On distribution of bivariate concomitants of records

Applied Mathematics Letters 23 (2010) 567–570 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On distribution of bivariate concomitants of records Muhammad Qaiser Shahbaz, Saman Shahbaz, Mohammad Mohsin, Arif Rafiq ∗ Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan article abstract info Article history: Received 13 October 2009 Received in revised form 18 January 2010 Accepted 25 January 2010 The distributions of two concomitants have been given when a random sample is available from a trivariate distribution. The illustration has been given by using a trivariate PseudoExponential distribution. © 2010 Elsevier Ltd. All rights reserved. Keywords: Concomitants Records Pseudo- distributions Exponential distribution 1. Introduction Record values and record statistics have been widely used in the theory of statistical sciences. The record values are simply referred to as the smallest (largest) observation among all the previously recorded values. The concept of record values was first given by [1]. Since the development of the theory of records, lot of work have been done by using various probability distributions. The records have also been used to characterize many probability distribution. A large number of characterizations of Exponential distribution have been done by [2–4] by using the theory of records. The relationship among moments of exponential distribution have been studied by [5] on the basis of records. Characterizations of several other distributions by using the distribution of record values have been given by [6]. The probability distribution of kth upper record from a sample of size n is given by [7] as: 1 f (xk ) [R (xk )]k−1 ; Γ (k) where R (xk ) = − ln [1 − F (x)]. fk:n (xk ) = (1.1) The joint distribution of kth and mth records is given by [7] as: fk,m:n (xk , xm ) = r (xk ) f (xm ) Γ (k) Γ (m − k) [R (xk )]k−1 [R (xm ) − R (xk )]m−k−1 ; (1.2) where r (x) = R/ (x) and −∞ < xm < xk < ∞. When a sample is available from a bivariate population with density function f (x, y) than the sample can be arranged with respect to records of random variable X . The random variable Y in this case is called the concomitants of record values. Specifically, the probability distribution of kth concomitant of record is given by [7] as: f (yk ) = Z ∞ f (yk |xk ) fk:n (xk ) dxk . −∞ ∗ Corresponding author. E-mail addresses: [email protected] (M.Q. Shahbaz), [email protected] (S. Shahbaz), [email protected] (M. Mohsin), [email protected] (A. Rafiq). 0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.01.012 (1.3) 568 M.Q. Shahbaz et al. / Applied Mathematics Letters 23 (2010) 567–570 The joint distribution of two concomitants of records is given as: f (yk , ym ) = Z ∞ −∞ Z xm f (yk |xk ) f (ym |xm ) fk,m:n (xk ) dxk dxm . (1.4) −∞ The distributions given in (1.3) and (1.4) can be found for any probability distribution. Recently Filus and Filus [8–10] have introduced a new class of probability distributions known as Pseudo-distribution as a linear combination of several random variables having the same distribution. These distributions are effectively useful in a number of situations where the standard distributions are not applicable. In the following section we have first introduced the concept of bivariate concomitants if the random sample of size n is available from a trivariate distribution. The concept has been applied to trivariate Pseudo-Exponential distribution in Section 4. 2. Distribution of bivariate concomitants of records Concomitants of record statistics have been widely studied by number of statisticians. It is often be the case that we have a random sample from a trivariate distribution with density function f (x, y, z ) and the sample is arranged with respect to records of random variable X . In this case the random variables Y and Z are automatically arranged and are referred to as the bivariate concomitants of record statistics. The probability density function of kth pair of concomitants of record values is easily derived parallel to (1.3) and the joint distribution of kth and mth pair of concomitants of record values can be derived parallel to (1.4). Specifically, the joint distribution of kth pair of concomitants of record values is given as: f (yk , zk ) = Z ∞ f (yk , zk |xk ) fk:n (xk ) dxk ; (2.1) −∞ where f (yk , zk |xk ) is the conditional distribution of (yk , zk ) given xk . The distribution of kth and mth pair of concomitants of record values is given as: f (wk , wm ) = Z ∞ −∞ Z xm f (yk , zk |xk ) f (ym , zm |xm ) fk,m:n (xk ) dxk dxm ; /  (2.2) −∞  / where wk = yk zk and wm = ym zm . The distribution of bivariate concomitants for trivariate Pseudo-Exponential distribution have been obtained in Section 4. 3. The trivariate Pseudo-Exponential distribution In this section we have introduced the trivariate Pseudo-Exponential distribution as a compound distribution of three random variables. The distribution is derived in the following: Suppose that the random variable X has the exponential distribution with parameter α . The density function of X is: f (x; α) = α e−α x ; x > 0. (3.1) Further, suppose that the random variable Y has the exponential distribution with parameter φ1 (x), where φ1 (x) is some function of random variable X . The density function of Y is: f (y|x) = φ1 (x) exp {−φ1 (x) y} ; y, φ1 (x) > 0. (3.2) Finally, suppose that the random variable Z also has the exponential distribution with parameter φ2 (xy), where φ2 (xy) is some function of random variables X and Y . The density function of Z is therefore: f (z |x, y) = φ2 (xy) exp {−φ2 (xy) z } ; z , φ2 (xy) > 0. (3.3) Now, the compound distribution of (3.1)–(3.3) is referred to as the trivariate Pseudo-Exponential distribution. The density function of the distribution is given as: f (x, y, z ) = αφ1 (x) φ2 (xy) exp [− {α x + φ1 (x) y + φ2 (xy) z }] φ1 (x) > 0; φ2 (xy) > 0; x, y, z > 0. (3.4) Several distributions can be derived from (3.4) for various choices of φ1 (x) and φ2 (xy). Using φ1 (x) = x and φ2 (xy) = xy we have obtained following trivariate Pseudo-Exponential distribution: f (x, y, z ) = α x2 y exp [−x {α + y + yz }] ; x, y, z > 0. The product moments for (3.4) are given as: / µr ,s,t = Z ∞ Z ∞ 0 = 0 Z Z ∞ 0 ∞ 0 Z Z ∞ xr ys z t f (x, y, z ) dxdydz 0 ∞ xr ys z t α x2 y exp [−x {α + y + yz }] dxdydz . 0 (3.4) M.Q. Shahbaz et al. / Applied Mathematics Letters 23 (2010) 567–570 569 After some calculus, the product moments turned out to be: / µr ,s,t = α s−r Γ (1 + r − s) Γ (1 + s − t ) Γ (1 + t ) . (3.5) The product moments (3.5) exist if r > s − 1 and s > t − 1. The marginal moments can be easily obtained from (3.5). The conditional distribution of X given Y and Z is obtained as: f (x, y, z ) f (x|y, z ) = f (y, z ) (3.6) · Now f (y, z ) = Z ∞ f (x, y, z ) dx = 0 = Z ∞ α x2 y exp [−x {α + y + yz }] dx 0 2α y · (α + y + yz )3 (3.7) Using (3.4) and (3.7) in (3.6), the conditional distribution of X given Y and Z is given as: 1 f (x|y, z ) = 2 (α + y + yz )3 x2 exp [−x {α + y + yz }] . The hth conditional moment of X is: h Z
E X |y, z = Z = ∞ xh f (x|y, z ) dx 0 ∞ 1 xh 2 0 (α + y + yz )3 x2 exp [−x {α + y + yz }] dx Γ (h + 3) = 2 (α + y + yz )h (3.8) · The conditional mean and variance can be readily obtained from (3.8). In the following section we have obtained the distribution of the concomitants of record statistics for (3.4). 4. Bivariate concomitants of upper records The trivariate Pseudo-Exponential distribution is given in (3.4). The distribution of bivariate concomitants can be obtained by using (2.1). To obtain the distribution (2.1) we first compute the conditional distribution of (Y , Z ) given X as: f (y, z |x) = f ( x, y , z ) f (x) 2 y, z > 0. = x y exp {−xy (1 + z )} ; (4.1) Also for the random variable X , the distribution of kth upper record is obtained by using (1.1). The distribution of kth upper record; Xk = X ; is: fk:n (xk ) = α k k−1 −αx x e ; Γ (k) α, x > 0. (4.2) Using (4.1) and (4.2) in (2.1), the joint distribution of the pair of concomitants; Yk = Y and Zk = Z ; is obtained below: f (yk , zk ) = Z =y ∞ x2 y exp {−xy (1 + z )} 0 αk Γ (k) Z α k k−1 −αx x e dx Γ (k) ∞ xk+1 exp {−x (α + y + yz )} dx. 0 Making the transformation x (α + y + yz ) = w and simplifying, the joint distribution of two concomitants; Yk = Y and Zk = Z ; is obtained as: f (yk , zk ) = k (k + 1) α k y (α + y + yz )k+2 ; y, z > 0 . (4.3) The product moments of (4.3) are given by:
/ µs,t = E Yks , Zkt = Z ∞ 0 Z ∞ ys z t 0 k (k + 1) α k y (α + y + yz )k+2 dydz . 570 M.Q. Shahbaz et al. / Applied Mathematics Letters 23 (2010) 567–570 Using [11] and some simplification, the product moments of two concomitants are given as: / µs,t = αk Γ (k − s) Γ (1 + s − t ) Γ (1 + t ) . Γ (k) (4.4) The means, variances and covariance can be easily obtained by using (4.4). It can be readily seen from (4.4) that mean and variance for Zk does not exist. Now using (4.3), the marginal distribution of Zk is given as f (zk ) = (1 + z )−2 ; z > 0. So the conditional distribution of Yk = Y given Zk = Z is given as: f (yk |zk ) = k (k + 1) α k y (1 + z )2 (α + y + yz )k+2 ; y, z > 0. (4.5) The conditional moments of Yk given Zk are given as: E Yks |zk =
= Z Z ∞ ysk f (yk |zk ) dyk 0 ∞ ysk 0 k (k + 1) α k y (1 + z )2 (α + y + yz )k+2 dyk . Using [11], the conditional moments are given as: E Yks |Zk = z =
α s Γ (k − s) Γ (s + 2) ; (1 + z )s Γ (k) k > s. (4.6) Using s = 1 in (4.6), the conditional expectation of Yk given z is E (Yk |z ) = 2α (k − 1)−1 (1 + z )−1 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] K.N. Candler, The distribution and frequency of record values, J. Roy. Statist. Soc., Ser. B 14 (1952) 220–228. M. Ahsanullah, Record values and exponential distribution, Ann. Inst. Statist. Math. 30 (1978) 429–433. M. Ahsanullah, Characterization of the Exponential distribution by some properties of the record values, Statist. Hefte 23 (1982) 326–332. M. Ahsanullah, S.N.U.A. Kirmani, Characterizations of the Exponential distribution through a lower record, Comm. Statist. Theory Methods 20 (4) (1991) 1293–1299. N. Balakrishnan, M. Ahsanullah, Relations for single and product moments of record values from exponential distribution, J. Appl. Statist. Sci 2 (1) (1995) 73–87. S.N.U.A. Kirmani, M.I. Beg, On Characterization of distribution by expected records, Sankhya, A 46 (3) (1984) 463–465. M. Ahsanullah, Record Statistics, Nova Science Publishers, USA, 1995. J.K. Filus, L.Z. Filus, A Class of generalized multivariate normal densities, Pakistan J. Statist 16 (1) (2000) 11–32. J.K. Filus, L.Z. Filus, On some bivariate pseudonormal densities, Pakistan J. Statist 17 (1) (2001) 1–19. J.K. Filus, L.Z. Filus, On some new classes of multivariate probability distributions, Pakistan J. Statist 22 (1) (2006) 21–42. M. Abramowitz, I.A. Stegum, Handbook of Mathematical Functions, in: Applied Mathematical Series, vol. 55, National Bureau of Standards, USA, 1965.