Chapter 5. Logistic Processes

Transcription

1 Chapter 5 Logistic Processes * 5.1 Introduction The logistic distribution is the most commonly used probability model for demographic contexts for modeling growth of populations. Oliver (1964) use it to nlodel agricultural production data. Fisk (1961) used it in studying lnconle distribution. Many authors used it to model growth of human population. Plackett (1959) used it to model survival data. Berkson (1953) used it in studying the growth of bioassays. Grizzle (1961) used it in the area of public health. Arnold (1989). (1993), Arnold and Robertson (1989), Sim (1993) use the logistic distribution in the construction of autoregressive time series models having different structures. In the present chapter we generalize the standard logistic distribution to obtain a wider class of distributions. In sectlon 5.2, the new generalized logistic distribution is introduced and its distributional properties are investigated. The graphs corresponding to probability density function distribution function and hazard rate function are given and it is shown to be of use * The results included in (hi> chapter were partly presented in the form of a paper entitled "Marshall Olkin generalized Logistic Distribution and its applications in time series modeling" jointly by Alice Thomas and K.K.Jose. at National Seminar on Recent Trends in Statistics and Workshop on Statistical Computing held at Nehm Arts and Science College, Kanjangad during January ,

2 to model different typeis of reliability data. It is shown that this distribution is geometrically extreme stable. We apply these to a more general family called semi logistic family as in the lines of Pillai (1991). In section 5.3, AK (1) and AR (k) model with Marshall Olkin generalized logistic (MOCiL) marginal is constructed and studied. In section 5.4, Marshall Olkin semi logistic (MOSL) distribution is introduced and its properties are studied. In section 5.5, first and kth order autoregressive minification structure with MOSL marginal is constructed and its properties are studied. In section 5.6, Marshall Olkin bivariate logistic (MOBL) family is considered and some characterizations are obtained. 5.2 Marshall Olkin generalized logistic family We consider the tvvo parameter logistic distribution with survival function Then substituting in (2.2.1 :I we get a new family of distributions, which we shall refer to as the MarshallOlkin Generalized Logistic (MOGL) family, whose survival function is given by a Table 5.1 gives the cumulative distribution function of MOGL distribution. The probability density function corresponding to G is given by

3 The hazard rate is given by The moment generating function of X is The expected value of X is If XI, X,,...,X, are independent and identically distributed random variables having the survival function of MOGL G(x), then [ 2r U, = min( XI. X2,...,X,,) has the survival function 1 +e. 'Table 5.2 shows the expected values of X for various combinations of a and 8 when h = 1. Figure 5.1 gives the graph of probability density function g (x). Figure 5.2 gives the graph of hazard rate function r (x). Theorem MarshallOlkin generalized logistci distribution is geometric extreme stable Proof Proceeding as in Theorern 3.4.1, we have G (x; p) = p F(x) I (1 p) F(X) Suppose F (x) =: w. /{a+o exp (h x))

4 which is the survival function of MOGL. Then G (x) =pa l{pa+q exp (h x)} Hence U is geometric minimum stable. Hence V is geometric maximum stable. Hence the family of MOGL distributions with the survival function of the form F (x) = ~1 /{IX+O exp (A x)), is geometricextreme stable. Theorem Let (Xi, i 2 1) be a sequence of independent and identically distributed random variables with common survival function F(x) and N be a geometric random variable with parameter p and P(N = n) = p q "~'; n = 1,2,..., 0 < p < 1, q =Ip, which is independent of {Xi)for all i 2 1. Let UN = min X,. Then {UN] is distributed as MOiSi'N GL if and only if {X,) is distributed as logistic. Proof Proceeding as in Theorem we have Suppose

5 H (x) = w /{a+e exp (A x)), which is the survi~al function of MOGL. This proves the sufficiency part of the theorem. Conversely suppose H (x) = a/{a+h exp (h x)). Then we get which is the survival function of logistic. 5.3 An AR (1) model with MOGL marginal distribution Now we consider an AR (1) model having stationary marginal distribution as the MOGL distribution. Theorem Consider an AR (1) structure given by (2.3.1), where {E,} is a sequence of independent and identically distributed random variables independent of {X,,, X,.,,... ), then {X,) is a stationary Markovian AR(1) process with MOGL marginals if and only if {E.) is distributed as logistic distribution. Proof Proceeding as in the case of theorem if we take then it easily follows that F. (x) =a /(a+@ exp (h x)), which is the survival function of MOGL.

6 Conversely. if we take, it is easy to show that F,,(x) is distributed as logistic and the process is stationary. In order to establish stationarity we proceed as follows. d Assume X,,., = MOGL and E, = logistic. Then Fx,, (x) =a /{u+h exp (h x)). This establishes that ( X is distributed as MOGL. Even if X,, is arbitrary, it is easy to establish that X } is stationary and is asymptotically marginally distributed as MOGL. Crollary Consider an autoregressive minification process X, of order k with structure (2.5.1). Then {Xn} has stationary marginal distribution as MOGL if and only if {E,] is distributed as logistic Sample path behaviour The sample path of' the process for different values of p and h are given in Figure 5.4. The simulated sample path using 100 observations generated from the MOCiLAR (1) process with p=0.3, 0.5 and 0.9 and h = 0.5, 1 and 2 are given respectively in the figure 5.4.The sample path behaviour of the MOGLAR (1) process seems to be distinctive and is adjustable through the parameters p and a. This makes the model very rich. Mainly there seems to be three cases: (4)runs of increasing values up runs (5)runs of decreasing values down runs (6) both (peaks). These observations can be verified by referring to the Table 5.3 showing P (X, < X,.,).These probabilities are obtained

7 through a Monte Carlo simulation procedure. Sequences of 100, 300, 500, observatxons from MOGLAR (1) process are generated repeatedly for ten tlrnes and for each sequence the probability is estimated. A table of such probabilities is provided with the average from ten trials along wilh an estimate of standard error (see Table 5.3). 5.4 Semi logistic family of distributions Pillai ( ) intrclduced the semipareto distribution. In a similar manner we introduce the semilogistic distribution as one having the survival function F(x) = P (X > X) = 1 I + exp(v(x)) where ~ (x) satisfies the functional equation ( ). Hence the semilogistic family is a wide class of distributions including the logistic distributions. When

8 Theorem Let {X,. i 2 I ) be a sequence of independent and identically distributed random variables with common survival function F(x) and N be a geometric random variable with parameter p and P(N = n) = p q "I; n =1,2,..., 0 < p <: 1, q =Ip, which is independent of {X,}for all i L 1. Let U, = min X,. Then {UN) is distributed as MOSL if and I<,<\ only if {X,) is distributed as semi logistic. Proof Proceeding as in Theorem we have 1 Suppose F(x) = I+ exp(w (x)) ' I c'v' 1' I' Then H (x) = which is the survival function of MOSL. This proves the sufficiency part of the theorem. Conversely suppose Then we get which is the survival function of semi logistic.

9 5.5 An AR (1) model with MOSL marginal distributions Theorem Consider an AR (I.) structure given by (2.3.1). where {E,] is a sequence of independent and identically distributed random variables independent of X,, then { X,) is a stationary Markovian AR (1) process with MOSL marginals if and only if (E,} is distributed as semilogistic distribution. Proof Proceeding as in the case of Theorem if we take then it easily follows that F X(~) = 1 1 I + ~ev'*' P which is the survival function of MOSL. Conversely if we take it is easy to show that F,,,(x) is distributed as semi logistic and the process is stationary. In order to establish stationarity we proceed as follows.,i,i Assume X,., =MOSI, and E.= semilogistic. I Then F x,, (x) =. I I+ (Y'" P

10 This establishes that ) is distributed as MOSL. Even if Xo is arbitrary, it is easy to establish that {XI is stationary and is asymptotically marginally distributed as MOSL. Theorem Consider an autoregressive minification process X, of order k with structure (2.5.1). Then X ) has stationary marginal distribution as MOSL if and only if {E.} is distributed as semilogistic. 5.6 Marshall Olkin bivariate logistic family of distributions function We consider the bivariate semilogistic distribution with survival 1 F (x, y) =P!X > x,y > y) = I1 + exp(v(x. Y)) where ylx. y) satisfies the functional equation (3.6.3) From (3.6.1) we can see that the new survival function is G (x, Y) = ll(l+(llp) exp (w(x,y))), (5.6.2) which we shall refer to as MarshallOlkin bivariate semilogistic distribution denoted as MOBSL. Consider the bivariate logistic distribution with survival function I F (x, y) = I + 'ti, \fi2 ;(x, Y)E R2, PI. 02 > 0. From (3.6. l), the new survival function is which we shall refer to as MarshallOlkin bivariate logistic distribution denoted as MOBL.

11 If (X,, Y,), i = 1,2,...,n are independent and identically distributed random variables having the survival function of MOBL G(X,~), then (minx,, miny,) has the survival function 1 Theorem Let N be a geometric random variable with parameter p and P(N = n) = p q"~', n =:I,2,...,0 < p < 1, q =1p. Consider a sequence {(Xi, Yi). i 2 1 ] be independently and identically distributed random vectors with common!survival function F(X,~). N and (Xi,Yi) are independent for all i 2 1. Let UN = min X, and VN = mink;. Then the IiiiN lc_i<n random vectors (UN, VN) and (X,, Y,) are distributed as MOBL if and only if (X,, Y,) have BL distribution. Proof Consider S (x, y) = P [IJN > x, VN > y] Let F (x, y) logistic:. ',which is the survival function of bivariate I +erd'."'? Substituting this in the above equation we have which is the survival function of MOBL. Conversely suppose that

12 s (x, y) x 1 I t (II~)G,~~~+~~. Then p F(x, y) I [ I ( 1p) F(x, y)] = On simplifying we get 1 I+ (l/a)er8'+"' ' Hence the proof is complete. Let (Nk, k 2 I } be a sequence of geometric random variables with parameters p,, 0 sr pk < 1. Define Fk(x,y) = PIUNk.I >x, VNk.[>y]r k=2,3... = pk~, FkI~x,~)~) I { I(~P~~) Fk,(x,y)~ (5.6.4) Here we refer ~k as ithe survival function of the geometric (P~.~) minimum of i.i.d. random vectors with Fk., as the common survival function. Theorem Let {(X,, Y,), i 21) be a sequence of i.i.d. bivariate random vectors with common survival function G(x,y). Define FI= 6 and Fk as the survival function of the geometric (~k.~) minimum of i.i.d.random vectors with common survival function Fk.lr k = Then R(x, y:) = G(X, Y) (5.6.5) if and only if (Xi,Y, ) has MOBL, distribution. Proof By definition, the survival function Fk satisfies the equation (5.6.4). 1 1 Lt(1,'p)e x"+,r ~+O(X,Y)' We have G(x, y)

13 J) is a monotonically increasing function in both x and y and lim lim $(.x,,v) =O andlimlim$(x, y) = m. v,*. I, So we can write F~x. Y) = +.k = 1,2,... ~ ~.. Substituting this in (5.6.4), we get Recursively using this relation we have This implies that Hence Fk (x. y) = G(x, y) This proves the sufficiency part. Conversely assume that equation (5.6.5) is true. By the hypothesis of the theorem equation (5.6.6) follows. Thus equation (5.6.5) and equation (5.6.6) together lead to the (.K>?.I This implies v) :=. PlP?...P{ I Hence the proof is complete. Theorem ( X Consider a bivariate autoregressive minification process Y ) having structure (3.7.1). Then (XY)) has stationary

14 marginal distribution as MOBL distribution if and only if {(U,,V,)] is jointly distributed as bivariate logistic distribution. Proof is similar to Theorem Theorem Consider a kth order bivariate autoregressive model having the structure (3.8.1). Then {(X,, Y,)] has stationary marginal distribution as MOBL if and only if {(U,, V,)] is jointly distributed as bivariate logistic distribution. Remark The abov~e theorems can be extended to bivariate semi logistic distributions in an analogous manner.

15 Table 4.1 c.d.f. table for the MOGP distribution for fi = 1

16 Table 5.2. Expected values of X for different values of a and 8 when h = 1

17 Fig 5.1 p.d.f. of MOGL for various values of a and h = 1,8 = 1

18 Fig 5.2 Hazard rate function of MOGL for various values of a and l. = 1,8 = 1

19 Table 5.3 P ( X, < X..,) for the MOGLAR(1) process where h = 1,8 = 1. (Standard error in brackets) Sam le size I

20 I Fig 5.3 Sample paths of MOGLAR (1) process for different values of p and P

21 Fig 5.4 Survival Function of MOBL