Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

Best Quadratic Unbiased Prediction in a General Linear Model with Stochastic Regression Coefficients

In this article, we discuss on how to predict a combined quadratic parametric function of the form β′ H β + hσ² in a general linear model with stochastic regression coefficients denoted by y  =  X β +  e . Firstly, the quadratic predictability of β′ H β + hσ² is investigated to obtain a quadratic unbiased predictor (QUP) via a general method of structuring an unbiased estimator. This QUP is also optimal in some situations and therefore we hope it will be a fine predictor. To show this idea, we apply the Lagrange multipliers method to this problem and finally reach the expected conclusion through permutation matrix techniques.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

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.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

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.     

Estimation of P(Y < X) for the Three-Parameter Generalized Exponential Distribution

This article considers the estimation of R = P(Y < X) when X and Y are distributed as two independent three-parameter generalized exponential (GE) random variables with different shape parameters but having the same location and scale parameters. A modified maximum likelihood method and a Bayesian technique are used to estimate R on the basis of independent complete samples. The Bayes estimator cannot be obtained in explicit form, and therefore it has been determined using an importance sampling procedure. An analysis of a real life data set is presented for illustrative purposes.     

Precise Large Deviations for Sums of Independent Random Variables with Consistently Varying Tails

In this article, we investigate the precise large deviations for a sum of independent but not identical distributed random variables. {X _n , n ≥ 1} are independent non-negative random variables with distribution functions {F _n , n ≥ 1}. We assume that the average of right tails of distribution functions F _n is equivalent to some distribution function F with consistently varying tails. In applications, we apply our main results to a realistic example (Pareto-type distribution) and obtain a specific result.     

On Thresholds of Moving Averages with Given On-Target Significant Levels

Let X ₁, X ₂,… be a sequence of independent and identically distributed random variables, and let Y _n , n = K, K + 1, K + 2,… be the corresponding backward moving average of order K. At epoch n ≥ K, the process Y _n will be off target by the input X _n if it exceeds a threshold. By introducing a two-state Markov chain, we define a level of significance (1 ? a)% to be the percentage of times that the moving average process stays on target. We establish a technique to evaluate, or estimate, a threshold, to guarantee that {Y _n } will stay (1 ? a)% of times on target, for a given (1 ? a)%. It is proved that if the distribution of the inputs is exponential or normal, then the threshold will be a linear function in the mean of the distribution of inputs μ _X . The slope and intercept of the line, in each case, are specified. It is also observed that for the gamma inputs, the threshold is merely linear in the reciprocal of the scale parameter. These linear relationships can be easily applied to estimate the desired thresholds by samples from the inputs.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

Prediction intervals for growth curves

The growth curve model Y^n×p = A^{n×p ξ mtimes;k}B^k×p+ E^nxp, where Y is an observation matrix, &sigma is a matrix of unknown parameters, A is a known matrix of rank m, B is a known matrix of rank k with 1'= (1, …, 1) as its first row, and the rows of E are independent each distributed as N^p(0,Σ,) is considered. The problem of constructing the prediction intervals for future observations using the above model is considered and approximate intervals assuming different structures on σ are derived. The results are illustrated with several data sets.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

On the Exponential Decay Rate of the Tail of a Discrete Probability Distribution

Abstract

We give a sufficient condition for the exponential decay of the tail of a discrete probability distribution π = (π _n ) _n≥0 in the sense that lim _n→∞(1/n) log∑ _i>n π _i  = ?θ with 0 < θ < ∞. We focus on analytic properties of the probability generating function of a discrete probability distribution, especially, the radius of convergence and the number of poles on the circle of convergence. Furthermore, we give an example of an M/G/1 type Markov chain such that the tail of its stationary distribution does not decay exponentially.     

Improved shrinkage estimators for the mean vector of a scale mixture of normals with unknown variance

The problem of estimating the mean θ of a not necessarily normal p-variate (p > 3) distribution with unknown covariance matrix of the form σ²A (A a known diagonal matrix) on the basis of n_i > 2 observations on each coordinate X_t (1 < i < p) is considered. It is argued that the class of scale (or variance) mixtures of normal distributions is a reasonable class to study. Assuming the loss function is quadratic, a large class of improved shrinkage estimators is developed in the case of a balanced design. We generalize results of Berger and Strawderman for one observation in the known-variance case. This methodology also permits the development of a new class of minimax shrinkage estimators of the mean of a p-variate normal distribution for an unbalanced design. Numerical calculations show that the improvements in risk can be substantial.     

Orthogonal polynomials generated by random vectors

Every random q-vector with finite moments generates a set of orthonormal polynomials. These are generated from the basis functions xn = xn11…xnqq using Gram–Schmidt orthogonalization. One can cycle through these basis functions using any number of ways. Here, we give results using minimum cycling. The polynomials look simpler when centered about the mean of X, and still simpler form when X is symmetric about zero. This leads to an extension of the multivariate Hermite polynomial for a general random vector symmetric about zero. As an example, the results are applied to the multivariate normal distribution.     

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.     

Locking the Wiener Process to its Level-Crossing Time

We consider the specific transformation of a Wiener process {X(t), t ≥ 0} in the presence of an absorbing barrier a that results when this process is “time-locked” with respect to its first passage time T _a through a criterion level a, and the evolution of X(t) is considered backwards (retrospectively) from T _a . Formally, we study the random variables defined by Y(t) ≡ X(T _a  ? t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series “response-locked” to their times of crossing a criterion level.