Mariusz Bieniek

On Families of Distributions for Which Optimal Bounds on Expectations of GOS Can be Derived

Let , r ≥ 1, denote rth generalized order statistics and for a given distribution function W let ℱ W denote the family of distributions succeeding W in convex ordering. We consider the problem of derivation of mean-variance bounds on expectations of , optimal in ℱ W , by application of the projection method. We show that the bounds can be determined if W = W α, for some fixed , where W α denotes generalized Pareto distribution.

Bounds for Expectations of Differences of Generalized Order Statistics Based on General and Life Distributions

Let X * (r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X * (s) − X * (r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.

On unimodality of the lifetime distribution of coherent systems with failure-dependent component lifetimes

We study the conditions for unimodality of the lifetime distribution of a coherent system when the ordered component lifetimes in the system are described by generalized order statistics. Results for systems with independent and identically distributed lifetimes of components are included in this setting. The findings are illustrated with some examples for different types of systems. In particular, coherent systems with strictly bimodal density functions are presented in the case of independent standard uniform distributed lifetimes of components. Furthermore, we use the results to derive a sharp upper bound on the expected system lifetime in terms of the mean and the standard deviation of the underlying distribution.

Comparison of the bias of trimmed and Winsorized means

ABSTRACT We derive sharp upper and lower projection bounds on the bias of two-sided Winsorized means. To determine the projection of appropriate function, we consider new analytic condition which describes the form of the corresponding greatest convex minorant. Then we compare numerically obtained bounds for trimmed and Winsorized means. We conclude that if we have no information about the underlying distribution then Winsorized means are better than the trimmed ones.

Projection bounds on expectations of spacings of generalized order statistics from DD and DDA families

We present sharp upper bounds for expectations of spacings of generalized order statistics based on distributions coming from restricted families of distributions. Two families are considered: distributions with decreasing density and with density decreasing on the average. The bounds are derived by application of the projection method.

Characterizations based on k-Th upper and lower record values

Let {Y^,\ n > l } and {z£, n > 1} denote respectively the sequences of fc-th upper and lower record values of the sequence {Xn, n > 1} of independent identically distributed random variables with distribution function F. Let n, fc and r be given positive integers. We characterize distributions for which one of the conditional expectations E ( Y ^ r | y„), | y W ) , E ( Z ^ r | Z ^ ) or | Z « r ) , is linear. For example, distributions for which E(Y^T \ Y^) has the form E(Y^r \ Y^) = aY^ + b for some o, 6 6 R.