Minimax subset selection for the multinomial distribution

Let (X₁,…,X_k) be a multinomial vector with unknown cell probabilities (p₁,?,p_k). A subset of the cells is to be selected in a way so that the cell associated with the smallest cell probability is included in the selected subset with a preassigned probability, P¹. Suppose the loss is measured by the size of the selected subset, S. Using linear programming techniques, selection rules can be constructed which are minimax with respect to S in the class of rules which satisfy the P¹-condition. In some situations, the rule constructed by this method is the rule proposed by Nagel (1970). Similar techniques also work for selection in terms of the largest cell probability.     

Test for Parameter Change in Linear Processes Based on Whittle's Estimator

In this article, we develop a cusum test for testing for parameter changes in linear processes based on Whittle's estimator. It is shown that under regularity conditions, the test statistic converges to the sup of a Brownian bridge. The result is particularly useful in handling the change point test in stationary ARMA processes. A simulation result is provided for illustration.     

A New Method for Testing Interaction in Unreplicated Two-Way Analysis of Variance

For a two-way ANOVA table, with a single observation per cell, the standard approach is to assume that interaction between the two factors is negligible, and to base inferences about the main factors on the model without interaction. But there is no totally satisfactory method for testing if interaction can be ignored. The classical approach is to specify a functional form for the interaction terms, involving a small number of parameters, and then use an appropriate test. But, such tests have low power if the functional form is inappropriate. This has led researchers to propose tests which do not assume a specific form for the interactions. In this article, we present a new approach for testing interaction which also does not assume a specific form for the interaction. This approach is fairly simple and flexible, and its usefulness is illustrated with several examples. We also present a general result which shows that there is no test of interaction with good power properties against all types of interaction.     

IPPS Sampling Plans Excluding Adjacent Units

The concept of inclusion probability proportional to size sampling plans excluding adjacent units separated by at most a distance of m (≥ 1) units {IPPSEA plans} is introduced. IPPSEA plans ensure that the first-order inclusion probabilities of units are proportional to size measures of the units, while the second-order inclusion probabilities are zero for pairs of adjacent units separated by a distance of m units or less. IPPSEA plans have been obtained by making use of binary, proper, and unequireplicated block designs and linear programing approach. The performance of IPPSEA plans using Horvitz–Thompson estimator of population total has been compared with existing sampling plans such as simple random sampling without replacement (SRSWOR), balanced sampling plans excluding adjacent units {BSA (m) plans}, probability proportional to size with replacement, Hartley and Rao's plan (1962 Hartley , H. O. , Rao , J. N. K. ( 1962 ). Sampling with unequal probabilities and without replacement . Ann. Math. Statist. 33 : 350 – 374 .[Crossref] , [Google Scholar]), Rao et al.'s strategy (1962 Rao , J. N. K. , Hartley , H. O. , Cochran , W. G. ( 1962 ). On a simple procedure of unequal probability sampling without replacement . J. Roy. Statist. Soc. B 24 : 482 – 491 . [Google Scholar]), and Sampford's IPPS plan (1967 Sampford , M. R. ( 1967 ). On sampling without replacement with unequal probabilities of selection . Biometrika 54 ( 3 ): 499 – 513 .[Crossref], [PubMed] , [Google Scholar]) using a real life population. Unbiased estimation of Horvitz–Thompson estimator of population total is not possible in these types of plans because some of the second-order inclusion probabilities are zero. To resolve this problem, one approximate variance estimation technique has been suggested.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

On the Use of Superpopulation Model in Estimation of Population Mean in Two-Occasion Rotation Patterns

The present work is an attempt to study the estimation of the population mean on the current occasion in two-occasion successive (rotation) sampling under a superpopulation model. Six different estimators are proposed for estimating the current population mean in two-occasion successive (rotation) sampling. Optimum replacement policies and performances of the proposed estimators have been discussed. Results are interpreted via empirical studies.     

Akaike Information Criterion for Selecting Variables in the Nested Error Regression Model

The Akaike Information Criterion (AIC) is developed for selecting the variables of the nested error regression model where an unobservable random effect is present. Using the idea of decomposing the likelihood into two parts of “within” and “between” analysis of variance, we derive the AIC when the number of groups is large and the ratio of the variances of the random effects and the random errors is an unknown parameter. The proposed AIC is compared, using simulation, with Mallows' C _p , Akaike's AIC, and Sugiura's exact AIC. Based on the rates of selecting the true model, it is shown that the proposed AIC performs better.     

On the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distribution

The Lomax (Pareto II) distribution has found wide application in a variety of fields. We analyze the second-order bias of the maximum likelihood estimators of its parameters for finite sample sizes, and show that this bias is positive. We derive an analytic bias correction which reduces the percentage bias of these estimators by one or two orders of magnitude, while simultaneously reducing relative mean squared error. Our simulations show that this performance is very similar to that of a parametric bootstrap correction based on a linear bias function. Three examples with actual data illustrate the application of our bias correction.