A study of generalized skew-normal distribution

Following the paper by Genton and Loperfido [Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ _p (z; ξ, Ω)π (z?ξ), z∈? ^p , where φ _p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈? ^p , Ω>0, and π is a skewing function from ? ^p to ?, that is 0≤π (z)≤1 and π (?z)=1?π (z), ? z∈? ^p . First the distribution of linear transformations of Z are studied, and some moments of Z and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈? ^p and κ is the normal, Laplace, logistic or uniform distribution function.     

A note on the admissibility of hammersley's estimator of an integer mean

Let X₁, X₂, …, X_n be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?_n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.     

Nonnegative unbiased estimability of linear combinations of two variance components

For a general mixed model with two variance components θ₁ and θ₂, a criterion for a function q₁θ₁+q₂θ₂ to admit an unbiased nonnegative definite quadratic estimator is established in a form that allows answering the question of existence of such an estimator more explicitly than with the use of the criteria known hitherto. An application of this result to the case of a random one-way model shows that for many unbalanced models the estimability criterion is expressible directly by the largest of the numbers of observations within levels, thus extending the criterion established by LaMotte (1973) for balanced models.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

On preliminary test almost unbiased two-parameter estimator in linear regression model with student's t errors

In this paper, the preliminary test approach to the estimation of the linear regression model with student's t errors is considered. The preliminary test almost unbiased two-parameter estimator is proposed, when it is suspected that the regression parameter may be restricted to a constraint. The quadratic biases and quadratic risks of the proposed estimators are derived and compared under both null and alternative hypotheses. The conditions of superiority of the proposed estimators for departure parameter and biasing parameters k and d are derived, respectively. Furthermore, a real data example and a Monte Carlo simulation study are provided to illustrate some of the theoretical results.     

The score function and a comparison of various adjustments of the profile likelihood

Several adjustments of the profile likelihood have the common effect of reducing the bias of the associated score function. Hence expansions for the adjusted score functions differ by a term, D_ξ, that has small asymptotic order (n ^?½). The effect of D_ξ on other quantities of interest is studied. In particular, we find the bias and variance of the adjusted maximum-likelihood estimate to be relatively unaffected, while differences in the Bartlett correction depend on D_ξ in a simple way.     

Quasi-minimax estimation in the partial linear model

This article discusses the minimax estimator in partial linear model y = Zβ + f + ε under ellipsoidal restrictions on the parameter space and quadratic loss function. The superiority of the minimax estimator over the two-step estimator is studied in the mean squared error matrix criterion.     

Limiting admissible estimators for variance components

Simultaneous estimation of the vector of the variance components for mixed and random models under the quadratic loss function is considered. For a large class of such models there are identified classes of admissible biased invariant quadratic estimators that are better than some admissible unbiased estimators. Numerous numerical results presented in the paper show that for many of the commonly used balanced models the improvements in the quadratic risk may be considerable over a large set of the parameter space.     

VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two‐phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s₁ is drawn according to a specific sampling design p(s₁) , and auxiliary data x are observed for the units i∈s₁ . Given the first‐phase sample s₁ , a second‐phase sample s₂ is selected from s₁ according to a specified sampling design {p(s₂∣s₁) } , and (y, x) is observed for the units i∈s₂ . In some cases, the population totals of some components of x may also be known. Two‐phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson‐type variance estimators are used for variance estimation. However, the Horvitz–Thompson ( Horvitz & Thompson, J. Amer. Statist. Assoc. 1952 ) variance estimator in uni‐phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non‐negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy ( Sen , J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy , J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two‐phase sampling, assuming fixed first‐phase sample size and fixed second‐phase sample size given the first‐phase sample. We apply the new variance estimators to two‐phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy‐type variance estimators of the two‐phase regression estimators that make use of the first‐phase auxiliary data and known population totals of some of the auxiliary variables.     

Estimating parameters of a selected Pareto population

Let Π₁,…,Π_k be k populations with Π_i being Pareto with unknown scale parameter α_i and known shape parameter β_i;i=1,…,k. Suppose independent random samples (X_i1,…,X_in), i=1,…,k of equal size are drawn from each of k populations and let X_i denote the smallest observation of the ith sample. The population corresponding to the largest X_i is selected. We consider the problem of estimating the scale parameter of the selected population and obtain the uniformly minimum variance unbiased estimator (UMVUE) when the shape parameters are assumed to be equal. An admissible class of linear estimators is derived. Further, a general inadmissibility result for the scale equivariant estimators is proved.     

Estimators of location based on Kolmogorov-Smirnov-type statistics

Two classes of estimators of a location parameter ø₀ are proposed, based on a nonnegative functional H₁^* of the pair (D₁^ø_N, G^ø_N), where and where F_N is the sample distribution function. The estimators of the first class are defined as a value of ø minimizing H₁^*; the estimators of the second class are linearized versions of those of the first. The asymptotic distribution of the estimators is derived, and it is shown that the Kolmogorov-Smirnov statistic, the signed linear rank statistics, and the Cramérvon Mises statistics are special cases of such functionals H₁^*;. These estimators are closely related to the estimators of a shift in the two-sample case, proposed and studied by Boulanger in B2 (pp. 271–284).     

Quadratic subspaces and construction of Bayes invariant quadratic estimators of variance components in mixed linear models

Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form where the matrices V _i , 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V _k is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV ₁ M,...,MV _k-1 M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV ₁ M,...,MV _k-1 M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV ₁ M,...,MV _k-1 M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. Dedicated to the memory of Professor Stanisław Gnot.     

Estimators of shift based on statistics of the Kolmogorov-Smirnov type

This paper is concerned with the estimation of a shift parameter δ_o, based on some nonnegative functional Hg₁ of the pair (D^δ_N(x), f?^δ_N(x)), where D^δ_N(x) = K_N/b {F_2,n(x)—F_1,m (x + δ)}, +^δ_N(x) = {mF_1,m (x + δ) + nF_2,n(x)}/N, where F_1,m and F_2,n are the empirical distribution functions of two independent random samples (N = m + n), and where K²_N = mn/N. First an estimator δ_N, is defined as a value of δ minimizing a functional H of the type of H₁. A second estimator δ¹_N is also defined which is a linearized version of the first. Finite and asymptotic properties of these estimators are considered. It is also shown that most well-known test statistics of the Kolmogorov-Smirnov type are particular cases of such functionals H₁. The asymptotic distribution and the asymptotic efficiency of some estimators are given.     

Estimating the mean of the selected uniform population

Let π_i(i=1,2,…K) be independent U(0,?_i) populations. Let Y_i denote the largest observation based on a random sample of size n from the i-th population. for selecting the best populaton, that is the one associated with the largest ?_i, we consider the natural selection rule, according to which the population corresponding to the largest Y_i is selected. In this paper, the estimation of M. the mean of the selected population is considered. The natural estimator is positively biased. The UMVUE (uniformly minimum variance unbiased estimator) of M is derived using the (U,V)-method of Robbins (1987) and its asymptotic distribution is found. We obtain a minimax estimator of M for K≤4 and a class of admissible estimators among those of the form cY_max. For the case K = 2, the UMVUE is improved using the Brewster-Zidek (1974) Technique with respect to the squared error loss function L₁ and the scale-invariant loss function L₂. For the case K = 2, the MSE'S of all the estimators are compared for selected values of n and ρ=?₁/(?₁+?₂).     

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.     

Some Estimators of Finite Population Variance of Stratified Sample Mean

We consider a continuous-time branching random walk on Z ^d , where the particles are born and die at a single lattice point (the source of branching). The underlying random walk is assumed to be symmetric. Moreover, corresponding transition rates of the random walk have heavy tails. As a result, the variance of the jumps is infinite, and a random walk may be transient even on low-dimensional lattices (d = 1, 2). Conditions of transience for a random walk on Z ^d and limit theorems for the numbers of particles both at an arbitrary point of the lattice and on the entire lattice are obtained.     

Estimation of the mean of the selected gamma population

Let л₁ and л₂ denote two independent gamma populations G(α₁, p) and G(α₂, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Y_i denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Y_i is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c₁x₍₁₎ +c₁x₍₂₎ and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)^-1x₍₁₎ is minimax and dominates the UMVUE. Also UMVUE is not minimax.     

Independent Quadratic Forms in 3-Variance-Component Models

This article addresses derivation and existence of quadratic forms that were suggested by Burch (2007 Burch , B. D. ( 2007 ). Generalized confidence intervals for proportions of total variance in mixed linear models . J. Statist. Plann. Infer. 137 : 2394 – 2404 .[Crossref], [Web of Science ®] , [Google Scholar]) for procedures for inference on variance components in mixed linear models in combination with generalized fiducial inference. A relatively simple algorithm leading to the required quadratic forms in a general 3-variance-component model is stated and designs for two-way ANOVA models without interactions that permit Burch's procedure are characterized. This complements developments in the original article by Burch.     

Estimating a positive normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal population with mean μ and variance σ ². In many real life situations, specially in lifetime or reliability estimation, the parameter μ is known a priori to lie in an interval [a, ∞). This makes the usual maximum likelihood estimator (MLE) ̄ an inadmissible estimator of μ with respect to the squared error loss. This is due to the fact that it may take values outside the parameter space. Katz (1961) and Gupta and Rohatgi (1980) proposed estimators which lie completely in the given interval. In this paper we derive some new estimators for μ and present a comparative study of the risk performance of these estimators. Both the known and unknown variance cases have been explored. The new estimators are shown to have superior risk performance over the existing ones over large portions of the parameter space.