Stein's Two-stage Procedure Research Articles

A Three-Stage Selection Procedure With an Exact Consistency

This article considers the problem of selecting the best population by using Bechhofer's indifference zone approach. Stein's two-stage procedure meets the probability requirement (exact consistency). However, it is shown that the procedure does not become asymptotically second-order efficient whatever the first sample sizes are chosen. Hall's three-stage procedure is asymptotically second-order efficient, but it loses the exact consistency. We propose a three-stage procedure that has an exact consistency and show that the procedure enjoys the asymptotic second-order efficiency.

Improving the post-experiment properties of stein's two-stage procedure

We show that the properties of stein's(1945) two-stage test conditional on an observed ancillary stastistic are such that one may be concerned aboutt its performance in particular individual experiments. We present a new two-stage t-test which has preferable conditional properties under the null hypothesis and we describe a group sequential Stein-based test with appealing properties under null and alternative hypotheses.Study of our new two-stage test reveals a surprising feature of Stein's original procedure which illustrates how the requirements of conditional acceptability and admissibility may come into conflict.

On some sample size formulae for controlling both size and power in clinical trials

Suppose that observations are taken from a population with normal distribution N(μ, σ2) where both μ and σ2 are unknown parameters. Our goal is to design a two-sided test of H0: μ = 0 against Ha: μ≠ 0 which has, at least approximately, size α and power β at |μ| = d > 0, where α, β and d are three preassigned constants. The classical solution is Stein's two-stage procedure, which tends to oversampling, however. Three new procedures are proposed and studied in this paper. They all have both size and power very close to the target values α and β and require only a few more observations than the necessary sample size to achieve our goal if σ2 had been completely known.

A two-stage rank test using density estimation

For the one-sample problem, a two-stage rank test is derived which realizes a required power against a given local alternative, for all sufficiently smooth underlying distributions. This is achieved using asymptotic expansions resulting in a precision of orderm −1, wherem is the size of the first sample. The size of the second sample is derived through a number of estimators of e. g. integrated squared densities and density derivatives, all based on the first sample. The resulting procedure can be viewed as a nonparametric analogue of the classical Stein's two-stage procedure, which uses at-test and assumes normality for the underlying distribution. The present approach also generalizes earlier work which extended the classical method to parametric families of distributions.

Short note: Stein's two-stage procedure for the intra-class model

In his nice paper (Mykhopadhyay, 1982) as well as in his significant monograph (Mykhopadhyay & Solanky, 1994) N. Mykhopadhyay considers the following application of STEIN's two-stage procedure: Suppose that (X 1,..., Xn ) T , n = 1, 2,..., is n-dimensional normal with mean vector µ = µ l and dispersion matrix Σ n =σ 2(ρij ) with ρij = 1, ρij = ρ *, i ≠ j = 1,..., n where (µ, Σ, ρ) ∈ ℝ × ℝ+ × (-1, 0); this is called the intra-class model. For given d > 0 and α ∈ (0, 1) one wants to construct a (sequential) confidence interval I for µ having width 2d and confidence coefficient at least (1 - α). It is claimed that where N is determined, according to Stein's two-stage procedure (Stein, 1945), as where m ⩾ 2 is the first stage sample size and denotes the sample variance, fulfills this aim.

Asymptotic expansions for two-stage rank tests

Stein's two-stage procedure produces a t-test which can realize a prescribed power against a given alternative, regardless of the unknown variance of the underlying normal distribution. This is achieved by determining the size of a second sample on the basis of a variance estimate derived from the first sample. In the paper we introduce a nonparametric competitor of this classical procedure by replacing the t-test by a rank test. For rank tests, the most precise information available are asymptotic expansions for their power to order n-1, where n is the sample size. Using results on combinations of rank tests for sub-samples, we obtain the same level of precision for the two-stage case. In this way we can determine the size of the additional sample to the natural order and moreover compare the nonparametric and the classical procedure in terms of expected additional numbers of observations required.

Second Order Analysis of Two-Stage Rank Tests for the One-Sample Problem

In this paper we present a rank analogue to Stein's two-stage procedure. We analyze its behavior to second order using existing asymptotic expansions for fixed sample size rank tests and recent results on combinations of independent rank statistics.

An analysis of five simulation methods for determining the number of replications in a complex Monte Carlo study

In this article we summarize the results of a simulation study that compares and contrasts five different methods for selecting the number of replications necessary to accurately estimate a group of parameters utilizing the simple Monte Carlo method. The five methods are: an extension to several dimensions of Stein's two-stage procedure, an extension of Hall's three-stage procedure, a purely-sequential procedure that continuously monitors the simulation output, a continuously-monitoring procedure with a correction term, and a maximum eigenvalue procedure due to Srivastava. This study varies three design factors: the dimension of the simulated response (two or three), the number of replications necessary to be accurate (small, moderate and large sample sizes), and the kind of accuracy sets (rectangular or spherical). The simulation methods are compared on the basis of their probability of being accurate and on the basis of their efficiency, as measured by the distribution of the number of replications needed to be accurate. Typical distributional and density estimates are presented in both table and graph form.

The Robustness of Stein's Two-Stage Procedure

The expressions for coverage probability and ASN of the Stein's two-stage confidence interval procedure for estimating the normal mean have been obtained under the assumption that the underlying distribution is, in fact, different from normal but could be approximated by the first four terms of Edgeworth series. The comparison of coverage probabilities with the corresponding probabilities obtained for the normal distribution shows that the procedure is quite insensitive to moderate departures from normality, and skewness of the parent population has very little effect on the coverage probability of the procedure. The Monte Carlo investigations which involve sampling from the gamma population confirm these conclusions.

Stein's two-stage procedure and exact consistency

Abstract Stein (1945 and 1949) achieved the exact probability coverage for a fixed-width confidence interval estimation of a normal mean when the variance is unknown. To achieve this type of exact result of “consistency” (in the Chow—Robbins (1965) sense), we notice that the assumption of normal distributions for the population is not essential. It can be replaced by less restrictive conditions on independence and pivotal nature of some suitable statistics. This is our Theorem 1. Examples are provided from negative exponential, symmetric normal (Rao, 1973, pp. 196–198) and normal populations. A similar type of problem is discussed for inverse Gaussian (see Folks & Chhikara, 1978) parameters. Modified two-stage procedures are proposed along the lines of Mukhopadhyay (1980) and are shown to be asymptotically “first-order efficient” (in the Ghosh—Mukhopadhyay (1981) sense). We also develop and study some properties of two-stage fixed-width confidence intervals constructed along the lines of Birnbaum & Healy ...

Robustness of Stein's Two-Stage Procedure for Mixtures of Normal Populations

Abstract Explicit expressions for the level of significance and power of Stein's two-stage hypothesis testing procedure are derived when the normal population is mixed with another normal population differing in the mean. Approximate expansions for these expressions have been obtained, and numerical studies of the size have been carried out using the expansions. Some numerical indication of the accuracy of the approximations has also been obtained. Stein's two-stage procedure is quite robust against mixtures of normal populations differing in location parameters. Similar conclusions can be drawn regarding the coverage probabilities of confidence intervals generated by this procedure.

Robustness of Stein's Two-Stage Procedure for Mixtures of Normal Populations

Abstract Explicit expressions for the level of significance and power of Stein's two-stage hypothesis testing procedure are derived when the normal population is mixed with another normal population differing in the mean. Approximate expansions for these expressions have been obtained, and numerical studies of the size have been carried out using the expansions. Some numerical indication of the accuracy of the approximations has also been obtained. Stein's two-stage procedure is quite robust against mixtures of normal populations differing in location parameters. Similar conclusions can be drawn regarding the coverage probabilities of confidence intervals generated by this procedure.