time t. By this we mean that it is the density of failure at time t given only that the entity existed at t=0.

The Hazard Rate Function

We now define a conditional density of failure at time t, such density to be conditional on survival to time t. This conditional instantaneous measure of failure at time t, given survival to time t, will be called the hazard rate at time t, or the Hazard Rate Function (HRF) when viewed as a function of t. It will be denoted by (t).If a conditional measure is multiplied by the probability of obtaining the condition, then the corresponding unconditional measure will result, or

(t) and f(t) compares

They are both instantaneous measures of the density of failure at time t;

they differ from each other in that (t) is conditional on survival to time t, whereas f(t) is unconditional (i.e., given only existence at time t=0).

In the actuarial context of human survival models, failure means death, or mortality, and the hazard rate is normally called the force of mortality.

Some important mathematical consequences

As it follows that

Cumulative Hazard Function

Moment of T

For 1st moment, T ≥ 0

Using Integration by part

For 2nd moment, T ≥ 0

Median of T

Complete Expectation of life Curtate expectation of life

Actuarial Survival Model

In actuarial context, the hazard rate is called force of mortality, x

The 1st moment of X,

It is the complete expectation of life at birth.For the select model S(t;x), t is a value of the random variable, T and x is the age at which the person selected.The expected value of T, E[T;x] gives the expected future lifetime for a person selected at age x is denoted by .

And HRF is given by

][ x

oe

Examples of Parametric Survival Models

Uniform Exponential Gompertz Makeham Weibull Others

Gompertz

x ≥ 0, B > 0, c > 1

Hazard Rate is defined as

SDF is given by

PDF is given by

Clearly not convenient

Makeham Distribution

PDF is not tractable that make calculation of probability, moment difficult

x ≥ 0, B > 0, c > 1, A > - B

Weibull

(x) = k xn, x ≥ 0, k > 0, n > 0

Distinction between npx, a conditional probability with the unconditional probability represented by S(n; x).

seek the probability that a person age x will survive to age x+ n.

When we determine this probability in accordance with the model S(x) it is conditional, it is

If it is S(n; x). It is unconditional, denoted as the PDF for death at age y, given alive at age x (y>x). the conditional HRF (or force of mortality) for death at age y,

given alive at age x (y>x). (There is no such thing as an unconditional HRF.)

y is conditional on survival to y,

Lower Truncation of the Distribution of X

When we speak of probabilities (or densities) conditional on survival to age x, we are dealing with the distribution of a subset of the sample space of the random variable X, namely those values of X which fall in excess of x. This distribution is called the distribution of X truncated below at x.

the probability that the age at death will exceed x+n, given that it does exceed x.

Upper and Lower Truncation of the Distribution of X

the probability that death will occur after age x, given that it does occur between y and z.

truncation only from below did not affect the HRF, truncation from above does (since the truncated HRF is a function of z).

Moments of Truncated Distributions

The Central Rate

A type of conditional measure over the interval from age x to age x+1 is called the central rate of death, and is denoted by mx.

It is defined as the weighted average value of the HRF (x) over the interval, using, as the weight for (y), the probability of survival to age y.

More generally, nmx is the average hazard, or central rate of death, over the interval from x to x+ n, and is given by

Example:If X has an exponential distribution, show that this implies mx = - ln Px

Use of Conditional Probabilities in Estimation

Suppose we wish to estimate, say, S(10), the probability of survival from t=0 to t=10.

The study (and the data) will suggest that we consider only the time interval from t=i to t=i+l, and estimate the conditional probability of survival over that interval.

i.e. we estimate

And S(10) = ….

In general S(t) = ….

)(ˆ)1(ˆˆ

iSiSpi

0p̂ 1p̂ 9p̂

0p̂ 1p̂ 1ˆ tp

Transformation of Random Variables

Given X as a random variable and y = g(x) with inverse x = h(y) where h(y) = g -1(y) exist.

If g(.) is a strictly increasingfunction then

Example: let X have a standard exponential distribution, so that Fx(x) = 1 - e-x, let y = g(x) =1/x. Find the SDF, PDF and HRF of Y.

Two special Transformation

Linear Transformation the hazard at a given value of the new variable is times the

hazard For certain kinds of distributions, a linear transformation of that

random variable produces a new random variable with the same kind of distribution as the original random variable

Probability Integral Transformations special case of a transformation is y = g(x) = Fx(x), where FX(X) is

a strictly increasing CDF of X. y has standard uniform distribution.

Show that if X has a Gompertz distribution, then the transformed variable y = (x-)/ also has a Gompertz distribution when =0

Let y = g(x) = - lnSx(x). Find the distribution of Y.

Since Sx(x) decreases from 1 to 0, then ln Sx(x) decreases from 0 to -∞, and the transformed variable y = g(x) = -lnSx(x) increases from 0 to ∞.

if z = Sx(x), then Z has a standard uniform distribution.

y = — ln z, inverse function z = h(y) = e -y, so that dz/dy= - e-y

Therefore Y has a standard exponential distribution with =1.

MEAN AND VARIANCE OF TRANSFORMED RANDOM VARIABLES

A rough approximation

Bivariate

THE LIFE TABLE

The life table is a table of numerical values of S(x) for certain values of x.

Typically a complete life table shows values of S(x) for all integral values of x, x = 0, I ,.. .. Since S(x) is represented by these values, it is clear that a practical upper limit on x must be adopted beyond which values of S(x) are taken to be zero.

Use as the smallest value such that S(x)=0 and

S(

Can calculate for integer x, n

Can not be determined without additional assumptions on S(x) between adjacent values

THE TRADITIONAL FORM OF THE LIFE TABLE the tabular survival model differs from Table in

two respects. Rather than presenting decimal values of S(x), it

is usual to multiply these values by, say, 100,000, and thereby present the S(x) values as integers.

As these integers are not probabilities (which S(x) values are), the column heading is changed from S(x) to lx, where l stands for number living, or number of lives. known as the life table.

As S(0) = 1, then clearly lo is the same as the constant multiple which transforms all S(x) into lxThis constant is called the radix of the table. Formally,

With radix of 100,000 the above table becomeslx is the expected number of survivors to age x out of an original group of l0 new-bornshas both a probabilistic and a deterministic interpretation.

Force of Mortality - relative instantaneous rate of death

Moment of T

Conditional Probability

Central Rate

The concept of exposure

is the total number of years lived by those deaths after attaining age x.

gives the aggregate number of years lived between ages x and x+ I by the lx persons who comprised the group at age x.

This quantity is measured in units of life-years, and is called exposure, since it is a measure of the extent to which the group is exposed to the risk of dying

Integrating by part

=, the limit of the Riemann sum

Method for non-integral ages

Uniform distribution

Method for non-integral ages

Constant force of death

Balducci assumption

Hyperbolic form

Select and Ultimate life table A life table is essentially a table of one-year death probabilities qx, which

completely defines the distribution. of If. Life tables are constructed for certain population groups, differentiated by

factors such as sex, race, generation and insurance type. The initial age x can have a significant influence in such tables. For instance, let x denote the age when the person bought life insurance.

Since insurance is only offered to individuals of good health (sometimes only after a medical test), it is reasonable to expect that a person who has just bought insurance, will be of better health than a person who bought insurance several years ago, other factors (particularly age) being equal.

This phenomenon is taken into account by select life tables. In a select life table, the probabilities of death are graded according to the age at entry.

The selection effect has usually worn off after some years, say r years after entry

The period r is called the select period, and the table used after the select period has expired, is called an ultimate life table. '