M-Estimates of regression when the scale is unknown and the error distribution is possibly asymmetric: A minimax result

Huber (1964) found the minimax-variance M-estimate of location under the assumption that the scale parameter is known; Li and Zamar (1991) extended this result to the case when the scale is unknown. We consider the robust estimation of the regression coefficients (β₁,…,β_p) when the scale and the intercept parameters are unknown. The minimax-variance estimates of (β₁,…,β_p) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over ?-contamination neighbourhoods of a central regression model with Gaussian errors (asymmetric contamination is allowed), and the minimum is taken over a large class of generalized M-estimates of regression of the Mallow type. The optimal choice of estimates for the nuisance parameters (scale and intercept) is also considered.     

Order statistics from extreme value distribution,ii: best linear unbiased estimates and some other uses

Let Y_r+1:n ≤ _Y:r+2:n ≤≤… <Y_n?6:n-<: TYPE-II censored sample from an extreme value population with µ and α as the location and scale parameters, respectively. Tables of coefficients for the best linear unbiased estimators (BLUEs) of µ and α are presented for various choices of censoring and sample sizes n = 2(1)15(5)30; variances and covariance of these estimators are also presented. The computational formulae and procedure used and some checks employed are explained. We finally illustrate some uses of the tables by taking examples.     

Recurrence relations for single and product moments of progressively Type-II censored order statistics from generalized logistic distribution with applications to inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a generalized logistic distribution. The use of these relations in a systematic manner allow us to compute all the means, variances, and covariances of progressively Type-II right censored order statistics from the generalized logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁, …, R_m). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the generalized logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made through Monte Carlo simulations. Finally, the best linear unbiased predictors of censored failure times is discussed briefly.     

Correctings tests for trend in proportions for skewness

We develop the score test for the hypothesis that a parameter of a Markov sequence is constant over time, against the alternatives that it varies over time, i.e., θ_t = θ + U_t; t = 1,2,…, where {U_t; t = 1,2,...} is a sequence of independently and identically distributed random variables with mean zero and variance σ^z _u and θ is a fixed constant. The asymptotic null distribution of the test statistic is proved to be normal. We illustrate our procedure by examples and a real life data analysis.     

Linear predictors of future order statistics

In this paper we establish an optimal asymptotic linear predictor which does not involve the finite-sample variance-covariance structure. Extensions to the problem of finding the best linear unbiased and simple linear unbiased predictors for k samples are given. Moreover, we obtain alternative linear predictors by modifying the covariance matrix by either an identity matrix or a diagonal matrix. For normal, logistic and Rayleigh samples of size 10, the alternative linear predictors with these modifications have high efficiency when compared with the best linear unbiased predictor.     

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.     

The covariance of the number of renewals in a fixed time and the ensuing excess life

The covariance of the number of renewals in a fixed time N ^t and the ensuing excess life time Y ^t is derived using matrix-analytic methods for the stationary PH-renewal process. Specific results for the Erlang and hyperexponential processes are provided to illustrate the ease of computation. Properties concerning the sign and the behavior of the covariance as t→∞ are provided throughout. Parameter estimation for renewal processes which cannot be fully observed serves as the motivation for our derivations. These statistical applications as well as links to estimation for service time distributions in queues shed light on the type of problems for which the covariance is useful.     

A LINEAR MODEL WITH ERRORS LACKING A VARIANCE II1

Numerical results are presented for estimates of the parameters in the linear model Y =βX +ε in which X is normally distributed and ε is symmetric stable. The study complements an earlier paper of the same title and the main concern is with numerical comparisons between four estimates of β; the least squares estimate, the minimum absolute deviations estimate, and two moment estimates of the form derived in Chambers and Heathcote (1975). The generation of fifty independent sets of observations (X_j, Y_j), j = 1,2, …, n for each of n = 100, 500 and selected combinations of parameter values provided the basis of the results. It is indicated that the moment estimators and the minimum absolute deviation estimator performed comparably, and are a significant improvement on the least squares estimator. The main conclusion is that one of the moment estimates, based on a two stage adaptive procedure and denoted by β¯_n(t_a) below, is generally the most useful of the four.     

Estimating the parameter of selected uniform population under the squared log error loss function

Let π₁, …, π_k be k (? 2) independent populations, where π_i denotes the uniform distribution over the interval (0, θ_i) and θ_i > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θ_i, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θ_L of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θ_L under the SLE loss function and two natural estimators of θ_L are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θ_L. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θ_L are compared through simulation.     

The Distribution of Sums of Certain I.I.D. Pareto Variates

ABSTRACT

Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)^α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.     

A comparison of some nonparametric tests for the simple tree alternative

This paper develops a new test statistic for testing hypotheses of the form H_O M₀=M₁=…=M_k versus H_a: M₀≦M₁,M₂,…,M_k] where at least one inequality is strict. M₀ is the median of the control population and M₁ is the median of the population receiving trearment i, i=1,2,…,k. The population distributions are assumed to be unknown but to differ only in their location parameters if at all. A simulation study is done comparing the new test statistic with the Chacko and the Kruskal-Wallis when the underlying population distributions are either normal, uniform, exponential, or Cauchy. Sample sizes of five, eight, ten, and twenty were considered. The new test statistic did better than the Chacko and the Kruskal-Wallis when the medians of the populations receiving the treatments were approximately the same     

Bayes Estimation of Shift Point in Geometric Sequence

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,… X _n were observed from geometric population with parameter q ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in parameter q ₂. The Bayes estimates of m, q ₁, q ₂, reliability R ₁ (t) and R ₂ (t) at time t are derived for symmetric and asymmetric loss functions under informative and non informative priors. A simulation study is carried out.     

A RENEWAL THEOREM IN MULTIDIMENSIONAL TIME

Let Y_l, Y₂,… be i.i.d., positive, integer-valued random variables with means, μ. Let the sequences {Y_ij, j= 1,2,…}, i= 1,…, r be independent copies of {Y₁, Y₂,…}. For n={n₁,…, n_r.}, n₁≥1, let S_n=S?ⁿ¹_k1=1= 1 …S?^nr_kr=1 Y_ik1… Y_rkr. We show that S?^N_k=1S?^∞_k1=1…S?^∞_nr=1 P[[S_n= k] ? [μ^-r N log^r-1 (N)/(r-1)!] as N →∞.     

On misspecification of the dispersion matrix in mixed linear models

The general mixed linear model can be written y = Xβ + Zu + e, where β is a vector of fixed effects, u is a vector of random effects and e is a vector of random errors. In this note, we mainly aim at investigating the general necessary and sufficient conditions under which the best linear unbiased estimator for \varvec r(\varvec l, \varvec m) = \varvec l\varvec ￠\varvec b+\varvec m\varvec ￠\varvec u{\varvec \varrho}({\varvec l}, {\varvec m}) = {\varvec l}{\varvec '}{\varvec \beta}+{\varvec m}{\varvec '}{\varvec u} is also optimal under the misspecified model. In addition, we offer approximate conclusions in some special situations including a random regression model.     

On estimating the reliability in a multicomponent stress-strength model based on Chen distribution

Abstract

In this article, we obtain point and interval estimates of multicomponent stress-strength reliability model of an s-out-of-j system using classical and Bayesian approaches by assuming both stress and strength variables follow a Chen distribution with a common shape parameter which may be known or unknown. The uniformly minimum variance unbiased estimator of reliability is obtained analytically when the common parameter is known. The behavior of proposed reliability estimates is studied using the estimated risks through Monte Carlo simulations and comments are obtained. Finally, a data set is analyzed for illustrative purposes.     

Hypothesis Testing and Confidence Regions for the Mean Sojourn Time of an M/M/1 Queueing System

This article discusses testing hypotheses and confidence regions with correct levels for the mean sojourn time of an M/M/1 queueing system. The uniformly most powerful unbiased tests for three usual hypothesis testing problems are obtained and the corresponding p values are provided. Based on the duality between hypothesis tests and confidence sets, the uniformly most accurate confidence bounds are derived. A confidence interval with correct level is proposed.     

Likelihood ratio inference for mean residual life

One of the basic parameters in survival analysis is the mean residual life M ₀. For right censored observation, the usual empirical likelihood based log-likelihood ratio leads to a scaled c₁²{\chi_1^2} limit distribution and estimating the scaled parameter leads to lower coverage of the corresponding confidence interval. To solve the problem, we present a log-likelihood ratio l(M ₀) by methods of Murphy and van der Vaart (Ann Stat 1471–1509, 1997). The limit distribution of l(M ₀) is the standard c₁²{\chi_1^2} distribution. Based on the limit distribution of l(M ₀), the corresponding confidence interval of M ₀ is constructed. Since the proof of the limit distribution does not offer a computational method for the maximization of the log-likelihood ratio, an EM algorithm is proposed. Simulation studies support the theoretical result.     

Explicit Formulae for the Queue Length Distribution of Batch Arrival Systems

Abstract

A G ^θ I/G/1-type batch arrival system is considered. Explicit formulae for the distribution of queue length both at the fixed time t and as t → ∞ are obtained. The study is based on the generalization of Korolyuk's method for semi-markov random walks.     

A Robust Variable Selection to t-type Joint Generalized Linear Models via Penalized t-type Pseudo-likelihood

Although the t-type estimator is a kind of M-estimator with scale optimization, it has some advantages over the M-estimator. In this article, we first propose a t-type joint generalized linear model as a robust extension to the classical joint generalized linear models for modeling data containing extreme or outlying observations. Next, we develop a t-type pseudo-likelihood (TPL) approach, which can be viewed as a robust version to the existing pseudo-likelihood (PL) approach. To determine which variables significantly affect the variance of the response variable, we then propose a unified penalized maximum TPL method to simultaneously select significant variables for the mean and dispersion models in t-type joint generalized linear models. Thus, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the mean and dispersion models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies are conducted to illustrate the proposed methods.