Estimation After Selection Under Reflected Normal Loss Function

Let X ₁ and X ₂ be two independent random variables from normal populations Π₁, Π₂ with different unknown location parameters θ₁ and θ₂, respectively and common known scale parameter σ. Let X ₍₂₎ = max (X ₁, X ₂) and X ₍₁₎ = min (X ₁, X ₂). We consider the problem of estimating the location parameter θ _M (or θ _J ) of the selected population under the reflected normal loss function. We obtain minimax estimators of θ _M and θ _J . Also, we provide sufficient conditions for the inadmissibility of invariant estimators of θ _M and θ _J .     

Prediction of Variables Via Their Order Statistics in Bivariate Elliptical Distributions with Application in the Financial Markets

Assuming that (X₁, X₂) has a bivariate elliptical distribution, we obtain an exact expression for the joint probability density function (pdf) as well as the corresponding conditional pdfs of X₁ and X₍₂₎ ? max?{X₁, X₂}. The problem is motivated by an application in financial markets. Exchangeable random variables are discussed in more detail. Two special cases of the elliptical distributions that is the normal and the student’s t models are investigated. For illustrative purposes, a real data set on the total personal income in California and New York is analyzed using the results obtained. Finally, some concluding remarks and further works are discussed.     

Nonnormality of Linear Combinations of Normal Random Variables

Hamedani and Tata (1975) showed that the bivariate normal distribution is determined uniquely by any countably infinite collection of distinct linear combinations of the variables and by no finite number of them. It is shown here that this characterization of bivariate normal distribution cannot be extended to the multivariate case. More specifically, it is shown that the multivariate normality of subsets (r < n) of the normal variables X ₁, X ₂, …, X_n together with the normality of an infinite number of linear combinations of them do not guarantee the joint normality of these variables.     

Limiting behavior of randomly weighted averages of symmetric heavy-tailed random variables

In this paper we consider a sequence of independent continuous symmetric random variables X₁, X₂, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages S_n = R⁽ⁿ⁾₁X₁ + ??? + R⁽ⁿ⁾_nX_n, where the random weights R⁽ⁿ⁾₁, …, R_n⁽ⁿ⁾ which are independent of X₁, X₂, …, X_n, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that c_nS_n converges in distribution to a symmetric α-stable random variable with c_n = n^{1 ? 1/α}/Γ^1/α(α + 1).     

Estimation of the Scale Parameter of the Selected Gamma Population Under the Entropy Loss Function

Let X ₁, X ₂,…, X _k be k (≥2) independent random variables from gamma populations Π₁, Π₂,…, Π _k with common known shape parameter α and unknown scale parameter θ _i , i = 1,2,…,k, respectively. Let X _(i) denotes the ith order statistics of X ₁,X ₂,…,X _k . Suppose the population corresponding to largest X _(k) (or the smallest X ₍₁₎) observation is selected. We consider the problem of estimating the scale parameter θ _M (or θ _J ) of the selected population under the entropy loss function. For k ≥ 2, we obtain the Unique Minimum Risk Unbiased (UMRU) estimator of θ _M (and θ _J ). For k = 2, we derive the class of all linear admissible estimators of the form cX ₍₂₎ (and cX ₍₁₎) and show that the UMRU estimator of θ _M is inadmissible. The results are extended to some subclass of exponential family.     

Exact distribution of the convolution of negative binomial random variables

In this note, we derive the exact distribution of S by using the method of generating function and BELL polynomials, where S = X₁ + X₂ + ??? + X_n, and each X_i follows the negative binomial distribution with arbitrary parameters. As a particular case, we also obtain the exact distribution of the convolution of geometric random variables.     

The uniform convergence of the nadaraya-watson regression function estimate

If (X₁,Y₁), …, (X_n,Y_n) is a sequence of independent identically distributed R_d × R-valued random vectors then Nadaraya (1964) and Watson (1964) proposed to estimate the regression function m(x) = ? {Y₁|X₁ = x{ by where K is a known density and {h_n} is a sequence of positive numbers satisfying certain properties. In this paper a variety of conditions are given for the strong convergence to 0 of ess_Xsup|m_n (X)-m(X)| (here X is independent of the data and distributed as X₁). The theorems are valid for all distributions of X₁ and for all sequences {h_n} satisfying h_n → 0 and nh/log n→0.     

A new class of bivariate lifetime distributions

This paper introduces a new class of bivariate lifetime distributions. Let {X_i}_{i ? 1} and {Y_i}_{i ? 1} be two independent sequences of independent and identically distributed positive valued random variables. Define T₁ = min?(X₁, …, X_M) and T₂ = min?(Y₁, …, Y_N), where (M, N) has a discrete bivariate phase-type distribution, independent of {X_i}_{i ? 1} and {Y_i}_{i ? 1}. The joint survival function of (T₁, T₂) is studied.     

On the strong Kotz approximation of Dirichlet random vectors

Let (X ₁, X ₂) be a bivariate L _p -norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α₁,α₂. If p=α₁=α₂=2, then (X ₁, X ₂) is a spherical random vector. The estimation of the conditional distribution of Z _u *:=X ₂ | X ₁>u for u large is of some interest in statistical applications. When (X ₁, X ₂) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Z _u * can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Z _u :=X ₂| X ₁=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Z _u * and Z _u can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function.     

Sums, products and ratios of generalized beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, products and ratios of independent beta random variables. In this paper, we extend professor Pham-Gia's results when X ₁ and X ₂ are independent random variables distributed according to two generalized beta distributions. For each of these distributions, we derive exact expressions for the densities of S=X ₁+X ₂, D=X ₁−X ₂, P=X ₁ X ₂, and R=X ₂/X ₁. The expressions turn out to involve the incomplete beta function as well as the hypergeometric functions of one and two variables.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Characterization of the mixtures of Rayleigh distributions by conditional expectation of order statistics

The mixture of Rayleigh random variables X ₁and X ₂ are identified in terms of relations between the conditional expectation of ( X_2:2² -X_1:2²)^r{\left( {X_{2:2}^2 -X_{1:2}^2}\right)^{r}} given X _1:2 (or X_2:2^2k{X_{2:2}^{2k}} given X_1:2,"k ￡ r){X_{1:2},\forall k\leq r)} and hazard rate function of the distribution, where X _1:2 and X _2:2 denote the corresponding order statistics, r is a positive integer. In addition, we also mention some related theorems to characterize the mixtures of Rayleigh distributions. Finally, we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems) and a simulated example is used to illustrate our results.     

New estimators of distribution functions

ABSTRACT

This article considers the estimation of a distribution function F_X(x) based on a random sample X₁, X₂, …, X_n when the sample is suspected to come from a close-by distribution F₀(x). The new estimators, namely the preliminary test (PTE) and Stein-type estimator (SE) are defined and compared with the “empirical distribution function” (edf) under local departure. In this case, we show that Stein-type estimators are superior to edf and PTE is superior to edf when it is close to F₀(x). As a by-product similar estimators are proposed for population quantiles.     

Contre-exemple au théorème de P.C. Consul sur la factorisation des distributions de Poisson généralisées

This note exhibits two independent random variables on integers, X₁ and X₂, such that neither X₁ nor X₂ has a generalized Poisson distribution, but X₁ + X₂ has. This contradicts statements made by Professor Consul in his recent book.     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

On the admissibility of ordered poisson parameter under the entropy loss function

Let X ₁, X ₂be two independent Poisson random variables with means θ ₁and θ ₂respectively. Assume 0 ≤ θ ₁ ≤ θ ₂ ≤ ∞. The problem is estimation of the ordered parameters which has received considerable attention in statistical literature during the last two decades, In this paper the main portion of the study is devoted to the problem of estimating the smallest of the two ordered Poisson means, when it is known which population corresponds to each mean under the entropy loss function. An extension of the problem to the k-ordered Poisson means is also discussed.     

Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x, y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁{(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.     

Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X₁₁,X₂₁),(X₁₂,X₂₂),… (X_1n,X_2n) having a bivariate elliptically contoured density of the form where θ₁ θ₂ are location parameters and Δ = ((λ_ik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X₁ and X₂ is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.     

On the maxima of heterogeneous gamma variables

In this article, we establish some new results on stochastic comparisons of the maxima of two heterogenous gamma variables with different shape and scale parameters. Let X₁ and X₂ [X*₁ and X*₂] be two independent gamma variables with X_i?[X*_i] having shape parameter r_i?[r*_i] and scale parameter λ_i?[λ*_i], i = 1, 2. It is shown that the likelihood ratio order holds between the maxima, X_{2: 2} and X*_{2: 2} when λ₁ = λ*₁ ? λ₂ = λ*₂ and r₁ ? r*₁ ? r₂ = r*₂. We also prove that, if r_i, r*_i ∈ (0, 1], (r₁, r₂) majorizes (r*₁, r₂*), and (λ₁, λ₂) is p-larger than (λ*₁, λ₂*), then X_{2: 2} is larger than X*_{2: 2} in the sense of the hazard rate order [dispersive order]. Some numerical examples are provided to illustrate the main results. The new results established here strengthen and generalize some of the results known in the literature.