Conditional properties of a random sample given an order statistic
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Conditional properties of a random sample given an order statistic Jafar Ahmadi1
· H. N. Nagaraja2
Received: 11 October 2017 / Revised: 31 March 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This paper deals with the conditional properties of unordered observations in a random sample given the ith order statistic of the same sample. A characterization of symmetric distribution is given based on the properties of conditional moments. It is proved that the joint distribution has negative quadrant dependence structure and an exact expression for the Pearson correlation coefficient is given. Stochastic ordering results are obtained and it is shown that some orderings are inherited by the conditional distribution of the sample values. The results are useful in studying the properties of component lifetimes of a system with n components given the ith failure time and associated repair costs. Keywords k-Out-of-n system · Moment generating function · Order statistics · Quadrant dependence · Stochastic orders Mathematics Subject Classification 62G30 · 60E15 · 62E1 · 62H20
1 Introduction Let X 1 , X 2 , . . . , X n be a sample of independent and identically distributed (iid) random variables with absolutely continuous cumulative distribution function (cdf) F(·) and probability density function (pdf) f (·). Denote the corresponding order statistics by X 1:n ≤ X 2:n ≤ · · · ≤ X n:n . These statistics appear frequently in the study of system lifetime in reliability and survival analysis, characterization of probability
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Jafar Ahmadi [email protected] H. N. Nagaraja [email protected]
1
Department of Statistics, Ordered and Spatial Data Center of Excellence, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
2
Division of Biostatistics, The Ohio State University, Columbus, OH 43210, USA
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J. Ahmadi, H. N. Nagaraja
distributions, analysis of censored samples, stochastic orderings, information theory, bootstrap estimation, ranked-set sampling and so on; see David and Nagaraja (2003), Arnold et al. (2008) and references therein. It is well-known that the order statistics in a random sample from a continuous parent form a Markov chain. Also, the conditional distribution of X r +1:n , . . . , X s−1:n given X i:n = xi , for i ≤ r and i ≥ s, (1 ≤ r < s ≤ n), is just the distribution of all order statistics in a random sample of s − r − 1 drawn from the parent distribution truncated in the tails at xr and xs , i.e. from the pdf f (x)/[F(xs ) − F(xr )], xr < x < xs . Consequently, X r +1:n , . . . , X s−1:n are independent of X 1:n , . . . , X r −1:n and X s+1:n , . . . , X n:n when X r :n and X s:n are given; see Theorem 2.5 in David and Nagaraja (2003). In a similar spirit, Iliopoulos and Balakrishnan (2009) considered a random variable D that counts the number of order statistics that are at most some pre-fixed number T . They proved that conditional on D = d, the vectors (X 1:n , . . . , X d:n ) and (X d+1:n , . . . , X n:n ) are mutually independent. It should be mention
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