Marshall-Olkin Univariate and Bivariate Logistic Processes

Transcription

1 CHAPTER 5 Marshall-Olkin Univariate and Bivariate Logistic Processes 5. Introduction The logistic distribution is the most commonly used probability model for modelling data on population growth, bioassay, medical diagnosis and public health. Oliver 964) used it to model agricultural production data. Fisk 96) used it in studying income distribution. Many authors used it to model growth of human population. Plackett 959) used it to model survival data. Berkson 953) used it in studying the growth of bioassays. Grizzle 96) used it in the area of public health. Arnold 989), 993), Arnold and Robertson 989), Sim 993), etc. used the logistic distribution in the construction of autoregressive time series models having different structures. Some of the results in this chapter are published in Alice et al 2007). 84

2 In section 2 of the present chapter the standard logistic distribution is generalized to obtain a wider class of distributions. Its distributional properties are investigated. The graphs corresponding to probability density function and hazard rate function are given and it is shown to be of use to model different types of reliability data. Autocorrelation function is also derived. Following Pillai 99), we developed a more general family called semi-logistic family. Some properties of this logistic process are obtained. The unknown parameters of the process are estimated and some numerical results of the estimators are given. AR) model with Marshall-Olkin logistic MO-L) marginal is constructed and studied. Marshall-Olkin semi-logistic MO-SL) distribution is introduced and its properties are studied. Bivariate extensions are also discussed. 5.2 Marshall-Olkin Logistic Distribution We consider the two parameter logistic distribution Lx; θ, λ) with survival function F x; θ, λ) =, < x <, θ, λ > ) + θeλx Then,applying 2..) the Marshall-Olkin logistic distribution has the survival function Gx; α, λ, θ) =, < x <, α, θ, λ > ) + θ eλx α Whenever F has a density, the survival function G.) given by 5.2.2) have easily computable densities. The probability density function corresponding to Ḡ.) is given by gx; α, λ, θ) = αλθeλx α + θe λx 2, < x <, α > 0, θ > 0, λ > 0. ) The probability density function has a unique mode at x = λ log α θ with g ) = g ) = 0. 85

3 The hazard rate is given by rx; α, λ, θ) = λθeλx, < x <, α, θ, λ > 0. α + θeλx The hazard rate function is an increasing function with r ) = 0 and r ) = λ. If X, X 2,..., X n are independent and identically distributed random variables having the survival function Gx) in 5.2.2), then U n function + θ α eλx) n. Now we have the following theorem. = minx, X 2,..., X n ) has the survival Theorem Let {X i, 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) = pq n, n =, 2,..., 0 < p <, q = p, which is independent of {X i } for all i. Let U N = min i N X i. Then {U N } is distributed as MO-L if and only if {X i } is distributed as logistic Lx; θ, λ). Proof. Consider Hx) = P U N > x). Then we have, Hx) = ) n F x) pq n = n= pf x) p)f x). Suppose that F x) = +θe λx. Then it follows that Hx) = sufficiency part of the theorem. Conversely suppose Hx) = p p+θe λx, which is the survival function of MO-L. This proves the p p+θe λx. Then we get F x) = +θe λx, which is the survival function of logistic random variable. 86

4 5.3 An AR) Model with MO-L Marginal Distribution Now we consider an AR) model having stationary marginal distribution as the MO-L distribution. Theorem Consider an AR) structure given by ɛ n, X n = minx n, ɛ n ), with probability p with probability p 5.3.) where {ɛ n } is a sequence of independent and identically distributed random variables independent of {X n, X n 2,... }. Then {X n } is a stationary Markovian AR) process with MO-L marginals if and only if {ɛ n } is distributed as logistic distribution Lx; θ, λ). Proof. From 5.3.) it follows that F Xn x) = pf ɛn x) + p)f Xn x)f ɛn x) ) Under stationary equilibrium, this gives F X x) = pf ɛ x) p)f ɛ x) ) 5.3.3) If we take then it follows that F ɛ x) = / + θe λx), F X x) = p/ p + θe λx), which is the survival function of MO-L. Conversely, if we take F Xn x) = p/ p + θe λx), 87

5 it is easy to show that F ɛ n x) is distributed as logistic and the process is stationary. In order to establish stationarity we proceed as follows. Assume X n d =MO-L and ɛ n d =logistic. Then F Xn x) = p/ p + θe λx). This establishes that {X n } is distributed as MO-L. Even if X 0 is arbitrary, it is easy to establish that {X n } is stationary and is asymptotically marginally distributed as MO-L. Now, we will consider some properties of AR) process with MO-L marginal distribution. First we consider the joint survival function S h of the random variables X n+h and X n, h. After some calculations, we obtain that S h x, y) = p2 + θe λx) h + p p) h θe λ minx,y) p + θe λx ) p + θe λy ) + θe λx ) h, x, y R. This is not an absolutely continuous distribution, since P X n+h = X n ) = p p)h + h 2F 2, h + ; h + 2; p) > 0. Consider now the probability of the event {X n+ > X n }.Calculations show that P X n+ > X n ) = p p + p log p) p) 2 0, 0.5). This probability is an increasing function of p. We consider the lag- autocovariance and autocorrelation function of {X n }. The conditional expectation of X n+ given X n = y is EX n+ /X n = y) = p)y p λ log + θeλy ) p log θ λ. 88

6 Then the moment of the product of the random variables X n+ and X n is EX n+ X n ) = p)exn) 2 p λ p)π2 = 6λ 2 where Li n z) = i= autocovariance function as E X n log + θe λxn)) p log θ λ EX n) + p Li λ 2 2 p) + 2 p) log2 p + 2λ 2 log θlog θ 2 log p) λ 2 z i i n is the polylogarithmic function. Using this we obtain the lag- CovX n+, X n ) = p)π2 6λ 2 + p Li λ 2 2 p) p log2 p. 2λ 2 Finally, the lag- autocorrelation function is obtained as CorrX n+, X n ) = p)π2 + 6 p)li 2 p) 3p log 2 p 2π 2. The lag- autocorrelation function lies between 0 and and is a decreasing function of p. 5.4 Estimation In the rest of this section, we consider the estimation of the unknown parameters p, λ and θ. As an estimator of the parameter p, we can use the solution of the equation p p + p log p) = n IX p) 2 i+ > X i ). n i= Since V arx n ) = π2 3λ 2, we can estimate the parameter λ by where S n = n n X i X n ) 2. i= λ = π 3 Sn Finally, from the expectation of the random variable X n, the estimator of the parameter θ 89

7 can be obtained as θ = p e λx n. In Table 5.) we give some numerical results of the estimation. We simulate 0000 realizations of the AR) process with MO-L distribution for different values of the parameters p, λ and θ. The simulations are repeated 00 times. For each sequence we estimate the unknown parameters using the subsamples of the first 00, 500, 000, 5000 and 0000 realizations. We computed the sample means and the standard errors of the obtained estimators. Table 5.: The estimators of the unknown parameters for different values of p, λ and θ. p = 0., λ = 0.5 and θ = 0.75 p = 0.8, λ =.5 and θ = 0.5 Sample size ˆp ˆλ ˆθ ˆp ˆλ ˆθ ) ) ) ) 0.706) ) ) ) ) ) ) ) ) ) 0.286) ) ) ) ) ) ) 0.042) ) 0.060) ) ) ) ) ) 0.075) 5.5 Generalization to the kth Order Autoregressive Model In this section we develop a kth order autoregressive model. Consider an autoregressive model of order k with structural form ɛ n, w.p. p 0 X n = minx n, ɛ n ), w.p. p minx n 2, ɛ n ), w.p. p 2.. minx n k, ɛ n ),. w.p. p k 90

8 where 0 < p 0, p,..., p k < and p + + p k = p 0. Then {X n } has a stationary marginal distribution as MO-L if and only if {ɛ n } is distributed as logistic Lx; θ, λ). 5.6 Semi-Logistic Distributions Similar to the semi-pareto distribution introduced by Pillai 99) we introduce the semilogistic distribution as one having the survival function F x) = + e ψ x )sgnx), x R where sgnx) is the sign function of x and ψx) satisfies the functional equation ρψx) = ψρ /β x), x > 0, β > 0, 0 < ρ <. Hence the semi-logistic family is a wide class of distributions including the logistic distributions. The survival function of the Marshall-Olkin semi-logistic distribution is Gx; α) = +, x R, α > 0. eψ x )sgnx) α The probability density function is obtained as The hazard rate function is gx; α) = αψ x ) e ψ x )sgnx) α + e ψ x )sgnx) 2, x R, α > 0. ) rx; α) = ψ x ) e ψ x )sgnx) α + e ψ x )sgnx), x R, α > 0. Remark family. Theorem 5.2. and 5.2.2) can be extended to the semi-logistic As a special case of the Marshall-Olkin semi-logistic distribution, we consider a Marshall- 9

9 Olkin logistic distribution when ψx) = x β. In this case, the survival function G.) is given by Gx; α, β) = The probability density function is α, x R, β > 0, α > 0. α + e x β sgnx) gx; α, β) = αβ x β e x β sgnx) α + e x β sgnx) ) 2, x R, β > 0, α > 0. Now we investigate the possible shapes of the Marshall-Olkin logistic distribution. First, we consider the probability density function gx; α, β) for x > 0. The following cases are possible: ) If β =, then gx) is the probability density function of the Marshall-Olkin Generalized logistic distribution with λ = θ =. 2) Let β < and αβ >. Let x 0 be the solutions of the equation + βx β ) e xβ = αβ. Then we have two possible cases: 2a) If αβx β 0 + β e xβ 0 > α, then gx; α, β) decreases on 0, m ) m 2, ) and increases on m, m 2 ), with g0) = and g ) = 0, where m and m 2 are the solutions of the equation αβ ) + β )e xβ + αβx β βx β e xβ = ) 2b) If αβx β 0 + β e xβ 0 < α, then gx) decreases with g0) = and g ) = 0. 3) Let β < and αβ <, then gx) decreases with g0) = and g ) = 0. 4) Let β >. Then gx) has a unique mode at x 0, where x 0 is the solution of 5.6.), with g0) = 0 and g ) = 0. Second, consider the probability density function gx) for x < 0. It is easy to show that in this case, gx) has the same shape as gx; /α, β) when x > 0. Figure 5. illustrates the 92

10 shape of the probability density function gx) of the Marshall-Olkin logistic distribution for different values of the parameters α and β. We investigate the hazard rate function of the Figure 5.: The probability density function of the Marshall-Olkin logistic distribution for different values of parameters α and β. Marshall-Olkin logistic distribution. The hazard rate is given by rx; α, β) = β x β e x β sgnx) α + e x β sgnx), x R, α, β > 0. As in the case of the probability density function, we first consider the case when x > 0. We have the following cases: 93

11 ) If β =, then rx) is the hazard rate function of the Marshall-Olkin generalized logistic distribution with λ = θ =. 2) Let β <. If α > β β e/β, then rx) decreases on 0, m ) m 2, ) and increases on m, m 2 ), with g0) = and g ) = 0, where m and m 2 are the solutions of the equation αβ ) + β )e xβ + αβx β = ) Otherwise, rx) is a decreasing function with r0) = and r ) = 0. 3) If β >, then rx) is an increasing function with r0) = 0 and r ) =. Finally, consider the hazard rate function when x < 0. Then we have two cases: ) If β <, then rx) is an increasing function with r ) = 0 and r0) =. 2) If β >, then rx) has a maximum at x 0, where x 0 is the solution of the equation 5.6.2). Also, r ) = r0) = 0. Figure 5.2) illustrates the hazard rate function hx) of the Marshall-Olkin logistic distribution for different values of the parameters α and β. 5.7 Estimation Now we consider the estimation of the parameters using the method of the maximum likelihood. The log-likelihood function is given by log Lα, β) =n log α + n log β + β ) n x i β sgnx i ) 2 i= n log i= n log x i + i= α + e x i β sgnx i ) ). 94

12 Figure 5.2: The hazard rate function of the Marshall-Olkin logistic distribution for different values of parameters α and β. 95

13 The partial derivatives are log Lα, β) α log Lα, β) β = n n α 2 α + e x i β sgnx i ) i= = n n β + log x i + 2 i= n x i β sgnx i ) log x i i= n x i β sgnx i ) log x i e x i β sgnx i ) i= α + e x i β sgnx i ) The estimators of the parameters α and β can be obtained using the nlm function in statistical package R. 5.8 Data Analysis We analyze a data set PORTw from the package extremes. The data are daily maximum and minimum Winter temperature for the North Atlantic Oscillation index from 927 through 995. Data is for Winter for Port Jervis, New York. First we consider the daily minimum Winter temperature. Using the method of the maximum likelihood estimation and the function nlm, we obtain the estimators ˆα = and ˆβ = with the negative logarithm of the maximum likelihood log L = Now we consider the daily maximum Winter temperature. We obtain the estimators ˆα = and ˆβ = with the negative logarithm of the maximum likelihood log L = Figure 5.3 shows the QQ plot for the daily minimum and maximum Winter temperature. 5.9 Marshall-Olkin Bivariate Logistic Family of Distributions Let X, Y ) be a random vector with joint survival function F x, y). Then Gx, y) = αf x, y) α)f x, y), x, y) R2 5.9.) 96

14 Figure 5.3: QQ Plots for daily minimum and maximum Winter temperature. is a proper bivariate survival function for 0 < α <. The family of distributions of the form 5.9.) is called Marshall-Olkin bivariate family of distributions. We consider the bivariate semi-logistic distribution with survival function F x, y) = where ψ x) and ψ 2 y) satisfy the functional equations + e ψ x )sgnx) + e ψ 2 y )sgny), x, y) R2, 5.9.2) ρψ x) = ψ ρ /β x) ρψ 2 y) = ψ 2 ρ /β 2 y). From 5.9.) we can see that the new survival function is Gx, y) = + α eψ x )sgnx) + e ψ 2 y )sgny) ), x, y) R ) which we shall refer to as Marshall-Olkin bivariate semi-logistic distribution denoted as 97

15 MO-BSL. Consider the bivariate logistic distribution with survival function F x, y) = + e x β sgnx) + e y β 2 sgny), x, y) R2, β, β 2 > ) From 5.9.), the new survival function is Gx, y) = + α e x β sgnx) + e ), x, y) y β 2 sgny) R2, 0 < α < β, β 2 > 0, which we shall refer to as Marshall-Olkin bivariate logistic distribution denoted as MO-BL. Figures 5.4 and 5.5 are the plots of MO-BL distributions. If X i, Y i ), i =, 2,..., n are independent and identically distributed random variables having the survival function of MO-BL Gx, y), then min X i, min Y i) has the survival function i n i n n )). + α e x β sgnx) +e y β 2 sgny) Theorem Let N be a geometric random variable with parameter p. Let {X i, Y i ), i } be a sequence of independently and identically distributed random vectors with common survival function F x, y). Let N and X i, Y i ) are independent for all i. Let U N = min X i and V N = min Y i. Then the random vector U N, V N ) is distributed i N i N as MO-BL if and only if X i, Y i ) have BL distribution. Proof. Consider Sx, y) = P U N > x, V N > y) = P U n > x, V n > y/n = n)p N = n) = n= F x, y)) n pq n = n= 98 pf x, y) p)f x, y).

16 Let F x, y) =. Substituting this in the above equation we have + e x β sgnx) +e y β 2 sgny) Sx, y) = + p e x β sgnx) + e ), y β 2 sgny) which is the survival function of MO-BL. Conversely suppose that Sx, y) = + p e x β sgnx) +e y β 2 sgny) ). Then pf x, y) p)f x, y) = + p e x β sgnx) + e ). y β 2 sgny) On simplifying we get F x, y) =. Hence the proof is complete. + e x β sgnx) +e y β 2 sgny) Let {N k, k } be a sequence of geometric random variables with parameters p k, 0 p k <. Define for k = 2, 3,... F k x, y) = P U Nk > x, V Nk > y) = p k F k x, y) p k )F k x, y) ) Here we refer F k as the survival function of the geometricp k ) minimum of i.i.d. random vectors with F k as the common survival function. Theorem Let {X i, Y i ), i } be a sequence of i.i.d. bivariate random vectors with MO-BLβ, β 2, p) distribution. Define F = G and F k as the survival function of the geometricp k ) minimum of i.i.d.random vectors with common survival function F k, k = 2, Then F k x, y) has a MO-BL distribution with parametersβ, β 2, p, p 2,... p k ). Proof. Theorem can be easily proved by induction. Theorem Consider a bivariate autoregressive minification process {X n, Y n )} 99

17 having the structure X n = Y n = U n, minx n, U n ), V n, miny n, V n ), w.p. p w.p. p w.p. p w.p. p 5.9.6) where {U n, V n )} are i.i.d. random vectors and are independent of {X i, Y i )} for all i < n. Then {X n, Y n )} has stationary marginal distribution as MO-BL distribution if and only if {U n, V n )} is jointly distributed as bivariate logistic distribution given by 5.9.4). Proof. From 5.9.6) we have that F Xn,Y n x, y) = p G Un,V n x, y) + p)f Xn,Y n x, y)g Un,V n x, y) 5.9.7) Under stationarity this simplifies to If we take we obtain that G U,V x, y) = F X,Y x, y) = p G U,V x, y) p)g U,V x, y). + e x β sgnx) + e y β 2 sgny), x, y) R2, β, β 2 > 0 F X,Y x, y) = + p e x β sgnx) + e ), x, y) y β 2 sgny) R2, 0 < p <, β, β 2 > 0 is the survival function of MO-BL. Conversely if we take 00

18 F X,Y x, y) = + p e x β sgnx) + e ), y β 2 sgny) it is easy to show that G U,V x, y) is distributed as bivariate logistic distribution given by 5.9.4) and the process is stationary. In order to establish stationarity we proceed as follows. Assume X n, Y n ) d = MO-BL and U n, V n ) d = BL. Then from 5.9.7) F Xn,Y n x, y) = + p e x β sgnx) + e ). y β 2 sgny) This establishes that {X n, Y n )} is distributed as MO-BL. Theorem Even if X 0, Y 0 ) is arbitrary, it is easy to establish that {X n, Y n )} is stationary and is asymptotically marginally distributed as MO-BL. Proof. From 5.9.7) we have that F Xn,Y n x, y) = G Un,V n x, y) p + p)f Xn,Y n x, y) ) = G Un,V n x, y) p + p p)g Un,V n x, y) + p) 2 G Un,V n x, y)f Xn 2,Y n 2 x, y) ) = + p + e x β sgnx) + e y β 2 sgny) p + + e x β sgnx) + e y β 2 sgny) ) p) 2 + e x β sgnx) + e ) y β 2 sgny) = + p e x β sgnx) + e ), y β 2 sgny) which is MO-BL distribution. Theorem Consider a kth order bivariate autoregressive model {X n, Y n )} hav- 0

19 ing the structure X n = Y n = U n, w.p. p 0 minx n, U n ), w.p. p... minx n k, U n ), w.p. p k V n, w.p. p 0 miny n, V n ), w.p. p... miny n k, V n ), w.p. p k 5.9.8) where 0 < p i <, k p i = p 0 and {U n, V n )} are i.i.d. and are independent of i= X i, Y i ) for all i < n. Then {X n, Y n )} has stationary marginal distribution as MO-BL if and only if {U n, V n )} is jointly distributed as bivariate logistic distribution given by 5.9.4). Remark The above theorems can be extended to bivariate semi-logistic distributions in an analogous manner. 5.0 q-logistic and Marshall-Olkin q-logistic Distributions Mathai and Provost 2006) developed models that generalize the type- and type-2beta distributions, along with their logistic counter parts. A q-analog of the generalized type-2 beta model and q-extended generalized logistic model were also discussed by them. Here we consider a special case of the logistic density given by them having the survival function given by F x) = + q )e x ) β q 02

20 Figure 5.4: Graph of bivariate logistic distribution. β =.2, β 2 =

21 Figure 5.5: Graph of bivariate logistic distribution. β =.2, β 2 =.5 04

22 and the corresponding density given by fx) = β q + )e x + q )e x ) β q where < x <,q > and β q > 0. The moments can be obtained by the method given in Mathai and Provost 2006). Now we state some properties of q-logistic distribution. Theorem Let {X i, i =, 2,..., n} be i.i.d. random variables following the q-logistic distribution, then min i n Proof: The survival function is X i and max i n X i have extreme value distributions. Then P r[ min i n X i > x] = F x) = [ + q ) e x ] β n P r[x i > x] i= q. = [ + q )e x )] nβ q e nβ e x as q. Now,the distribution function is F x) = [ + q ) e x ] β q Then P r[ max i n X i x] = n i= [ + q )e x ] β q = { [ + q ) e x ] β q } n [ e β ex ] n as q. Applying 2..) we can construct Marshall-Olkin q- logistic distribution similar to Marshall- Olkin q-gamma distribution and it has the survival function given by Gx; α, β, q) = α + q )e x ) β q + q )e x ) β q α) We can also develop AR) models with q logistic distribution as the stationary 05

23 marginal distribution. 5. Conclusions Both univariate and bivariate semi-logistic distributions are introduced and the properties of these distributions are studied in detail. These distributions are extended to a more general class of Marshall- Olkin distributions. They can be used to model different types of reliability data. Minification processes are developed with univariate and bivariate M- O logistic distributions. Estimation problems are also discussed. Some data sets are analyzed and some simulation studies also done. q- logistic distribution is also introduced. References Alice, T. and Jose, K.K. 2004). Bivariate semi-pareto minification processes. Metrika, 59, 3), Alice, T., Jose, K.K., Ristić M. and Ancy, J. 2008). Marshall-Olkin univariate and bivariate logistic processes Proceedings of the 8th International Conference of the Society for Special Functions and their Applications, Arnold, B.C. 989). A logistic process constructed using geometric minimization. Statist. Prob. Letters, 7, Arnold, B.C. 993). Logistic processes involving Markovian minimization. Statist. Theory Methods, 22,6), Commun. Arnold, B.C. and Robertson, C.A. 989) Autoregressive logistic processes. Appl. Prob., 26, Balakrishnan, N. 992). Handbook of the Logistic Distribution. New York, Dekker. Berkson, J. 953). A statistically precise and relatively simple method of estimating the bio-assay and quantal response, based on the logistic function. J. Amer. Stat. Assoc. 48,

24 Fisk, P.R. 96). The graduation of income distributions. Econometrica,29, Grizzle, J.E. 96). A new method for testing hypotheses and estimating parameters for the logistic model. Biometrics, 7, Marshall, A.W. and Olkin, I. 997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrica, 84,3), Oliver, F.R. 964). Methods of estimating the logistic growth function. Applied Statistics, 3, Plackett, R.L. 959). The analysis of life-test data. Technometrics,, 9 9. Pillai, R.N. 99). Semi-Pareto processes. J. Appl. Prob., 28, Sim, C.H. 993). First order autoregressive logistic process. J. Appl. Prob., 30,