Summary: Let \(X\) be a nonnegative and continuous random variable having the probability density function (pdf) \(f(\cdot)\). Let \(X_{k:n}\) \((k=1,2, \dots,n)\) denote the \(k\) th order statistic based on \(n\) independent observations on \(X\) and, for a given positive integer \(m\) \((\leq n)\), let \[ D^{(m)}_{ k,n}=X_{k+ m-1:n}-X_{k-1:n},\quad k=1,2,\dots, n-m+1, \] denote the successive (overlapping) spacings of gap size \(m\) (to be referred as \(m\)-spacings); here \(X_{0:n}\equiv 0\).
It is shown that if \(f(\cdot)\) is log convex, then the pdf of corresponding simple (gap size one) spacings \(D^{(1)}_{k,n}\), \(k=1,2,\dots,n\), are also log convex. It is also shown that the \(m\)-spacings \(D^{(m)}_{k,n}\), \(k= 1,2, \dots,n-m+1\), preserve the log concavity of the parent pdf \(f(\cdot)\). Under the log convexity of the parent pdf \(f(\cdot)\), we further show that, for \(k=1,2, \dots,n-m\), \(D^{(m)}_{k,n}\) is smaller than \(D^{(m)}_{k+1,n}\) in the likelihood ratio ordering and that, for a fixed \(1\leq k\leq n-m+1\) and \(n\geq k+m-1\), \(D^{(m)}_{k,n+1}\) is smaller than \(D^{(m)}_{k,n}\) in the likelihood ratio ordering.
Finally, we show that if \(X\) has a decreasing failure rate then, for \(k=1\), \(2,\dots, n-m\), \(D^{(m)}_{k,n}\) is smaller than \(D^{(m)}_{k+1,n}\) in the failure rate ordering and that, for a fixed \(1\leq k\leq n-m+1\) and \(n\geq k+m-1\), \(D^{(m)}_{k,n+1}\) is smaller than \(D^{(m)}_{k,n}\) in the failure rate ordering.