Sequential estimation is a well recognized approach to inference in statistical theory. In sequential estimation the sample size to use is not specified at the start of the study, and instead study outcomes are used to evaluate a predefined stopping rule, if sampling should continue or stop. In this article we develop a general theory for sequential estimation procedure for constructing a narrow confidence interval for a general class of effect sizes with a specified level of confidence (e.g., 95%) and a specified upper bound on the confidence interval width. Our method does not require prespecified, yet usually unknowable, population values of certain parameters for certain types of distributions, thus offering advantages compared to commonly used approaches to sample size planning. Importantly, we make our developments in a distribution-free environment and thus do not make untenable assumptions about the population from which observations are sampled. Our work is thus very general, timely due to the interest in effect sizes, and has wide applicability in the context of estimation of a general class of effect sizes. (PsycINFO Database Record

The virtual dimensionality (VD) estimates the number of spectrally distinct materials in the hyperspectral data. The VD is generally considered as the number of signal sources under binary hypothesis, based on the Neyman-Pearson detection criteria. We observe that the hypothesis testing procedure used in many approaches is prone to inflated Type I (false positive) error. This is due to carrying out the binary hypothesis test individually on each band image, i.e., more than 200 images in hyperspectral data. In this paper, we propose multiple hypothesis testing to control the expected proportion of falsely rejected null hypotheses, i.e., false discovery rate (FDR), and in turn, improve the probability of better performance in estimating the VD. To this end, we employ Benjamini and Hochberg procedure that controls the FDR. We provide multiple hypothesis testing based algorithms to estimate VD wherein the hypothesis can be formulated according to eigenanalysis, the target specified by statistical approach, and by geometric analysis. The efficacies of the proposed algorithms are evaluated by estimating number of endmembers for the spectral unmixing application. We conduct experiments on four synthetic hyperspectral data sets at different noise levels as well as on two well-known real hyperspectral data sets. Time complexity and execution time are discussed to study the algorithmic aspects while sensitivity analyses of parameters are carried out for better performance analysis of the proposed approach. We found that the use of multiple hypothesis testing improves estimation of number of endmembers in hyperspectral data.

The coefficient of variation is an effect size measure with many potential uses in psychology and related disciplines. We propose a general theory for a sequential estimation of the population coefficient of variation that considers both the sampling error and the study cost, importantly without specific distributional assumptions. Fixed sample size planning methods, commonly used in psychology and related fields, cannot simultaneously minimize both the sampling error and the study cost. The sequential procedure we develop is the first sequential sampling procedure developed for estimating the coefficient of variation. We first present a method of planning a pilot sample size after the research goals are specified by the researcher. Then, after collecting a sample size as large as the estimated pilot sample size, a check is performed to assess whether the conditions necessary to stop the data collection have been satisfied. If not an additional observation is collected and the check is performed again. This process continues, sequentially, until a stopping rule involving a risk function is satisfied. Our method ensures that the sampling error and the study costs are considered simultaneously so that the cost is not higher than necessary for the tolerable sampling error. We also demonstrate a variety of properties of the distribution of the final sample size for five different distributions under a variety of conditions with a Monte Carlo simulation study. In addition, we provide freely available functions via the MBESS package in R to implement the methods discussed.

We revisit fixed-width (= 2d) confidence interval procedures with a preassigned confidence coefficient (≥1 − α) for the mean μ of a normal distribution when its variance σ² is unknown. Had σ been known, the required optimal fixed sample size would be C ≡ a ²σ²/d ² where a ≡ a _α is the upper 50α% point of N(0, 1). In his fundamental two-stage procedure, Stein ( 1945 Stein , C. ( 1945 ). A Two Sample Test for a Linear Hypothesis Whose Power Is Independent of the Variance , Annals of Mathematical Statistics 16 : 243 – 258 . [Crossref] [Google Scholar] , 1949 Stein , C. ( 1949 ). Some Problems in Sequential Estimation , Econometrica 17 : 77 – 78 . [Google Scholar] ) estimated C by replacing σ² with a sample variance from pilot data of size m(≥2) and a with t _{m−1, α/2}. Mukhopadhyay ( 1982 Mukhopadhyay , N. ( 1982 ). Stein's Two-Stage Procedure and Exact Consistency, Skandinavisk Aktuarietdskrift pp. 110 – 122 . [Google Scholar] ) opened the possibility of incorporating less traditional estimators of σ². We focus on estimating σ² by a statistic defined via Gini's mean difference (GMD), mean absolute deviation (MAD), and range. Such modifications are warranted especially when we suspect one or more outliers but normality of the data may not be questioned by a standard test for normality assumption. Obviously, then, t _{m−1, α/2} must be replaced by the upper 50α% point corresponding to a pivotal distribution of the sample mean standardized appropriately by U _m . This way, we explore the role of Mukhopadhyay ( 1982 Mukhopadhyay , N. ( 1982 ). Stein's Two-Stage Procedure and Exact Consistency, Skandinavisk Aktuarietdskrift pp. 110 – 122 . [Google Scholar] ) two-stage confidence interval procedure when the requisite sample size is determined through GMD, MAD, or range. Associated exact and some asymptotic first-order properties are developed first. Next, we revisit Mukhopadhyay and Duggan ( 1997 Mukhopadhyay , N. and Duggan , W. T. ( 1997 ). Can a Two-Stage Procedure Enjoy Second-Order Properties? Sankhya, Series A 59 : 435 – 448 . [Google Scholar] ) updated two-stage methodology that was proposed when a known positive lower bound was available for σ² in the present light. We highlight associated (i) first-order efficiency properties for all proposed fixed-width confidence interval procedures and (ii) second-order efficiency property of the GMD-based procedure. These are accompanied with extensive data analyses via simulations as well as real data.

The standardized mean difference is a widely used effect size measure. In this article, we develop a general theory for estimating the population standardized mean difference by minimizing both the mean square error of the estimator and the total sampling cost. Fixed sample size methods, when sample size is planned before the start of a study, cannot simultaneously minimize both the mean square error of the estimator and the total sampling cost. To overcome this limitation of the current state of affairs, this article develops a purely sequential sampling procedure, which provides an estimate of the sample size required to achieve a sufficiently accurate estimate with minimum expected sampling cost. Performance of the purely sequential procedure is examined via a simulation study to show that our analytic developments are highly accurate. Additionally, we provide freely available functions in R to implement the algorithm of the purely sequential procedure. (PsycINFO Database Record

Gini index is a widely used measure of economic inequality. This article develops a theory and methodology for constructing a confidence interval for Gini index with a specified confidence coefficient and a specified width without assuming any specific distribution of the data. Fixed sample size methods cannot simultaneously achieve both specified confidence coefficient and fixed width. We develop a purely sequential procedure for interval estimation of Gini index with a specified confidence coefficient and a specified margin of error. Optimality properties of the proposed method, namely first order asymptotic efficiency and asymptotic consistency properties are proved under mild moment assumptions of the distribution of the data.