Chapter 4 Parametric Families of Lifetime Distributions

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1 Chapter 4 Parametric Families of Lifetime istributions In this chapter, we present four parametric families of lifetime distributions Weibull distribution, gamma distribution, change-point model, and mixture exponential distribution. They represent parametric extensions of exponential distribution under its two fundamental characteristics constant failure rate and memoryless property. The notions of increasing (decreasing) failure rate along with a more general mixture Erlang distribution family are also introduced. 4. Weibull istribution A randomvariable X is said to have Weibull distribution with parameters and, denoted by WEI.; / if its distribution function has the form F ;.x/ e.x/ ; x ; where and are positive real numbers. Its density function is and its failure rate function is given by f ;.x/ x e.x/ ; x > ; r.t/ f.t/ F.t/ t e.t/ e.t/ t ; for t>. Thus, if >, the failure rate is increasing; and if < <, the failure rate is decreasing. Here, increasing means nondecreasing and decreasing means nonincreasing. A.K. Gupta et al., Probability and Statistical Models Foundations for Problems in Reliability and Financial Mathematics, OI.7/ , c Springer Science+Business Media, LLC 7

2 7 4 Parametric Families of Lifetime istributions The parameter is a shape parameter that determines the shape of the density function. Three basic shapes of the density function are represented by <,,or >, respectively, and corresponds to the exponential model. The parameter is a scale parameter as every Weibull distribution is related to WEI.; / by rescaling F ;.x/ e.x/ F ;.x/ That means, if X has Weibull distribution WEI.; /, then X has Weibull distribution WEI.; /. Hence, WEI.; / can be regardedas the standardized Weibull distribution. The special case is known as the Rayleigh distribution. Theorem 4.. If X has a Weibull distribution WEI.; /, then its mean EŒX,variance VarŒX, and the r th moment EŒX r, r>, are given by EŒX C ; VarŒX C C ; EŒX r C r ; r where.t/ defines the gamma function as.t/ x t e x dx; t > Before we prove Theorem 4., we point out the following properties of gamma functions. Lemma 4....t/.t /.t /, t>...n/.n /Š, n ; ;. 3. p. Proof. For t>,.t/ R xt e x dx x t e x j C.t / R x.t / e x dx.t /.t / It follows from./ and induction that.n/.n /Š. To show (3), let t p x, and change of variable leads to. / e t dt. Thus,

3 4. Weibull istribution 73 e s ds 4 e.s Ct / dsdt e t dt Z 4 e dd (in Polar coordinates) Therefore, p. ut Proof of Theorem 4.. We only need to show that for r>, EŒX r C r r In fact, EŒX r x r x e.x/ dx r y= e y dy (letting y.x/ / y.c r / e y dy r C r r It follows that EŒX C ; and VarŒX EŒX.EŒX / C C The moment generating function of Weibull distribution is too complicated to be tractable. The Weibull distribution has been used successfully to describe fatigue failure (Weibull 939, the distribution is named after him), vacuum tube failure (Kao 956), and ball-bearingfailure (Liebleinand Zelen 956). The simplicity of the distribution ut

4 74 4 Parametric Families of Lifetime istributions function of Weibull distribution and the parameters and providequite some flexibility in applications. It is perhaps the most popular parametric family of lifetime (failure) distributions at the present time in reliability of electronic and mechanical systems and components, but not so much in biological components and systems. This will be seen in the next section. It has also been shown that the Weibull distribution is one of the limiting (extreme) distributions of the minimum of independent random variables. The special case of exponential distribution has been studied in Chap., where nx ;n has the same exponential distribution as the identically and independently distributed exponential random variables X ; ;X n have. In fact, the Weibull distribution has the similar property. If X ; ;X n have Weibull distribution F ;.x/ exp..x/ /; then X ;n has the Weibull distribution F = ;.x/. In reliability terms, X n ;n is the lifetime for a serial system with n similar components. Thus, the Weibull family is closed under the series system formation (minimum operation). 4. Gamma istribution A random variable X is said to have gamma distribution with parameters and, denoted by Gam.; /, if its density function is given by f ;.x/ x. / e x ; x > ; where the parameters and are positive real numbers. For an arbitrary, thereis no closed form for the distribution function F ;.x/.when is an integer, F ;.x/ is called an Erlang distribution and its form can be obtained by integration by parts as below F ;.x/ X k.x/ k e x ; x kš To calculate the failure rate function r.t/, we consider r.t/ F.x/ f.t/ t t x e x dx. /. / t e t x e.x t/ dx t

5 4. Gamma istribution 75 u C t e u du t C u e u du t Thus r.t/ C u t e u du If >, C u is decreasing in t, and hence r.t/ is increasing. If < <, t C u is increasing in t, andsor.t/ decreases. When, the distribution t reduces to exponential with constant failure rate. Thus, gamma distribution generalizes the exponential model similar to Weibull distribution in terms of the failure rate function. For any integer n, F ;n.x/ is simply the distribution of the sum of n independent and identically distributed exponential random variables, with parameter. Similar to Weibull distribution, the parameter is a shape parameter, and is a scale parameter. The three basic shapes of the density function are given according to <, and >. Every gamma distribution Gam.; / is related to Gam.; / by rescaling, i.e., if X has gamma distribution Gam.; /,thenx has gamma distribution Gam.; /. Thus, Gam.; / can be viewed as the standard gamma distribution. A special case of gamma distribution with = and n= is referred to as -squared distribution with n degree of freedom, denoted by.n/. Theorem 4.. If X has a gamma distribution Gam.; /, then its mean EŒX,variance VarŒX, and moment generating function M X.t/ EŒe tx are given by EŒX ; VarŒX ; and M X.t/ t Proof. We start with the moment generating function M X.t/. M X.t/ EŒe tx t t t e tx. / x e x dx. / x e. t/x dx. / p t;.x/dx ; for t<

6 76 4 Parametric Families of Lifetime istributions It follows that and Hence, M X.t/. t/ M X.t/. C /. t/ EŒX M X./ VarŒX M X./ M X./. C / ut In general, we have the following result for any moments of X. Theorem 4.3. Let X have a gamma distribution Gam.; /. For any real number r>, EŒX r. C r/. / r Proof. EŒX r x r x e x dx. /. / x Cr e x dx. C r/. / r Cr. C r/ x Cr e x dx. C r/. / ; r since, again, the integrand is the density function of Gam.; C r/. ut The usefulness of the Theorem 4.3 can be illustrated by the following example. Let X have.4/ distribution, i.e., the distribution of X is Gam ;. Then, we have. / E X./ I

7 4. Gamma istribution 77 hp i E X C./ = p 3 3 p 4 Theorem 4.4. Let X i have gamma distribution Gam.; i /, i n, and let X ;X ; ;X n be independent. Then, Y X C X CCX n has gamma distribution Gam.; C CC n/. Proof. Since X ;X ; ;X n are independent, the moment generating function of Y is C CC n M Y.t/ M X.t/M X.t/ M Xn.t/ t The uniqueness of the moment generating function yields that Y has gamma distribution Gam.; C CC n/. ut In reliability terms, if X i represents the lifetime for the i th component for i ; ;n and the system is formed as cold-redundant, that means, if the i th component fails, the.i C / th component will replace it like a new one. Then, the system lifetime for this n cold-standby components will be X C C X n. Thus, the gamma family is closed under the formation of cold standby system (or convolution) with same scale parameter. Corollary 4.. Let X ;X ; ;X n be a random sample from a population, and Y X C X CCX n.. If the population has gamma distribution Gam.; /, theny has gamma distribution Gam.; n /.. If the population has distribution./,theny has distribution.n/. Proof. By letting n in Theorem 4.4, we get (), and () follows from = and =. ut Example 4.. Let X ;X ; ;X n be a random sample from a normal population N.; /. When is known, Z i X i has N.; / distribution, so that Z i.x i /

8 78 4 Parametric Families of Lifetime istributions has./ distribution. Thus, nx.x i / i has.n/ distribution. When is unknown, it is known that follows.n / distribution..n /S nx.x i X/ N On the one hand, the Weibull and gamma distributions with shape parameter >are of great interests. For example, they may be more suitable as models for life length of electronic components, where few will have very short life lengths, many will have something close to an average life length, and very few will have extraordinary long life lengths. In application, Weibull paper is used to plot data set, like the normal paper, to verify whether the data are drawn from a Weibull population. On the other hand, the Weibull distribution with shape parameter <has been used to model large claim in insurance risk model (Chap. 9). i 4.3 Change-Point Model A lifetime X with failure rate function r.t/ follows a change-point model if r.t/ ; for t I ; for t>; where is called the change-point, the pre-change failure rate, and the postchange failure rate. Obviously, when, it reduces to the exponential distribution. If >, the failure rate is increasing, and if <, the failure rate is decreasing. Under the change-point model, F.x/ N e R x r.t/dt e x ; for x I e.x / ; for x> Consequently, the probability density function is given by f.x/ e x ; for x I e.x / ; for x>

9 4.4 Mixture Exponential istribution 79 The mean of X can be calculated as EŒX. F.x//dx Z e x dx C e e.x / dx. e / C e e x dx. e / C e That means, the mean is a weighted average of = and =. The calculation of variance is left as an exercise. The change-point model has been widely used in online quality control, where the change represents a sudden change in system dynamic structures. More recent applications can be found in survival analysis where the baseline hazard function may follow a change-point model (Matthews and Farewell 98). Further extensions of the single change-point model to multiple change-point model have been used as alternatives to bathtub failure rate functions and epidemic models. 4.4 Mixture Exponential istribution The simple two-component mixture exponential distribution is defined as F.x/. /. e x / C. e x /; where Œ; is the mixing proportion. The corresponding mixture exponential density function is given by and its failure rate function is equal to f.x/. / e x C e x ; r.x/ f.x/ G.x/. / e x C e x. /e x C e x By taking the derivative of r.x/,wehave r.x/. / e x C e x. /e x C e x.. / e x C e x /.. /e x C e x / Thus, its failure rate function is always decreasing.

10 8 4 Parametric Families of Lifetime istributions Theorem 4.5. The mean and variance of mixture exponential distribution are given by E.X/ Var.X/ C I C C. / Proof. We only give the details for the variance and leave the proof for the mean as an exercise. First, the second moment of X can be calculated as Thus, E.X /. / x h. / e x C e x i dx. / x e x dx C Var.X/ E.X /.EX/. / C C C C C x e x dx C C C C C. /. / C. / C C C. /. / C. / Generally, let H./ be a mixing distribution for <. We call F.x/ a mixture exponential distribution. Its failure rate function can be calculated as r.x/ d dx ln N F.x/. e x /dh./ R e y dh./ R e y dh./ ut

11 4.5 IFR (FR) and Mixture Erlang istribution 8 To analyze r.x/, we note that H y./ e y dh./ R e y dh./ ; defines a new distribution function for. Itsk th moments can be calculated as k H y./ R k e y dh./ R e y dh./ The derivative of r.x/ is equal to r.x/ ; R R R! e y dh./ e y dh./ C R e y dh./ e y dh./ dh y./ C dh y./ since the last equality gives the negative variance of H y./. Thus, the mixture exponential distribution always has a decreasing failure rate function. The mixture exponential distribution has been used as a model for claim amounts in insurance mathematics and lifetime distribution for systems under shock models when there are multiple shock sources or lifetime distribution for components when components are produced from different assembly lines. 4.5 IFR (FR) and Mixture Erlang istribution The four parametric families are mainly generalized by extending the constant failure rate for the exponential distribution to increasing or decreasing failure rate functions. To understand how they also generalize the exponential distribution in terms of memoryless property, we recall that the residual life has the distribution function F.x C t/ F.t/ F.xjt/ P.X t C xjx >t F.t/ Z xct exp r.s/ds t Since d dt Z tcx t r.s/ds r.t C x/ r.t/;

12 8 4 Parametric Families of Lifetime istributions thus, r.t/ is increasing if and only if R tcx t r.s/ds is increasing in t for any x>. So we have the following result. Theorem 4.6. If the failure rate function r.t/ for X exists, then r.t/ is increasing (decreasing) if and only if the residual life distribution F.xjt/ F.xCt/ F.t/ is F.t/ increasing (decreasing) in t for any x>. Therefore, the IFR (FR)(Increasing Failure Rate (ecreasing Failure Rate)) class can be defined based on increasing or decreasing residual life distribution F.xjt/ in t no matter whether the failure rate function r.t/ exists or not. Remark. A typical probability model popular in reliability and biostatistics is a combination of IFR and FR. There are many data sets in reliability testing, clinical trials, and biostatistics surveys that reveal the so-called bath-tup shape failure rate. The failure rate is initially decreasing during the infant mortality phase, then remains relatively constant during the useful life phase, and finally reaches the wear-out phase, i.e., the failure rate increases. The following is a typical example Example4. (Mixture Erlang istribution). Let f k.x/.x/k e x ; for k ; ; ;.k /Š be the Erlang density function. The Erlang density can be seen as a special case of gamma distribution with integer shape parameter and it can also seen as a convolution of k identical exponential density functions. Its cumulative distribution can be calculated by integrating by parts k times as F k.x/ Z x Z x f k.y/dy.y/ k e y dy.k /Š k X e x j.x/ j jš Obviously, every Erlang distribution is IFR. Our interest is on the following mixture Erlang density f.x/ rx q k f k.x/; k where q CCq r are the mixing proportions. enote by Q j q Cq j ; and NQ j q j C Cq r ;

13 4.5 IFR (FR) and Mixture Erlang istribution 83 with NQ. Then the corresponding survival function N F.x/can be calculated as N F.x/ rx k e x q k N F k.x/ rx k Xr e x j Xr e x j k X q k j.x/ j jš NQ j.x/ j jš.x/ j jš rx kj C We shall denote by E ;;;r the above mixture Erlang distribution of order k. Its failure rate function can thus be written as By taking derivative, we obtain r.x/ r.x/ f.x/ P r F.x/ N j q j C.x/ j =j Š P r N j Q j.x/ j =j Š " Pr 3 j Q N.x/ j Pr j C jš j P r j P r j Pr j q k Q N # j C.x/ j =j Š Q N j.x/ j =j Š Q N.x/ j j jš N Q j.x/ j jš Pr N.x/ j Q j j C jš That means, r.x/ has the same sign as the term 6 Xr 3.x/ 4@ NQ j Xr j C jš j j NQ j.x/ j jš Xr j A3.x/ NQ j 7 j C 5 jš Let us consider two special cases (a) r. The above term becomes NQ >. Therefore, E ; is always IFR. (b) r 3. The sign of r.x/ is the same as the sign of Œ NQ NQ NQ NQ.x/ NQ.x/ = q 3 C.q C q 3 / C q 3.q C q 3 /.x/ C q 3.x/ =

14 84 4 Parametric Families of Lifetime istributions Thus, we see that E ;;3 is IFR, if and only if, q 3 C.q C q 3 / That means, if q 3 >.q C q 3 / ; then r.x/ changes from negative to positive. That means, r.x/ is first decreasing and then becomes increasing. Mixture Erlang distributions define a much broad parametric distribution class since any absolutely continuous distribution on.; / may be approximated arbitrarily accurately by a distribution of this type. In fact, for F./ and arbitrary >, the distribution function defined by the density function F ı.x/ X k p k. /.= /k x k e x= ;.k /Š where p k. / F.k / F..k / /, satisfies lim! F.x/ F.x/ for any continuous point of F.x/ (Tijms 994). Thus, mixture Erlang distributions can be used to approximate any continuous distribution by matching a certain number of finite moments. We conclude this section by noticing that if X has a distribution that is both IFR and FR, then X must have an exponential distribution. A more systematic discussion for various nonparametric lifetime distribution classes based on the two fundamental properties of exponential distribution will be given in the next chapter. Problems. Consider the following two Weibull distributions as survival models (a), 5 (b) 5, For each distribution, find (a) The survival distribution (b) The failure rate function (c) The mean and variance. Which distribution gives the larger survival probability of at least three units of time?. Suppose the pain relief time follows the gamma distribution with, 5. Find the mean and variance.

15 4.5 IFR (FR) and Mixture Erlang istribution Suppose that, for the gamma distribution. Find (a) The survival function (b) The failure rate function (c) The mean and variance. 4. Consider a life-testing distribution with density function of the form f.x/ xe x for x>. (a) Find the cumulative distribution function F(x) and the survival function. (b) Find the failure rate function r.x/. (c) oes this model have the IFR property? 5. Suppose items are produced from two different assembly lines. An item is produced from Line with probability.7 and its lifetime distribution is exponential with mean.. It is produced from Line with probability.3 and its lifetime distribution is exponential with mean.5. (a) Find the mixture exponential distribution for the lifetime. (b) Find its corresponding density function. (c) Calculate the failure rate function. (d) Show that the failure rate function is always decreasing. 6. Suppose the survival times for a group of leukemia patients has failure rate following a change point model r.t/ for t 5 and 5 for t>5. (a) Write down the survival function. (b) Find the probability that a patient will survive more than.. (c) What is the mean survival time. 7. Suppose the O-rings on the spaceship Columbia have the Weibull lifetime distribution with failure rate function r.t/ t =. What is the probability that this O-ring is still in use after two years? 8. The lifetime, T, of a semiconductor device has a Gamma distribution with and 5. (a) Find the probability that the device fails before 5 months. (b) What is the failure rate of the device at age t 5. (c) What are the mean and standard deviation of the life distribution? 9. A large number of identical relays have times to first failure that follows a Weibull distribution with parameters and 5 years. What is the probability that a relay will survive (a) year, (b) 5 years, and (c) years without failure and what is the mean time to the first failure?. The variation in the output power of a motor is found to follow a gamma distribution with and 3. What is the probability that the power output is less than KW?

16 86 4 Parametric Families of Lifetime istributions. Suppose a component with mean operating time of h has five spares. (a) What is the expected operation time to be obtained from the component and spares? (b) If the components are distributed as exponential, find the reliability R(), the probability that the system is still in operation after h. (c) If we want the reliability R./ 95, how many spares are needed to achieve it?. Prove that a necessary and sufficient condition for a random variable with support Œ; / to be distributed with F.x/ exp..x=ˇ/ / is that EŒX jx >y y C ˇ ; > ; y< 3. Prove that the failure rate function is r.t/ t ; t > if and only if the lifetime distribution is WEI.; /. 4. Find the density function of the minimum of n i.i.d. Weibull random variables. 5. Let X and Y be independent, nondegenerate and positive random variables. Then X C Y and X=Y are independent if and only if both X and Y are gamma distributions with the same scale parameter. 6. Let X i, i ; ;nbe i.i.d. random variables. Prove that if the distribution of X i is IFR, then so is the distribution of i th order statistic X i;n. 7. Let X and Y be independent exponential random variables with same rate. (a) Show that the conditional density function of X,givenX C Y c for c>is uniform f XjXCY.xjc/ c ; < x < c (b) What is EŒXjX C Y c? 8. Let X and Y be independent exponential random variables with rate and ı, respectively. Calculate the conditional density function and conditional mean of X given X C Y c for c>. 9. Show that the mixture Erlang distribution E ; is always IFR.. Show that the mixture Erlang distribution E ;;3 with q has the failure rate which is first decreasing and then increasing.. Show that a plot of logœlog. F.x// against log x will be a straight line with slope for a Weibull distribtion.. Calculate the variance of X whose failure rate follows the change-point model.

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