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 (less than or equal to n), let D-k,n((m)) = Xk+m-1:n - Xk-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 equivalent to 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 less than or equal to k less than or equal to n - m + 1 and n greater than or equal to 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 less than or equal to k less than or equal to n - m + 1 and n greater than or equal to k + m - 1, D-k, n+1((m)), is smaller than D-k, n((m)) in the failure rate ordering. |