A note on exact moments of order statistics from exponentiated log-logistic distribution

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1 ProbStat Forum, Volume 7, July 214, Pages ISSN ProbStat Forum is an e-ournal. For details please visit A note on exact moments of order statistics from exponentiated log-logistic distribution Haseeb Athar, Nayabuddin Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh-22 2, India Abstract. In this paper we establish explicit expressions for single and product moments of order statistics from exponentiated log-logistic distribution. These expressions are used to calculate the mean and variances. 1. Introduction Let X 1, X 2,..., X n be a random sample of size n from a continuous population having probability density function (pdf) f(x) and distribution function (df) F (x). Let X 1:n X 2:n... X n:n be the corresponding order statistics. The pdf of X r:n, the rth order statistic is given by David and Nagaraa (23) f r:n (x) = (r 1)!(n r)! [F (x)]r 1 [1 F (x)] n r f(x), < x <, (1) and oint pdf of X r:n, X s:n, 1 r < s n is given as f r,s:n (x, y) = (r 1)!(s r 1)!(n s)! [F (x)]r 1 [F (y) F (x)] s r 1 [1 F (y)] n s f(x)f(y), (2) < x < y <. A random variable X is said to have exponentiated log-logistic distribution (Rosaiah et al., 26) if its pdf is given by x f(x) = θx (+1)[ ] θ+1, θ, >, < x <, (3) 1 + x and the corresponding df of X is [ x ] θ, F (x) = θ, >, < x <. (4) 1 + x For more details on this distribution and its application one may refer to Rosaiah et al. (26, 27). Keywords. Order statistics, Exponentiated log-logistic distribution, Single and product moments. Received: 5 December 213; Accepted: 6 June 214 Corresponding author address: [email protected] (Haseeb Athar )

2 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Moments of order statistics for some specific distributions are investigated by several authors in the literature. Malik (1966) derived the expression for exact moments of order statistics from Pareto distribution. Malik (1967) obtained the explicit expression for moments of order statistics of power function distribution. Khan and Khan (1987) obtained the moments of order statistics from Burr distribution. Khan and Ali (1995) derived the ratio and inverse moments of order statistic from Burr distribution. Further, Ali and Khan (1996) established the ratio and inverse moments of order statistic from Weibull and exponential distribution. In this paper we have obtained simple expressions for the exact moments of order statistic from exponentiated log-logistic distribution. Also means and variances are tabulated. 2. Single moments Lemma 2.1. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a and b, we have that ( I (a, ) = θb ) + θ(a + 1), 1, I (a, b) = Proof. From (5), we have I (a, ) = x [F (x)] a [1 F (x)] b f(x)dx. (5) x [F (x)] a f(x)dx = θ x 1[ ( ) θ x Set 1+x = t θ in (6), after simplification we get required result. x 1 + x ] θ(a+1)+1 dx. (6) Lemma 2.2. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a and b, we have that I (a, b) = b ( ) b ( 1) k I (a + k, ) = θ k k= b ( ) b ( ( 1) k B k ) + θ(a + k + 1), 1. Proof. It can be proved by expanding [1 F (x)] b binomially in (5) and using Lemma 2.1. k= Theorem 2.3. For exponentiated log-logistic distribution as given in (3) and 1 r n,, θ >, we have that n r ( n r n r ( ) n r ( E(Xr:n) = C r:n ( 1) )I k (r +k 1, ) = θc r:n ( 1) k B k k ) +θ(r +k), 1, (7) C r:n = k= (r 1)!(n r)!. Proof. From (1), we have E(X r:n) = (r 1)!(n r)! On application of Lemma 2.2 in (8), we get required result. k= x [F (x)] r 1 [1 F (x)] n r f(x)dx. (8)

3 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Identity 2.1. For 1 r n 1, we have that C r:n n r ( ) n r 1 ( 1) k k r + k k= = 1. (9) Proof. At = in (7), we have n r ( ) n r 1 1 = C r:n ( 1) k k r + k k= and hence the result given in (9). 3. Product moments Lemma 3.1. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a and c, we have that θ 2 ( i + ) I i, (a,, c) = I i, (a, b, c) = + θ(c + 1)B 3 F 2 x + θ(c + 1), + θ(c + 1) + 1, and a 1, a 2,..., a p [ p pf q, 1 = b 1, b 2,..., b q r= =1 + θ(a + c + 2), 1 i, + θ(a + c + 2) + 1 i+ + θ(a + c + 2) ; 1 x i y [F (x)] a [F (y) F (x)] b [F (y)] c f(x)f(y)dydx (1) Γ(a + r) ][ q Γ(b ) ]. a b =1 + r For p = q + 1 and q =1 b p =1 a >, see Mathai and Saxena (1973). Proof. We have from (1) I i, (a,, c) = x In view of (3), (4) and (11), we have By setting I i, (a,, c) = (θ) 2 ( y x i y [F (x)] a [F (y)] c f(x)f(y)dydx. (11) x i 1[ x ] θ(a+1)+1[ x 1 + x y 1[ y ] θ(c+1)+1 ] dy 1 + y 1+y ) θ = t θ in (12), we get after simplification I i, (a,, c) = θ 2 B x (p, q) = x x i 1[ t p 1 (1 t) q 1 dt, dx. (12) x ] θ(a+1)+1 ( B 1 + x x 1+x + θ(c + 1), 1 ) dx (13)

4 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages is the incomplete beta function. Also note that Mathai and Saxena (1973) proved that and B x (p, q) = xp p 2 F 1 (p, 1 q, p + 1; x) 1 t a 1 (1 t) b 1 2 F 1 (c, d ; e; t)dt = B(a, b) 3 F 2 (c, d, a ; e, a + b ; 1). Substituting these value in (13), we get the required result. Lemma 3.2. For exponentiated log-logistic distribution as given in (3) and any non negative finite integers a, b and c, we have that I i, (a, b, c) = = b ( ) b ( 1) k I i, (a + k,, b + c k) k k= θ 2 ( i + + θ(c + 1)B + θ(a + c + 2), 1 i ) + θ(c + 1),, 3 F 2 + θ(c + 1) + 1, + θ(a + c + 2) + 1 i+ + θ(a + c + 2) ; 1. Proof. Lemma can be proved by expanding [F (y) F (x)] b binomially in (1) and using Lemma 2.1. Theorem 3.3. For exponentiated log-logistic distribution as given in (2) and, θ > and 1 r < s n 1, we have that E(X i, n s s r 1 ( )( s r 1 n s r,s:n) = C r,s:n ( 1) k+l k l l= k= n s s r 1 ( )( ) s r 1 n s = C r,s:n ( 1) k+l k l l= k= ( i + B + θ(s + l + k), 1 i ) + θ(s + l r),, Proof. From (2), we have 3 F 2 E(X i, r,s:n) = C r,s:n x + θ(s + l r) + 1, On application of Lemma 3.2, (14) can be proved. o ) I i, (r + k 1,, s + l r 1) θ 2 + θ(s + l r) + θ(s + l + k) + 1 i+ + θ(s + l + k), 1 [F (x)] r 1 [F (y) F (x)] s r 1 [1 F (y)] n s f(x)f(y)dydx.. (14)

5 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Table 1: Mean of order statistics n r θ = 2, = 2 θ = 3, = 3 θ = 4, = 4 θ = 5, = Here in Table 1, it may be noted that the well known property of order statistics n i=1 E(X i:n) = ne(x) (David and Nagaraa, 23) is satisfied. Table 2: Variance of order statistics n r θ = 2, = 3 θ = 2, = 4 θ = 3, = 4 θ = 3, = References Ali, M.A., Khan, A. H. (1996) Ratio and inverse moments of order statistics from Weibull and exponential distribution, J. Applied statistical science 4(1), 1 7. David, H.A., Nagaraa H.N. (23) Order Statistics, John Wiley, New York. Khan, A.H., Khan, I.A. (1987) Moments of order statistics from Burr distribution and its characterizations, Metron XLV,

6 Athar and Nayabuddin / ProbStat Forum, Volume 7, July 214, Pages Khan, A.H., Ali, M.A. (1995) Ratio and inverse moments of order statistics from Burr distribution, J. Ind. Soc. Prob. Statist. 2, Malik, H.J. (1966) Exact moments of order statistics from Pareto distribution, Skand. Aktuarie Tidskr 49, Malik, H.J. (1967) Exact moments of order statistics from a power function distribution, Skand. Aktuar. 5, Mathai, A.M., Saxena, R.K. (1973) Generalized hyper-geometric functions with applications in statistics and physical science, Lecture Notes in Mathematics 348, Springer-Verlag, Berlin. Rosaiah, K., Kantam, R.R.L., Santosh Kumar (26) Reliability test plan for exponentiated log-logistic distribution, Economic Quality Control 21, Rosaiah, K., Kantam, R.R.L., Santosh Kumar (27) Exponentiated log-logistic distribution an economic reliability test plan, Pakistan J. Statist. 23,