A BIVARIATE MARSHALL AND OLKIN EXPONENTIAL MINIFICATION PROCESS

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1 Faculty of Sciences Mathematics, University of Niš, Serbia Available at: Filomat 22: (2008), A BIVARIATE MARSHALL AND OLKIN EXPONENTIAL MINIFICATION PROCESS Miroslav M Ristić, Biljana Č Popović, Aleksar Nastić Miodrag -Dord ević Abstract In this paper we present a stationary bivariate minification process with Marshall Olkin exponential distribution The process is given by X n = K min(x n, Y n, η n), Y n = K min(x n, Y n, η n2 ), where (η n, η n2 ), n } is a sequence of independent identically distributed rom vectors, the rom vectors (X m, Y m) (η n, η n2) are independent for m < n λ > 0, λ 2 > 0, λ 2 > 0, K > (λ + λ 2 + λ 2 )/λ 2 The innovation distribution, the joint distribution of rom vectors (X n, Y n) (X n j, Y n j ), j > 0, the autocovariance the autocorrelation matrix are obtained The unknown parameters are estimated their asymptotic properties are obtained INTRODUCTION A minification process of the first-order is given by X n = K min(x n, ε n ), n, where K > ε n, n } is an innovation process of independent identically distributed (iid) rom variables Several authors have introduced minification processes with given marginals Tavares 3 introduced the minification process with exponential marginal distribution Sim 0 introduced the minification process with Weibull marginal distribution Yeh, Arnold Robertson 4 introduced a Pareto minification process Arnold Robertson introduced a logistic minification process Pillai 7 Pillai, Jose Jayakumar 8 introduced semi-pareto minification processes Balakrishna 2 considered some properties of 2000 Mathematics Subject Classifications 62M0, 60G0 Key words Phrases Minification process, Bivariate Marshall Olkin Exponential Distribution, Estimation, Ergodic, Uniformly mixing Received: February 8, 2007 Communicated by Dragan S Djordjević

2 70 M M Ristić, B Č Popović, A Nastić, M -Dord ević the semi-pareto minification process of Pillai 7 estimated the unknown parameters of the model Lewis McKenzie 5 introduced the minification process with marginal distribution function F X0 (x) Some bivariate multivariate minification processes are introduced by Balakrishna Jayakumar 3, Thomas Jose, 2 Ristic 9 In this paper we consider a stationary bivariate minification process of Ristić 9 with the bivariate Marshall Olkin exponential distributions BVE(λ, λ 2, λ 2 ) K = L Motivated by situations that arise in reliability theory such as the failure of paired jet engines or the registration of an event by two adjacent geiger counters, Marshall Olkin 6 introduced the bivariate exponential distribution with survival function P X > x, Y > y} = e λ x λ 2 y λ 2 max(x,y), x, y > 0, where λ > 0, λ 2 > 0, λ 2 > 0 The rom variables are constructed such that X Y are dependent exponentially distributed rom variables with parameters λ + λ 2 λ 2 + λ 2, respectively The important property of this distribution is that it is not absolutely continuous distribution, since the probability P X = Y } = λ 2 /(λ + λ 2 + λ 2 ) is non-negative The density function f(x, y) of the BVE(λ, λ 2, λ 2 ) distribution is given by λ (λ 2 + λ 2 )e λx (λ2+λ2)y, y > x > 0, f(x, y) = λ 2 (λ + λ 2 )e (λ +λ 2 )x λ 2 y, x > y > 0, λ 2 e (λ +λ 2 +λ 2 )x, x = y > 0 This paper is organized as follows The properties of the process are considered in Section 2 In Section 3 we give the estimates of the parameters of the process 2 PROPERTIES OF THE PROCESS In this section we consider a stationary bivariate minification process with bivariate Marshall Olkin exponential distribution BVE(λ, λ 2, λ 2 ) The process is given by X n = K min(x n, Y n, η n ), () Y n = K min(x n, Y n, η n2 ), where λ > 0, λ 2 > 0, λ 2 > 0, K > λ/λ 2, λ = λ + λ 2 + λ 2, (η n, η n2 ), n } is a sequence of iid rom vectors the rom vectors (X m, Y m ) (η n, η n2 ) are independent for m < n Ristić 9 derived the innovation distribution of the rom vector (η n, η n2 ) The rom vector (η n, η n2 ) has the bivariate Marshall Olkin exponential distribution BVE(λ K, λ 2 K, λ 2 K λ) The marginal distributions of the rom variables η n η n2 are ε(c ) ε(c 2 ), respectively, where c = (λ + λ 2 )K λ c 2 = (λ 2 +λ 2 )K λ Following Ristić 9, we obtain the joint survival function

3 A bivariate Marshall Olkin exponential minification process 7 of the rom vectors (X n, Y n ) (X n j, Y n j ), j > 0 Denote the joint survival function of (X n, Y n ) (X n j, Y n j ) by S j (x, y, x 2, y 2 ; K) = P X n > x, Y n > y, X n j > x 2, Y n j > y 2 } (2) The joint survival function S j (x, y, x 2, y 2 ; K), j, can be obtained recursively as S j (x, y, x 2, y 2 ; K) = F ( max ( x ) ( K, y j K, x j 2, max x )) K, y j K, y j 2 F (x, y ) F ( max ( x ) ( K, y j K, max x )), j K, y j K j = S (x, y, x 2, y 2 ; K j ) It is obvious that the joint distribution the properties of the rom vector (X n, Y n, X n j, Y n j ) can be derived from the joint distribution the properties of the rom vector (X n, Y n, X n, Y n ) replacing K by K j Now we discuss the autocovariance structure of the bivariate Marshall Olkin exponential minification process We define the autocovariance matrix of a bivariate process (X n, Y n ), n 0} by Γ(j) = Cov(Xn, X n j ) Cov(X n, Y n j ) Cov(Y n, X n j ) Cov(Y n, Y n j ) To derive the autocovariance matrix Γ(j) it suffices to derive the autocovariance matrix Γ() In order to compute the moment E(X n X n ), we consider the conditional expectation E(X n X n, Y n ) From the definition of the process (X n, Y n ), n 0}, we have that conditional distribution for X n, given X n = x Y n = y, is given by e c z K, z < K min(x, y), P X n z X n = x, Y n = y} =, z K min(x, y) Note that this is not an absolutely continuous distribution, since the probability P X n = K min(x n, Y n ) X n = x, Y n = y} = P η n > min(x, y)} = c min(x,y) e is non-negative Now, the conditional expectation is E(X n X n = x, Y n = y) = c K Using this it is easy to verify that K min(x,y) 0 ze c z K dz + K min(x, y)e c min(x,y) = K c ( e c min(x,y) ) E(X n X n ) = K c E X n ( e c min(x n,y n ) ) (3)

4 72 M M Ristić, B Č Popović, A Nastić, M -Dord ević In order to compute the moment E(X n X n ) we will need the following lemma Lemma Let (X, Y ) be a rom vector with bivariate Marshall Olkin exponential distribution BV E(λ, λ 2, λ 2 ) Let U = X, W = Y V = min(x, Y ) Then: (i) the rom vector (U, V ) has the survival function P U > x, V > y} = e (λ +λ 2 ) max(x,y) λ 2 y, P U = V } = λ + λ 2, λ (ii) the rom vector (W, V ) has the survival function P W > x, V > y} = e λ y (λ 2 +λ 2 ) max(x,y), P W = V } = λ 2 + λ 2 λ Proof (i) From the definition of the rom variables U V, we have that P U > x, V > y} = P X > max(x, y), Y > y} = e λ max(x,y) λ 2 y λ 2 max(max(x,y),y) = e (λ +λ 2 ) max(x,y) λ 2 y, P U = V } = P X Y } = λ + λ 2 λ (ii) The proof is very similar to the proof of (i) Now, setting U = X n V = min(x n, Y n ) in (3) using Lemma, we have that E(X n X n ) = K E U ( ) cv e c = K c λ 2 (λ + λ 2 ) + K c (λ + λ 2 ) = 0 0 K + K(λ + λ 2 ) 2 Using this result, we conclude that u Cov(X n, X n ) = 0 u( e c v )e (λ +λ 2 )u λ 2 v dvdu u( e cu )e λu du K(λ + λ 2 ) 2

5 A bivariate Marshall Olkin exponential minification process 73 Similarly, we can verify that E(X n Y n ) = K c E W ( e cv ) = K(λ + λ 2 ) + λ 2 + λ 2 K(λ + λ 2 ) 2 (λ 2 + λ 2 ), Cov(X n, Y n ) = K(λ + λ 2 ) 2 Let us consider now Cov(Y n, X n ) Cov(Y n, Y n ) The conditional distribution for Y n, given X n = x Y n = y, is given by P Y n z X n = x, Y n = y} = e c 2 z K, z < K min(x, y),, z K min(x, y) Also, we have P Y n = K min(x n, Y n ) X n = x, Y n = y} = e c 2 min(x,y) Finally, we obtain Cov(Y n, X n ) = Cov(Y n, Y n ) = K(λ 2 + λ 2 ) 2 in a similar way as we have obtained Cov(X n, X n ) Cov(X n, Y n ) Thus we obtain the autocovariance matrix Γ() as Γ() = K (λ +λ 2 ) 2 (λ +λ 2 ) 2 (λ 2+λ 2) 2 (λ 2+λ 2) 2 If we replace K by K j in Γ(), then we will obtain the autocovariance matrix Γ(j) as Γ(j) = (λ +λ 2) 2 (λ +λ 2) 2 K j (λ 2+λ 2) 2 (λ 2+λ 2) 2 We will now discuss the autocorrelation structure of the bivariate minification process with bivariate Marshall Olkin exponential distribution We define the autocorrelation matrix by R(j) = After elementary calculation we get R(j) = K j Corr(Xn, X n j ) Corr(X n, Y n j ) Corr(Y n, X n j ) Corr(Y n, Y n j ) λ 2 +λ 2 λ +λ 2 λ +λ 2 λ 2+λ 2 Now, we will derive the range of the correlations Corr(X n, X n ), Corr(X n, Y n ), Corr(Y n, X n ) Corr(Y n, Y n ) Since K > λ/λ 2, it follows that 0 < Corr(X n, X n ) = Corr(Y n, Y n ) < λ 2 λ <, 0 < Corr(X n, Y n ) < λ 2(λ 2 + λ 2 ) (λ + λ 2 )λ <, 0 < Corr(Y n, X n ) < λ 2(λ + λ 2 ) (λ 2 + λ 2 )λ <

6 74 M M Ristić, B Č Popović, A Nastić, M -Dord ević 3 ESTIMATION OF THE PARAMETERS In this section we will estimate the unknown parameters K, λ, λ 2 λ 2 Ristić 9 showed that our bivariate minification process is ergodic uniformly mixing Let us consider the estimation of the unknown parameters Let (X 0, Y 0 ), (X, Y ),, (X N, Y N )} be a sample of size N First, we estimate the parameter K Ristić 9 used the estimate ˆK N = max n N X n min(x n, Y n ) He showed that the estimate ˆK N is strongly consistent estimate is not asymptotically normal As an alternative strongly consistent estimator of K, we can consider } Y n K N = max n N min(x n, Y n ) Both estimators K N K N can be used in practical situation, since the true values of the parameters can be obtained for small N Now we consider the estimation of the parameters λ, λ 2 λ 2 We will use the estimates where X N = N Y N = N I N = N i=0 N i=0 X i, Y i, } N I(X i > min(x i, Y i )), N i= I(X i > min(x i, Y i )) =, Xi > min(x i, Y i ), 0, X i min(x i, Y i ) Since the bivariate minification process with bivariate Marshall Olkin exponential distribution is ergodic, it follows that the estimates X N, Y N I N are strongly consistent estimates of the parameters /(λ + λ 2 ), /(λ 2 + λ 2 ) Kλ/(Kλ + K(λ + λ 2 ) λ) Also, since the bivariate minification process is stationary uniformly mixing φ /2 (h) <, it follows from Theorem 20 (Billingsley 4) that i= N XN λ +λ 2 Y N λ 2 +λ 2 has asymptotically bivariate normal distribution N 2 (0, Σ), as N, where Σ = K+ (K )(λ +λ 2 ) 2 σ xy K+ σ xy (K )(λ 2+λ 2) 2

7 A bivariate Marshall Olkin exponential minification process 75 σ xy = λ 2 λ(λ + λ 2 )(λ 2 + λ 2 ) + K (λ + λ 2 ) 2 + (λ 2 + λ 2 ) 2 Following Balakrishna Jacob (2003), we can show that N (I N λk ) c + λk has asymptotically normal distribution with zero mean variance σ 2 c λk = (c + λk) 2 + K h + 2c2 (c + λ)((c + λ)k h + c ) (c + λk) 2 > 0 h= So, we can take the estimates of the parameters λ, λ 2 λ 2 as the solutions of the system of the equations X N = Y N = I N =, λ + λ 2, λ 2 + λ 2 Kλ Kλ + K(λ + λ 2 ) λ Now we present some numerical results We simulated 0000 realizations of our process for the true values: a) K = 2, λ = 02, λ 2 = 05, λ 2 = 08, b) K = 3, λ = 05, λ 2 = 5, λ 2 = 25 The simulation was replicated 00 times for each data set we computed sample means of the estimates ˆK N, KN, ˆλ, ˆλ 2, ˆλ 2 the stard errors (SE) The results are summarized in Table 4 ACKNOWLEDGEMENTS The authors would like to thank Professor N Balakrishna Professor KK Jose for sending copies of their published works on this subject Also, the authors wish to thank to a referee for the suggestions which have improved the article This work was partially supported by Grant of MNTR REFERENCES B C Arnold, C A Robertson: Autoregressive logistic processes J Appl Prob 26 (989),

8 76 M M Ristić, B Č Popović, A Nastić, M -Dord ević N KN KN λ SE(λ ) λ 2 SE(λ 2 ) λ 2 SE(λ 2 ) N KN KN λ SE(λ ) λ 2 SE(λ 2 ) λ 2 SE(λ 2 ) Table : Some numerical results of the estimations (a) K = 2, λ = 02, λ 2 = 05, λ 2 = 08, b) K = 3, λ = 05, λ 2 = 5, λ 2 = 25) 2 N Balakrishna: Estimation for the semi-pareto processes Commun Statist-Theory Meth 27 (998), N Balakrishna, T M Jacob: Parameter estimation in minification processes Commun Statist-Theory Meth 32 (2003), N Balakrishna, K Jayakumar: Bivariate semi-pareto distributions processes Statistical Papers 38 (997), P Billingsley: Convergence of Probability Measures, 968, Wiley, New York 6 P A W Lewis, E McKenzie: Minification processes their transformations J Appl Prob 28 (99), A W Marshall, I Olkin: A multivariate exponential distribution J Amer Stat Assoc 62 (967), R N Pillai: Semi-Pareto processes J Appl Prob 28 (99), R N Pillai, K K Jose, K Jayakumar: Autoregressive minification process universal geometric minima J Indian Statist Assoc 33 (995), M M Ristić: Stationary bivariate minification processes Statist Probab Lett 76 (2006), C H Sim: Simulation of Weibull Gamma autoregressive stationary process Commun Stat-Simul Computat 5 (986), A Thomas, K K Jose: Bivariate semi-pareto minification processes Metrika 59 (2004), A Thomas, K K Jose: Multivariate minification processes STARS 3 (2002), 9 4 V L Tavares: An exponential Markovian stationary process J Appl Prob 7 (980), H C Yeh, B C Arnold, C A Robertson: Pareto process J Appl Prob 25 (988), 29 30

9 A bivariate Marshall Olkin exponential minification process 77 Address Department of Mathematics Informatics, Faculty of Sciences Mathematics, University of Niš, Serbia Miroslav Ristić: Biljana Popović: