A Marshall-Olkin Gamma Distribution and Process

Transcription

1 CHAPTER 3 A Marshall-Olkin Gamma Distribution and Process 3.1 Introduction Gamma distribution is a widely used distribution in many fields such as lifetime data analysis, reliability, hydrology, medicine, meteorology, etc. (see for example the references in Mosino and Garcia (1981), Gupta and Kundu (1999), Aksoy (2000) and Nadarajah and Gupta (2007)). In recent years many authors generalized the gamma distribution by various ways. Some new generalizations can be found in Nadarajah and Kotz (2006) and Nadarajah and Gupta (2007). Nadarajah and Kotz (2006) introduced the exponentiated Some results included in this chapter have appeared in the paper Ristic et al. (2007) 48

2 gamma distribution with the density function f(x) = αxa 1 e x { } α 1 γ(a, x), x > 0, a > 0, α > 0. Those authors discussed the shape properties of the probability density function and the hazard rate function of the exponentiated gamma distribution. Also, they derived the moment generating function and the moments when the parameters a and α are positive integers. Nadarajah and Gupta (2007) introduced a generalized gamma distribution with the density function f(x) = Cx α 1 (x + z) ρ e λx, x > 0, where C is a normalizing constant, α > 0, λ > 0, z > 0 and ρ R. They discussed the shape properties of the probability density function of the generalized gamma distribution, derived the moments and estimated the unknown parameters by the method of maximum likelihood. Also, they discussed the application of this distribution to drought data. In this chapter we generalize the gamma distribution using the method proposed by Marshall and Olkin (1997). These authors introduced two new distributions: an exponential distribution with two parameters and a Weibull distribution with three parameters. Jose and Alice (2001) introduced a Marshall-Olkin generalized Weibull distribution. Alice and Jose (2003) introduced a Marshall-Olkin Pareto distribution. Alice and Jose (2004) introduced a Marshall-Olkin bivariate semi-pareto distribution. Alice and Jose (2005a) introduced a Marshall-Olkin Semi-Weibull distribution. Alice and Jose (2005b) introduced a Marshall-Olkin logistic distribution. Jayakumar and Thomas (2007) generalized a family of two parameters Marshall-Olkin distributions to a family of three parameters Marshall-Olkin distributions. In section 2 the Marshall-Olkin gamma distribution is introduced and its shape properties are studied in detail.the geometric minimum and geometric maximum are also found to have the Marshall-Olkin gamma distribution. Maximum likelihood estimation of the parameters are also considered. Expressions for moments are derived and some simulation 49

3 studies are also done. A minification process with Marshall-Olkin gamma distribution is constructed and its properties are studied. The new distribution is fitted to a real data set and it is established that the new distribution is a good fit. 3.2 A Marshall-Olkin Gamma Distribution and its Properties In this section, we introduce a Marshall-Olkin gamma distribution. Let F (x) = γ(a, x) = 1 x 0 t a 1 e t dt be the distribution function of gamma distribution with a parameter a > 0. Replacing F in (2.1.1) we obtain that a Marshall-Olkin gamma distribution has distribution function given by G(x) = γ(a, x), x > 0, a > 0, α > 0. (3.2.1) α + (1 α)γ(a, x) Let us consider the density function of a Marshall-Olkin gamma distribution with parameters a and α. We have g(x) = αx a 1 e x, x > 0, a > 0, α > 0. (3.2.2) (α + (1 α)γ(a, x)) 2 If α = 1, then we obtain gamma distribution with parameter a > 0. If a = 1, then we obtain a Marshall-Olkin exponential distribution with parameter α > 0. Let a be a positive integer. Then the distribution function G is given by G(x) = 1 a 1 m=0 x m m! e x 1 (1 α) a 1 m=0, x m m! e x and the density function g is given by g(x) = αx a 1 e x ( (a 1)! 1 (1 α) a 1 m=0 ) 2. x m m! e x 50

4 Figure 3.1: The probability density function of Marshall-Olkin gamma for different values of parameters a and α. The parameters a and α are shape parameters. Now we discuss the shape properties of a Marshall-Olkin gamma density function. If a > 1, then the function g(x) has a unique mode at x = x 0 with g(x) increasing for all x < x 0 and decreasing for all x > x 0, where x 0 is the solution of the equation (a 1 x)(α + (1 α)γ(a, x)) = 2(1 α)x a e x. (3.2.3) Furthemore, g(0) = 0 and g( ) = 0. If a < 1 and α < 1, then the function g(x) is a decreasing function with g(0) = and g( ) = 0. 51

5 Let a < 1 and α > 1. Let x 1 be the solution of the equation (1 α)x a 1 e x (x a 1) = α + (1 α)γ(a, x). a) If x 1 < a 1 + 2(1 a), then the density function g(x) is a decreasing function with g(0) = and g( ) = 0. b) If x 1 > a 1+ 2(1 a), then the equation (3.2.3) has two real roots x 2 < x 3. The density function g(x) has a unique mode at x = x 3 and decreases for all x (0, x 2 ) (x 3, ) and increases for all x (x 2, x 3 ). Furthermore, g(0) = and g( ) = 0. Figure (3.1) illustrate the shape of the Marshall-Olkin gamma density function for different values of the parameters a and α. Let us consider the hazard rate function of the Marshall-Olkin gamma distribution. The hazard rate function of the Marshall-Olkin gamma distribution is given by r(x, a, α) = x a 1 e x, x, a, α > 0. (3.2.4) ( γ(a, x))(α + (1 α)γ(a, x)) Now we discuss the shape properties of the hazard rate function of the Marshall-Olkin gamma distribution. For various values of the parameters it is found to be IFR or DFR as given below.: If a < 1 and α < 1, then the hazard rate function r(x) is a decreasing function (DFR) with r(0) = and r( ) = 1. If a > 1 and α > 1, then the hazard rate function r(x) is an increasing function (IFR) with r(0) = 0 and r( ) = 1. Let a < 1 and α > 1. Let p(x) = (a 1)[(2α 1) + 2(1 α)γ(a, x)] + 4(1 α)(a x)x a e x 52

6 Figure 3.2: The hazard rate function of Marshall-Olkin gamma for a = 0, b = 1 and different values of parameters m, n, α. and s(x) = (a 1)( γ(a, x))(α + (1 α)γ(a, x)) +x a e x [(2α 1) + 2(1 α)γ(a, x)]. If p((2a a + 3)/2) > 0, s (x 1 ) < 0 and s(x 2 ) > 0, where x 1 and x 2 are the lower solutions of the equations p(x) = 0 and s (x) = 0, respectively, then r(x) has a minimum at x = x 3 and a maximum at x = x 4, where x 3 < x 4 are the solutions of the equation s(x) = 0. In other cases, the hazard rate function r(x) is a decreasing function with r(0) = and r( ) = 1. 53

7 Let a > 1 and α < 1/2. If p((2a a + 3)/2) < 0, s (x 1 ) > 0 and s(x 2 ) < 0, where x 1 (x 01, x 02 ), x 01, x 1 and x 02 are the solutions of the equation p(x) = 0 and x 2 is the lower solution of the equation s (x) = 0, then r(x) has a maximum at x = x 3 and a minimum at x = x 4, where x 3 < x 4 are the solutions of the equation s(x) = 0. In other cases, the hazard rate function r(x) is an increasing function with r(0) = 0 and r( ) = 1. Let a > 1 and 1/2 < α < 1. If p((2a a + 3)/2) < 0, s (x 1 ) > 0 and s(x 2 ) < 0, where x 1 and x 2 are the lower solutions of the equations p(x) = 0 and s (x) = 0, respectively, then r(x) has a maximum at x = x 3 and a minimum at x = x 4, where x 3 < x 4 are the solutions of the equation s(x) = 0. In other cases, the hazard rate function r(x) is an increasing function with r(0) = 0 and r( ) = 1. Figure (3.2) illustrate the shape of the hazard rate function of the Marshall-Olkin gamma distribution for different values of the parameters a and α. Now we establish the following properties of the Marshall-Olkin gamma distribution. Theorem Let {X n, n 1} be a sequence of i.i.d. random variables with a Marshall-Olkin gamma distribution with parameters a and α. Let N be a random variable with a geometric(p) distribution and suppose that N and X i are independent. Then: a) min(x 1, X 2,..., X N ) has a Marshall-Olkin gamma distribution with parameters a and αp, b) max(x 1, X 2,..., X N ) has a Marshall-Olkin gamma distribution with parameters a and α/p. Proof. Follows from the Proposition 5.1 and 5.2 of Marshall and Olkin (1997). Corollary Let {X n, n 1} be a sequence of i.i.d. random variables with a gamma(a) distribution. Let N be a random variable with a geometric(p) distribution and suppose that N and X i are independent. Then: 54

8 a) min(x 1, X 2,..., X N ) has a Marshall-Olkin gamma distribution with parameters a and p, b) max(x 1, X 2,..., X N ) has a Marshall-Olkin gamma distribution with parameters a and 1/p. 3.3 Estimation We consider estimation of the unknown parameters by the method of maximum likelihood. The log-likelihood function for some realizations x 1, x 2,..., x N is N N log L(a, α) = N log α + N log x i + (a 1) log(x i ) 2 i=1 i=1 N log (α + (1 α)γ(a, x i )). The first derivatives of log L(a, α) with respect to a and α are i=1 log L a = Nψ(a) + N log x i 2 i=1 N i=1 1 α + (1 α)γ(a, x i ) αψ(a) + (1 α)x a i 2 F 2 (a, a; a + 1, a + 1; x i ) +Γ(a, x i, 0) [log(x i ψ(a)] The maximum likelihood estimates can be obtained setting the above expressions to zero and solving them numerically. Function nlm from the statistical software R can be used to finding the maximum likelihood estimates. 3.4 Moments Now we consider the moments of the Marshall-Olkin gamma distribution when the parameter a is a positive integer. The moments can be evaluated using Corollary (3.2.1). First consider the case when α < 1. Let N be a random variable with geometric(α) distribution. Then the random variable Z N = min(x 1, X 2,..., X N ) has a Marshall-Olkin gamma 55

9 distribution with parameters a and α < 1. Then the kth moment is E(Z k N) = α E(X(1),n)(1 k α) n 1, n=1 where X (1),n is a 1th order statistic in a sample of size n from a gamma distribution with parameter a. Now, using the expression of the moments of the 1th order statistic from Gupta (1960), we derive the moments of random variable Z N as E(Z k N) = α n=1 (a 1)(n 1) n(1 α) n 1 m=0 where b m (a, p) is the coefficient of t m in the expansion of as b m (a, p) = m j=0 Γ(a + k + m) b m (a, n 1), n a+k+m ( a 1 p t j!) j/ given recursively j=0 b j (a,p 1), (m j)! m a 1 m b j (a,p 1), (m j)! a m (a 1)(p 1) j=m a+1 a 1 j=m (a 1)(p 1) b m j (a,p 1) j!, (a 1)(p 1) + 1 m (a 1)p with initial conditions b m (a, 1) = 1/m! for m {0, 1,..., a 1}. Since G(x) Γ(a, x)/, for all x > 0, when α < 1, it follows that E(ZN k ) Γ(a + k)/. Let us consider the case α > 1. Let N be a random variable with geometric(1/α) distribution. Then the random variable W N = max(x 1, X 2,..., X N ) has a Marshall-Olkin gamma distribution with parameters a and α > 1. The kth moment of the random variable W N is E(W k N) = 1 α n=1 E(X k (n),n) ( 1 1 α) n 1, where X (n),n is a nth order statistic in a sample of size n from a gamma distribution with parameter a. Using the expression of the moments of the nth order statistic from Gupta 56

10 (1960), we derive the moments of random variable W N as E(W k N) = 1 α n=1 ( n 1 1 ) n 1 n 1 ( n 1 ( 1) s α s s=0 ) (a 1)s m=0 Γ(a + k + m) b m (a, s) (s + 1). a+k+m In Table (3.1), the 1st moment, the standard deviation, the skewness and the kurtosis of the Marshall-Olkin distribution are given for some values of a and α. The computations are available to eight significant figures and rounded to four decimal places. 3.5 A Marshall-Olkin Gamma Minification Process In this section we introduce a minification process with Marshall-Olkin gamma distribution. Let us consider a process {X n, n 0} given by ε n, w.p. α, X n = min(x n 1, ε n ), w.p. 1 α, (3.5.1) where {ε n, n 1} is a sequence of i.i.d. random variables, X n 1 and ε n are independent random variables and 0 < α < 1. Suppose that {ε n, n 1} is a sequence of i.i.d. random variables with Gamma(a) distribution and let X 0 has a Marshall-Olkin gamma distribution with parameters a and α. Then it follows from (3.2.1) and (3.5.1) that F X1 (x) = αf ε (x) + (1 α)f X0 (x)f ε (x) = 1 γ(a, x) α + (1 α)γ(a, x), i.e. it follows that X 1 has a Marshall-Olkin gamma distribution with parameters a and α. By induction we can show that X n has a Marshall-Olkin gamma distribution with parameters a and α. Thus {X n, n 0} is a stationary process with Marshall-Olkin gamma distribution with parameters a and α. Conversely, suppose that {X n, n 0} is a stationary process with a Marshall-Olkin gamma distribution with parameters a and α and let {ε n, n 1} is a sequence of i.i.d. 57

11 random variables. Then F ε (x) = F X (x) α + (1 α)f X (x) = 1 γ(a, x), i.e. ε n has a Gamma(a) distribution. Thus, we have proved the following theorem. Theorem Let {ε n, n 1} is a sequence of i.i.d. random variables. A random process {X n, n 0} given by (3.5.1) is a stationary process with Marshall-Olkin gamma distribution with parameters a > 0 and α (0, 1) if and only if ε n has a Gamma(a) distribution. Now we consider some properties of the Marshall-Olkin gamma minification process. First, we derive the joint survival function of a random vector (X n+1, X n ). The joint survival function is given by F Xn+1,X n (x, y) = F ε (x) [ αf X (y) + (1 α)f X (max(x, y)) ] ( Γ(a,x) 1 α2 γ(a,y)+α(1 α)γ(a,x)+(1 α)γ(a,x)γ(a,y) (α+(1 α)γ(a,x))(α+(1 α)γ(a,y)) = ( 1 Γ(a,x) γ(a,y) α+(1 α)γ(a,y) This is not an absolutely continuous survival function, since ), x > y ), x < y. P (X n+1 = X n ) = 1 α + α log α 1 α (0, 1). Now, we consider the probability of the event {X n+1 > X n }. We have P (X n+1 > X n ) = α(1 α + α log α) (1 α) 2 (0, 0.5). 3.6 Autocorrelation The autocovariance between the random variables X n+1 and X n can be determined using the same technique used for Marshall-Olkin beta process in chapter 2 and the result can 58

12 be obtained as where { Cov(X n+1, X n ) = (1 α) µ 2 I 1 + I 2 µ } 1 aα 1 α µ 1 Similarly, I 1 E(XnI(X 2 n > ɛ n+1 )) = 1 = 1 ( 1) k (a + k)k! µ a+k+2. k=0 I 2 E(X n ɛ n+1 I(X n > ɛ n+1 )) = 1 0 k=0 x 2 f(x)γ (a, x) dx ( 1) k (a + k + 1)k! µ a+k+2. The autocorrelations of the Marshall-Olkin gamma minification process are difficult to calculate. We derive the first autocorrelation using Monte Carlo method. For different values of the parameters a and α, we simulate 100 sequences of 1000 observations from Marshall-Olkin gamma minification process. For each sequence we estimate the sample first order autocorrelation. In Table (3.2), we illustrate the averages of these sample first order autocorrelations with standard deviations in brackets. From Table (3.2), it is clear that the first order autocorrelation increases as a increases and decreases as α increases. 3.7 Data Analysis In this section we analyze a real data set and compare Marshall-Olkin gamma distribution with gamma distriution. The data is on the effect of an analgesic (Gross and Clark (1975)) 1.1,1.4,1.3,1.7,1.9, 1.8,1.6,2.2,1.7,2.7,4.1,1.8,1.5,1.2,1.4,3,1.7,2.3,1.6,2. The Q-Q and P- P plots for the two distributions are given in figure 3.3 and 3.4. The estimated values are given in table 3.3. We can conclude that Marshall-Olkin gamma distribution is a better fit. 59

13 Table 3.1: The first four moments of the Marshall-Olkin gamma distribution for some values of a and α. a α Mean SD Skewness Kurtosis a α Mean SD Skewness Kurtosis

14 Table 3.2: The sample first order autocorrelations of the Marshall-Olkin gamma minification process for different values of a and α. a\α (0.0625) (0.0524) (0.0404) (0.0452) (0.0385) (0.0356) (0.0333) (0.0363) (0.0313) (0.0700) (0.0544) (0.0471) (0.0355) (0.0402) (0.0334) (0.0368) (0.0336) (0.0337) (0.0691) (0.0456) (0.0433) (0.0380) (0.0342) (0.0336) (0.0354) (0.0315) (0.0288) (0.0740) (0.0479) (0.0467) (0.0409) (0.0368) (0.0362) (0.0325) (0.0347) (0.0306) (0.0692) (0.0443) (0.0385) (0.0407) (0.0399) (0.0345) (0.0310) (0.0356) (0.0309) (0.0665) (0.0579) (0.0487) (0.0430) (0.0369) (0.0337) (0.0369) (0.0321) (0.0344) (0.0767) (0.0510) (0.0446) (0.0396) (0.0419) (0.0301) (0.0303) (0.0353) (0.0295) (0.0612) (0.0492) (0.0375) (0.0373) (0.0348) (0.0380) (0.0341) (0.0298) (0.0323) (0.0573) (0.0431) (0.0378) (0.0379) (0.0433) (0.0289) (0.0351) (0.0302) (0.0289) (0.0618) (0.0466) (0.0402) (0.0359) (0.0327) (0.0373) (0.0337) (0.0322) (0.0311) (0.0560) (0.0421) (0.0336) (0.0401) (0.0344) (0.0366) (0.0332) (0.0340) (0.0318) (0.0492) (0.0469) (0.0386) (0.0359) (0.0344) (0.0349) (0.0326) (0.0322) (0.0328) (0.0496) (0.0381) (0.0436) (0.0374) (0.0375) (0.0331) (0.0370) (0.0321) (0.0335) (0.0462) (0.0425) (0.0341) (0.0330) (0.0338) (0.0327) (0.0316) (0.0323) (0.0366) (0.0441) (0.0355) (0.0411) (0.0386) (0.0374) (0.0286) (0.0343) (0.0294) (0.0350) (0.0484) (0.0357) (0.0341) (0.0384) (0.0373) (0.0363) (0.0335) (0.0328) (0.0366) 61

15 Figure 3.3: Q-Q and P-P plot of Gamma distribution. Figure 3.4: Q-Q and P-P plot of Marshall-Olkin gamma distribution. 62

16 Table 3.3: Estimated values, log-likelihood,kolmogorov-smirnov statistics and p value for the data set. Distribution Estimates log L K-S p-value Gamma â = Marshall-Olkin gamma â = ˆα = Conclusions In this chapter we have introduced the Marshall- Olkin gamma distribution and studied its properties. Reliability properties are also discussed. Moments are derived and estimation problem is addressed. A minification process is constructed and its covariance structure is derived. The new distribution is fitted to a survival data and it is established that the Marshall- Olkin gamma distribution a better fit to the data. References Aksoy, H. (2000). Use of gamma distribution in hydrological analysis. Turk J. Engin. Environ. Sci., 24, Alice, T., Jose, K.K. (2003). Marshall-Olkin Pareto Processes. Far East Journal of Theoretical Statistics 9, Alice, T., Jose, K.K. (2004). Bivariate semi-pareto minification processes. Metrika 59, Alice, T., Jose, K.K. (2005a). Marshall-Olkin Semi-Weibull Minification Processes. Recent Advances in Statistical Theory and Applications I, Alice, T. and Jose, K.K. (2005b). Marshall-Olkin logistic processes. STARS Int. Journal 6, Gross,A.J.,Clark,V. (1975). Survival distributions: Reliability applications in the Biomedical Sciences. Wiley, New York. 63

17 Gupta, S.S. (1960). Order statistics from the gamma distribution. Technometrics 2, Gupta, R.D., Kundu, D. (1999). Generalized exponential distributions. Austral. New Zealand J. Statist. 41, Jayakumar, K., Thomas, M. (2007). On a generalization to Marshall-Olkin scheme and its application to Burr type XII distribution. Statistical Papers, (accepted) Jose, K.K., Alice, T. (2001). Marshall-Olkin generalized Weibull distributions and applications. STARS Int. Journal 2, 1 8. Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84, Mosino, A.P., Garcia, E. (1981). The variability of rainfall in Mexico and its determination by means of the gamma distribution. Geografiska Annaler. Series A, Physical Geography 63, Nadarajah, S., Gupta, A.K. (2007). A generalized gamma distribution with application to drought data. Math. Comp. Sim. 74, 1 7. Nadarajah, S., Kotz, S. (2006). The exponentiated type distributions. Acta Appl. Math. 92, Ristic, M.M., Jose, K.K. and Ancy, J. (2007). A Marshall-Olkin gamma distribution and minification process, STARS: International Journal (Sciences), 1(2). 64