Approximations for a fixed sample size selection procedure for negative binomial populations

A large sample approximation of the least favorable configuration for a fixed sample size selection procedure for negative binomial populations is proposed. A normal approximation of the selection procedure is also presented. Optimal sample sizes required to be drawn from each population and the bounds for the sample sizes are tabulated. Sample sizes obtained using the approximate least favorable configuration are compared with those obtained using the exact least favorable configuration. Alternate form of the normal approximation to the probability of correct selection is also presented. The relation between the required sample size and the number of populations involved is studied.     

Multinomial Selection for Comparison with a Standard

The multinomial selection problem is considered under the formulation of comparison with a standard, where each system is required to be compared to a single system, referred to as a “standard,” as well as to other alternative systems. The goal is to identify systems that are better than the standard, or to retain the standard when it is equal to or better than the other alternatives in terms of the probability to generate the largest or smallest performance measure. We derive new multinomial selection procedures for comparison with a standard to be applied in different scenarios, including exact small-sample procedure and approximate large-sample procedure. Empirical results and the proof are presented to demonstrate the statistical validity of our procedures. The tables of the procedure parameters and the corresponding exact probability of correct selection are also provided.     

A stochastic approximation algorithm for maximum-likelihood estimation with incomplete data

We propose a new stochastic approximation (SA) algorithm for maximum-likelihood estimation (MLE) in the incomplete-data setting. This algorithm is most useful for problems when the EM algorithm is not possible due to an intractable E-step or M-step. Compared to other algorithm that have been proposed for intractable EM problems, such as the MCEM algorithm of Wei and Tanner (1990), our proposed algorithm appears more generally applicable and efficient. The approach we adopt is inspired by the Robbins-Monro (1951) stochastic approximation procedure, and we show that the proposed algorithm can be used to solve some of the long-standing problems in computing an MLE with incomplete data. We prove that in general O(n) simulation steps are required in computing the MLE with the SA algorithm and O(n log n) simulation steps are required in computing the MLE using the MCEM and/or the MCNR algorithm, where n is the sample size of the observations. Examples include computing the MLE in the nonlinear error-in-variable model and nonlinear regression model with random effects.     

Analyzing continuous randomized response data with an indifference-zone selection procedure

A review of the randomized response model introduced by Warner (1965) is given, then a randomized response model applicable to continuous data that considers a mixture of two normal distributions is considered. The target here is not to estimate any parameter, but rather to select the population with the best parameter value. This article provides a study on how to choose the best population between k distinct populations using an indifference-zone procedure. Also, this article includes tables for the required sample size needed in order to have a probability of correct selection higher than some specified value in the preference zone for the randomized response model considered.     

A subset selection procedure for multinomial distributions

A subset selection procedure is developed for selecting a subset containing the multinomial population that has the highest value of a certain linear combination of the multinomial cell probabilities; such population is called the ‘best’. The multivariate normal large sample approximation to the multinomial distribution is used to derive expressions for the probability of a correct selection, and for the threshold constant involved in the procedure. The procedure guarantees that the probability of a correct selection is at least at a pre-assigned level. The proposed procedure is an extension of Gupta and Sobel's [14] selection procedure for binomials and of Bakir's [2] restrictive selection procedure for multinomials. One illustration of the procedure concerns population income mobility in four countries: Peru, Russia, South Africa and the USA. Analysis indicates that Russia and Peru fall in the selected subset containing the best population with respect to income mobility from poverty to a higher-income status. The procedure is also applied to data concerning grade distribution for students in a certain freshman class.     

Analyzing Hybrid Randomized Response Data with a Binomial Selection Procedure

The operating characteristics (OCs) of an indifference-zone ranking and selection procedure are derived for randomized response binomial data. The OCs include tables and figures to facilitate tradeoffs between sample size and a stated probability of a correct selection, i.e., correctly identifying the binomial population (out of k ≥ 2) characterized by the largest probability of success. Measures of efficiency are provided to assist the analyst in selection of an appropriate randomized response design for the collection of the data. A hybrid randomized response model, which includes the Warner model and the Greenberg et al. model, is introduced to facilitate comparisons among a wider range of statistical designs than previously available. An example comparing failure rates of contraceptive methods is used to illustrate the use of these new results.     

Selection of the best with a preliminary test for two-parameter exponential distributions

This paper deals with the problem of selecting the best population from among k(≥ 2) two-parameter exponential populations. New selection procedures are proposed for selecting the unique best. The procedures include preliminary tests which allow the xperimenter to have an option to not select if the statistical evidence is not significant. Two probabilities, the probability to make a selection and the probability of a correct selection, are controlled by these selection procedures. Comparisons between the proposed selection procedures and certain earlier existing procedures are also made. The results show the superiority of the proposed selection procedures in terms of the required sample size.     

SHOULD WE ESTIMATE THE PROBABILITY OF CORRECT SELECTION?

Various authors, given k location parameters, have considered lower confidence bounds on (standardized) dserences between the largest and each of the other k - 1 parameters. They have then used these bounds to put lower confidence bounds on the probability of correct selection (PCS) in the same experiment (as was used for finding the lower bounds on differences). It is pointed out that this is an inappropriate inference procedure. Moreover, if the PCS refers to some later experiment it is shown that if a non-trivial confidence bound is possible then it is already possible to conclude, with greater confidence, that correct selection has occurred in the first experiment. The short answer to the question in the title is therefore ‘No’, but this should be qualified in the case of a Bayesian analysis.     

On a sequential subset selection procedure for the least probable multinomial cell

This paper studies a sequential procedure R for selecting a random size subset that contains the multinomial cell which has the smallest cell probability. The stopping rule of the proposed procedure R is the composite of the stopping rules of curtailed sampling, inverse sampling, and the Ramey-Alam sampling. A reslut on the worst configuration is shown and it is employed in computing the procedure parameters that guarantee certain probability requirements. Tables of these procedure parameters, the corresponding probability of correct selection, the expected sample size, and the expected subset size are given for comparison purpose.     

Ultimate Extinction of the Promiscuous Bisexual Galton—Watson Metapopulation

A variant of a sexual Gallon–Watson process is considered. At each generation the population is partitioned among n‘hosts’ (population patches) and individual members mate at random only with others within the same host. This is appropriate for many macroparasite systems, and at low parasite loads it gives rise to a depressed rate of reproduction relative to an asexual system, due to the possibility that females are unmated. It is shown that stochasticity mitigates against this effect, so that for small initial populations the probability of ultimate extinction (the complement of an ‘epidemic’) displays a tradeoff as a function of n between the strength of fluctuations which overcome this ‘mating’ probability, and the probability of the subpopulation in one host being ‘rescued’ by that in another. Complementary approximations are developed for the extinction probability: an asymptotically exact approximation at large n, and for small n a short‐time probability that is exact in the limit where the mean number of offspring per parent is large.     

Smooth-Threshold GEE Variable Selection in High-Dimensional Partially Linear Models with Longitudinal Data

We consider the problem of variable selection in high-dimensional partially linear models with longitudinal data. A variable selection procedure is proposed based on the smooth-threshold generalized estimating equation (SGEE). The proposed procedure automatically eliminates inactive predictors by setting the corresponding parameters to be zero, and simultaneously estimates the nonzero regression coefficients by solving the SGEE. We establish the asymptotic properties in a high-dimensional framework where the number of covariates p_n increases as the number of clusters n increases. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedure.     

A distribution-free approach to qdantal response assays

A nonparametric method is developed to estimate the minimum dosage level required to induce a given response rate in an experiment. The only assumption used about the response rate is that it is a nondecreasing function with respect to the dosage level. Let n_isubjects be independently tested at dosage level x_ix₁x₂x_k. This paper presents methodology for the estimation of the smallest i such that the response probability at x_i is no less than a required level p. A comparison with well-known nonparametric methods shows that this method is better in some cases. A design of minimum required sample size for a given accuracy is also developed.     

A selection procedure for estimating the number of signal components

This paper proposes a selection procedure to estimate the multiplicity of the smallest eigenvalue of the covariance matrix. The unknown number of signals present in a radar data can be formulated as the difference between the total number of components in the observed multivariate data vector and the multiplicity of the smallest eigenvalue. In the observed multivariate data, the smallest eigenvalues of the sample covariance matrix may in fact be grouped about some nominal value, as opposed to being identically equal. We propose a selection procedure to estimate the multiplicity of the common smallest eigenvalue, which is significantly smaller than the other eigenvalues. We derive the probability of a correct selection, P(CS), and the least favorable configuration (LFC) for our procedures. Under the LFC, the P(CS) attains its minimum over the preference zone of all eigenvalues. Therefore, a minimum sample size can be determined from the P(CS) under the LFC, P(CS|LFC), in order to implement our new procedure with a guaranteed probability requirement. Numerical examples are presented in order to illustrate our proposed procedure.     

Some Aspects of Probability of Correct Selections

In an experiment of treatment selections, random samples are drawn from k populations with ordered means. The probability that a sample statistic from the population with the highest mean turns out to be ranked the highest is referred to as the probability of correct selection (PCS). An inequality was proved previously that shows the monotonicity of PCS with respect to change in variance of the samples. In this article, we first present a more general form of the probability inequality to be used to investigate PCS. An extension of the monotonicity of PCS to order statistics is considered. We show that the PCS of the smallest order statistic preserves the monotonicity. Additionally, a normal approximation method is used to further generalize the theory. The general order statistics will not enjoy the same properties, as we reveal the obstacles, and a numerical counter example.     

One-sided tolerance limits for unbalanced one-way anova andomc effects model

In this article, we investigate techniques for constructing tolerance limits such that the probability is γ that at least _p proportion of the population would exceed that limit. We consider the unbalanced case and study the behavior of the limit as a function of n_i 's (where n_i is the number of observations in the ith batch), as well as that of the variance ratio. To construct the tolerance limits we use the approximation given in Thomas and Hultquist (1978). We also discuss the procedure for constructing the tolerance limits when the variance ratio is unknown. An example is given to illustrate the results.     

News about Members

In this article a natural extension of the beta-binomial distribution is developed. Forced binary choice situations are modeled such that each individual has a probability p of knowing the correct answer. (This probability is distributed f(p) across the population.) Hence each individual will guess at the correct answer with probability 1 – p. The observable random variable R, the total number of correct answers (both by knowing and guessing) out of k trials has a rather complicated distribution. However, when f(p) is distributed beta with parameters m and n, the distribution P(r; k, m, n) can be expressed in terms of the well-known Gaussian hypergeometric function. This distribution has implications for true-false tests, taste tests, and virtually every other forced binary choice situation.     

Selecting the normal population with the smallest variance: A restricted subset selection rule

Consider k( ? 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ²_[1] ? ??? ? σ_[k]² denote the ordered variances. Our goal is to select a non empty subset of the k populations whose size is at most m(1 ? m ? k ? 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probability P* whenever σ²_[2]/σ_[1]² ? δ* > 1, where P* and δ* are specified in advance of the experiment. Based on samples of size n from each of the populations, we propose and investigate a procedure called R_BCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example.     

Some results on bootstrap prediction intervals

We investigate the construction of a BC_a-type bootstrap procedure for setting approximate prediction intervals for an efficient estimator θ_m of a scalar parameter θ, based on a future sample of size m. The results are also extended to nonparametric situations, which can be used to form bootstrap prediction intervals for a large class of statistics. These intervals are transformation-respecting and range-preserving. The asymptotic performance of our procedure is assessed by allowing both the past and future sample sizes to tend to infinity. The resulting intervals are then shown to be second-order correct and second-order accurate. These second-order properties are established in terms of min(m, n), and not the past sample size n alone.     

GMM estimation in partial linear models with endogenous covariates causing an over-identified problem

ABSTRACT

We study partial linear models where the linear covariates are endogenous and cause an over-identified problem. We propose combining the profile principle with local linear approximation and the generalized moment methods (GMM) to estimate the parameters of interest. We show that the profiled GMM estimators are root? n consistent and asymptotically normally distributed. By appropriately choosing the weight matrix, the estimators can attain the efficiency bound. We further consider variable selection by using the moment restrictions imposed on endogenous variables when the dimension of the covariates may be diverging with the sample size, and propose a penalized GMM procedure, which is shown to have the sparsity property. We establish asymptotic normality of the resulting estimators of the nonzero parameters. Simulation studies have been presented to assess the finite-sample performance of the proposed procedure.     

Truncation of the Bechhofer-Kiefer-Sobel sequential procedure for selecting the multinomial event which has the largest probability

In this article we study the effect of truncation on the performance of an open vector-at-a-time sequential sampling procedure (P^* _B) proposed by Bechhofer, Kiefer and Sobel , for selecting the multinomial event which has the largest probability. The performance of the truncated version (P^{* B} _T) is compared to that of the original basic procedure (P^* _B). The performance characteristics studied include the probability of a correct selection, the expected number of vector-observations (n) to terminate sampling, and the variance of n. Both procedures guarantee the specified probability of a correct selection. Exact results and Monte Carlo sampling results are obtained. It is shown that P^* _{B T}is far superior to P^* _B in terms of E{n} and Var{n}, particularly when the event probabilities are equal.The performance of P^* _{B T} is also compared to that of a closed vector-at-a-time sequential sampling procedure proposed for the same problem by Ramey and Alam; this procedure has here to fore been claimed to be the best one for this problem. It is shown that p^* _{B T} is superior to the Ramey-Alam procedure for most of the specifications of practical interest.