On stochastic properties of m-spacings

Abstract

Let X be a nonnegative and continuous random variable having the probability density function (pdf) f(.). Let X k:n (k=1,2,…,n) denote the kth order statistic based on n independent observations on X and, for a given positive integer m (⩽n) , let D k,n (m)=X k+m−1:n−X k−1:n, k=1,2,…,n−m+1 , denote the successive (overlapping) spacings of gap size m (to be referred as m-spacings); here X 0: n ≡0. It is shown that if f(.) is log convex, then the pdf of corresponding simple (gap size one) spacings D k,n (1), k=1,2,…,n , are also log convex. It is also shown that the m-spacings D k,n (m), k=1,2,…,n−m+1 , preserve the log concavity of the parent pdf f(.). Under the log convexity of the parent pdf f(.), we further show that, for k=1,2,…,n−m, D k,n (m) is smaller than D k+1, n ( m) in the likelihood ratio ordering and that, for a fixed 1⩽ k⩽ n− m+1 and n⩾k+m−1, D k,n+1 (m) is smaller than D k, n ( m) in the likelihood ratio ordering. Finally, we show that if X has a decreasing failure rate then, for k=1,2,…,n−m, D k,n (m) is smaller than D k+1, n ( m) in the failure rate ordering and that, for a fixed 1⩽ k⩽ n− m+1 and n⩾k+m−1, D k,n+1 (m) is smaller than D k, n ( m) in the failure rate ordering.