Fernando López-Blázquez

Distribution of a Sum of Weighted Noncentral Chi-Square Variables

We derive Laguerre expansions for the density and distribution functions of a sum of positive weighted noncentral chi-square variables. The procedure that we use is based on the inversion of Laplace transforms. The formulas so obtained depend on certain parameters, which adequately chosen will give some expansions already known in the literature and some new ones. We also derive precise bounds for the truncation error.

Random translations, contractions and dilations of order statistics and records

We introduce a new tool to investigate some distributional properties of order statistics and records related by a random translation (contraction or dilation) scheme. This technique is based on the property of uniqueness of solutions of certain non-linear integral equations of Volterra type. We show how this tool is used to obtain new characterizations of distributions.

Discrete distributions for which the regression of the first record on the second is linear

The linearity of regression of the first record on the second is examined for discrete random variables. Both ordinary and weak records are considered. The analysis involves the determination of all possible linear relationships and all possible probability distributions. Several characterizations of geometric distributions are also shown.

Linearity of regression for the past weak and ordinary records

In this paper we study the property of linearity of backward regression for non-adjacent records. In the case of weak records, a characterization of the geometric distribution is obtained. It also appears that a related characterization for ordinary records does not hold, showing the difference in conditional behaviour between weak and ordinary records.

A Characterization of Geometric Distribution Through kth Weak Records

ABSTRACT Let be the sequence of kth weak records from a distribution F having support on non negative integers. We show that is a constant almost surely (a.s.) iff the underlying distribution is geometric. We also prove that for n ≥ 1 and k ≥ 2 the property (a.s.) does not hold in the geometric case.

On Terrel's characterization of uniform distribution

Terrel (1983) (The Annals of Probability, Vol. 11, No. 3, 823–826) showed that the coefficient of correlation between the smaller and larger of a sample of size two is at most one-half, and this upper bound is attained only for continuous uniform distributions. His proof is of computational nature and is based on the properties of Legendre polynomials. We give an easier proof of Terrels characterization and we show how our method can be used for obtaining sharper bounds within the class of discrete distributions onN points and also a characterization of the equidistant uniform distribution.

Bounds for the expected value of spacings from discrete distributions

We give upper bounds for expected spacings from discrete distributions and characterize the random variables attaining those bounds. In the case of the range, we give tables from which we can obtain the values of the proposed bounds and the probability functions of the extremal distributions. In particular, when the sample size is 2 or 3, the discrete distributions with maximum standardized range are discrete uniform.

Binomial approximation to hypergeometric probabilities

Abstract We give an orthogonal expansion for hypergeometric probabilities in terms of Krawtchoucks polynomials. An adequate choice of the parameters involved in the expansion and truncation yield binomial approximations to hypergeometric probabilities. We also investigate the limit behavior of these approximations.

Discrete distributions with maximum expected value of the maximum

Abstract We give an upper bound for the expected value of the largest order statistic of a simple random sample of size n from a discrete distribution on N points. We also characterize the distributions that attain such bound. In the particular case n =2, we obtain a characterization of the discrete uniform distribution.

Random Translation of Records from Geometric Distributions

It is well known that the distributions of two records from a sequence of iid geometric random variables are related by a random translation. The distribution of the random translation is a negative binomial distribution. In this article, we investigate if this property characterizes the geometric parent distribution. We show that the answer is positive when weak records are considered. On the other hand, when ordinary records are considered, it is possible to characterize tail-geometric distributions if the records are consecutive, but in the case of non consecutive ordinary records it is possible to find out distributions that do not seem to be tail-geometric satisfying the property of random translation.