Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics

Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a₁<a₂<a₃<…. In particular, a_i could be equal to i for all i. Let X_1n≦X_2n≦?≦X_nn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦k≦n), independence of the event {X_kn=X_1n} and X_1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{X_kn = X_1n | X_1n = a_i} is equivalent to X having the IFR (DFR) property. Let a_i = i and $\text{G(i) = P(X≧i), i = 1, 2, …}$. We prove that the independence of {X_2n ? X_1n ∈B} and X_1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = q^i?1, 1≦i≦m+2 (1≦i≦m+1), where 0<q<1.     

On UMVU estimators and Bhattacharya bounds in exponential distributions

A subfamily of exponential distributions is considered and it is shown that the variance of the UMVU estimator of an estimable function g(θ) having power series expansion is the limit of Bhattacharya bounds.     

Some Bayesian lower bounds on reliability in the lognormal distribution

The two-parameter lognormal distribution with density function f(y: γ, σ²) = [(2πσ²)^1/2y] ¹exp[?(ln y ? γ)²/2σ²], y > 0, is important as a failure-time model in life testing. In this paper, Bayesian lower bounds for the reliability function R(t: γ, σ²) = ?[(γ ? ln t)/σ] are obtained for two cases. First, it is assumed that γ is known and σ² has either an inverted gamma or “general uniform” prior distribution. Then, for the case that both γ and σ² are unknown, the normal-gamma prior and Jeffreys' vague prior are considered. Some Monte Carlo simulations are given to indicate some of the properties of the Bayesian lower bounds.     

A formula for P(bi≤Ri≤ai, 1≤i≤m) under a general class of alternatives

Let X_i≤?≤X_m and Y_i≤?≤Y_n be two sets of independent order statistics from continous distributions with distribution functions F and G respectively. Let R_i denote the rank of X_i in the combined order sample. Steck (1980) has found an expression for P(b_i≤R_i≤a_i, all i) when F = h(G), h being the incomplete beta function with parameters (α,β?α+1). An alternative expression for the same probability is obtained which is computationally a substantial improvement on Steck's result.     

Structural probability bounds for the strong Pareto law

By using the structural density function (Fraser 1979, Ch. 7) of the parameters of a Pareto distribution, the structural distribution function of the strong Pareto law is derived. Its fractiles have been evaluated numerically for special cases, and the results are displayed through graphs from which structural one-sided probability bounds may be found. It is shown that these graphs may also be used to find structural tolerance bounds for the Pareto distribution.     

Collective and Unitary Models: A Clarification

In this note we identify and clarify a confusion that has arisen in the literature about the exact relationship between unitary and collective models and what enters the Pareto weight and the sharing function. We suggest that we should denote as ‘unitary’ any model that leads to outcomes that satisfy the Slutsky conditions whether or not these outcomes depend on distribution factors. In particular, income pooling is neither necessary nor sufficient for a unitary model. We also show that the presence of prices or total expenditure in the sharing rule cannot be used as a test for a unitary model.

Search for a majority element

A new short proof of a theorem of Saks and Werman is given about the number of weighings necessary and sufficient to determine a majority type coin among n coins of two possible weights by comparisons. A lower bound on the number of good algorithms is proved and further questions are posed about determining the majority in different weighing models.     

Moment Information for Probability Distributions,Without Solving the Moment Problem. I: Where is the Mode?

How much information does a small number of moments carry about the unknown distribution function? Is it possible to explicitly obtain from these moments some useful information, e.g., about the support, the modality, the general shape, or the tails of a distribution, without going into a detailed numerical solution of the moment problem? In this paper a theoretical result of Johnson and Rogers is generalized to be valid for all moment problems and is exploited to demonstrate that a few moments are able to provide us with valuable information about the position of the mode of an unknown (unimodal) distribution.     

Renyi Entropy Properties of Order Statistics

In this article, we discuss some properties of Renyi entropy and Renyi information of order statistics. Some bounds for Renyi entropy of order statistics are obtained. Also, we relate Renyi entropy ordering of order statistics to Renyi entropy ordering and other well known orderings of parent random variables. Then it is proved that the Renyi information between order statistics and parent random variable is distribution free, and it is shown, as expected, the distance is minimum for the median.     

Lower and Upper Bounds for Rectangular Probabilities

This article presents some results showing how rectangular probabilities can be studied using copula theory. These results lead us to develop new lower and upper bounds for rectangular probabilities which can be computed efficiently. The new bounds are compared with the ones obtained from the generalized Fréchet–Hoeffding bounds and Bonferroni-type inequalities.