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We have designed multistage minimum risk point estimation (MRPE) strategies for a function of an unknown mean in a normal population with its variance unknown. These are developed under a weighted ...

Purely sequential estimation for unknown mean ( μ ) in a normal population having an unknown variance ( σ 2 ) when observations are gathered in groups has been recently discussed in Mukhopadhyay an...

Two fundamental problems on purely sequential estimation are revisited—(i) the fixed-width confidence interval (FWCI) estimation problem and (ii) the minimum risk point estimation (MRPE) problem—in...

We have designed a sequential minimum risk point estimation (MRPE) strategy for the unknown mean of a normal population having its variance unknown too. This is developed under a Linex loss plus li...

A purely sequential minimum risk point estimation (MRPE) methodology with associated stopping time N is designed to come up with a useful estimation strategy. We work under an appropriately formula...

In this article, two-stage and fixed sample size procedures are developed for constructing confidence intervals for the common correlation ρ of an equi-correlated multivariate normal distribution. ...

In this paper, we develop appropriate sampling methodologies for testing hypotheses regarding the difference of mean values from two independent (or dependent) normal populations when their varianc...

A vector-valued autoregressive time series model is considered. The autoregressive coefficients of the model are random with possible dependencies among them. Estimation of the large number of para...

Revisiting Stein (1945, 1949) as well as Mukhopadhyay and Duggan (1997), we have proposed new two-stage procedures under both minimum risk point estimation and fixed-width confidence interval confi...

A negative binomial (NB) distribution is often used to model over dispersed count data arising from areas including ecology, agriculture, health, etc. We have designed purely sequential bounded-ris...

We have developed purely sequential methodologies for problems associated with both fixed-width confidence interval estimation and minimum risk point estimation for the normal mean μ when the variance σ2 is assumed unknown. New stopping rules are constructed by replacing the sample variance with appropriate multiples of Gini’s mean difference (GMD) and mean absolute deviation (MAD) in defining the conditions for boundary crossing. A number of asymptotic first-order consistency, efficiency, and risk efficiency properties associated with these new estimation strategies have been investigated. These are followed by summaries obtained from extensive sets of simulations by drawing samples from (i) normal universes or (ii) mixture-normal universes where samples may be reasonably treated as observations from a normal universe in a large majority of simulations. We also include illustrations using sales data and horticulture data. Overall, we empirically feel confident that our newly developed GMD-based o...

Fisher’s “Nile” example is a classic that involves a bivariate random variable (X,Y) having a joint probability density function given by f(x,y;𝜃) = exp(−𝜃x−𝜃−1y), 0 0 is a single unknown parameter. We develop (i) fixed-width and (ii) fixed-accuracy confidence intervals for 𝜃 with a preassigned confidence coefficient. In problem (i), we develop a purely sequential estimation strategy along with its asymptotic properties. In problem (ii), we determine that a fixed-sample-size estimation strategy will suffice and yet the requisite sample size would have to be found. We have done that both exactly as well as approximately and we report that for all practical purposes the approximations nearly provide the exact sample size whether it is small, moderate, or large. The last problem we address is bounded-accuracy fixed-sample-size estimation of P𝜃{X>Y}. All theoretical properties are adequately validated by large-scale simulations.

We have designed modified two-stage and purely sequential strategies to estimate the difference of location parameters from two independent negative exponential populations having unknown but proportional scale parameters under a modified Linex loss function. This article extends one-sample methodologies of Mukhopadhyay and Bapat (2016, Sequential Analysis). Some preliminary results are established along the lines of Mukhopadhyay and Hamdy (1984, Canadian Journal of Statistics) and Mukhopadhyay and Darmanto (1988, Sequential Analysis). We have resorted to Mukhopadhyay and Duggan (1997, Sankhya, Series A) in developing asymptotic second-order properties for the modified two-stage methodology and to nonlinear renewal theory of Lai and Siegmund (1977, 1979, Annals of Statistics) and Woodroofe (1977, Annals of Statistics) in addressing analogous properties under the purely sequential methodology. Then, we supplement with extensive sets of data analysis via computer simulations validating that both mod...

We have designed Stein-type (Stein, 1945, Annals of Mathematical Statistics) two-stage, modified two-stage (Mukhopadhyay and Duggan, 1997, Sankhya, Series A), and purely sequential strategies (Chow and Robbins, 1965, Annals of Mathematical Statistics) to estimate an unknown location parameter of a negative exponential distribution having an unknown scale parameter under a newly defined and modified Linex loss function. We aim at controlling the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, ω, and we emphasize both asymptotic first-order and asymptotic second-order properties for the modified two-stage and purely sequential estimation strategies. In developing asymptotic second-order properties for the modified two-stage methodology, we have heavily relied upon basic ideas rooted in Mukhopadhyay and Duggan (1997). In developing asymptotic second-order properties for the purely sequential methodology, however, we have heavily relied upon n...

Confidence interval estimation following a sequential probability ratio test (SPRT) is an important and difficult problem with applications in clinical trials. Difficulties arise because following termination of SPRT, a customary estimator of an unknown parameter of interest obtained from the randomly stopped data is often biased. As a result, coverage probability of a naive confidence interval based on the randomly stopped version of the customary estimator often falls below the target confidence level. We address this problem for a normal distribution having mean and variance unknown but equal and propose a methodology based on random central limit theorem that is remarkably easy to implement with bias-corrected estimators. We have also explored limited bootstrapped versions of our parametric resolutions. With the help of extensive sets of simulations, we have concluded that our data-driven bias-corrected parametric confidence intervals with a slight variance inflation perform remarkably well to...

Two-stage procedures are proposed to construct fixed-accuracy confidence intervals for the common variance of an equi-correlated normal distribution. The exact distributions of the stopping variable and the estimator of the common variance at stopping are derived. We also derive exact formulas for the expectations of functions of the stopping variable and the estimator of the common variance at stopping. The coverage probabilities of the proposed interval estimator are computed exactly and are shown to be nearly the same as the prescribed level.

Consider independent observations having a common normal probability density function with x ∈ R, unknown mean μ(∈R), and unknown variance σ2(∈R +). We propose estimating f(x; μ, σ2) with both two-stage and purely sequential methodologies under the mean integrated squared error (MISE) loss function. Our goal is to make the associated risk not to exceed a preassigned positive number c, referred to as the risk bound. No fixed-sample-size methodology would handle this estimation problem. We show that both density estimation methodologies satisfy an asymptotic (a) first-order efficiency property and a (b) first-order risk-efficiency property. Interestingly, purely sequential density estimation methodology has a better second-order efficiency property than that associated with two-stage methodology. Some robustness issues have been addressed. Small, moderate, and large sample performances are examined with the help of simulations. Illustrations are given with real data sets.

Our main focus is on count data arising from statistical ecology. Anscombe (1949) emphasized negative binomial (NB) modeling for overdispersed count data. A large majority of existing methodologies, both sequential and nonsequential, were reviewed by Mukhopadhyay and Banerjee (2012). We assume that the thatch parameter remains known and revisit a sequential confidence interval estimation method for an NB mean proposed by Willson and Folks (1983) that may not always guarantee a positive lower confidence limit. Moreover, any postsampling adjustment of their lower confidence limit would compromise the preset confidence coefficient. In this article, we provide an appropriate resolution by estimating the NB mean with a fixed-accuracy confidence interval such that both confidence limits are positive. Our proposed purely sequential and two-stage estimation methodologies enjoy asymptotic consistency and asymptotic efficiency properties. Next, we consider estimating the ratio of two NB means under a two-s...

Among other issues, Sriram and Iaci (2014) have elegantly drawn attention to the notion of “negative regret” for a number of interesting sequential minimum risk point estimation problems under a time series model. We briefly lay down our thoughts and understanding with regard to possible negative regret in some of the point estimation problems.

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, 1949) 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) 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 standardize...

The Anscombe-Ray-Chow-Robbins purely sequential procedure provided a breakthrough for constructing a fixed-width (= 2d) confidence interval for a normal population mean μ with a confidence coefficient 1 − α when the standard deviation σ was unknown. They proposed a purely sequential procedure of the following kind: with the associated confidence interval for μ. But, a purely sequential procedure does not meet the exact consistency property. While Simons (1968) proved the existence of a universal fixed non negative integer r such that for all fixed μ, σ, α, and d, the magnitude of r has remained unknown. We introduce a new methodology along with appropriate truncation in which the number of additional observations required beyond Q 1 is determined by the sequential sampling process itself. Interesting properties and performances of this methodology under successive crossings are explored theoretically and compared with those of the existing procedures by large-scale simulations.

Incomplete data is a common complication in applied research. This complication has been mostly ignored by researchers using sequential test methodology. In this article we bring together the theoretical and practical benefits of both fields by combining them together. This is a first attempt to develop methodology for the sequential probability ratio test when data are incomplete.

We suppose that there are two independent normal populations, π1: N(μ1, σ2) and π2: N(μ2, a 2σ2), where μ1, μ2, σ2 are unknown but a (0

In this discussion, I have added some comments and observations that arose after I read the masterfully prepared article of Professor Albert Shiryaev.

A (co-)founding editor of this prestigious journal, Professor B. K. Ghosh (BKG), passed away at the age of 72 on August 3, 2008. This tribute to BKG includes my reminiscences of him and what I believe to be his most important legacy. It ends with some personal recollections from BKG's niece, Mallika Ghosh, in her own words.

We consider two independent N(μ1, σ2) and N(μ2, a 2σ2) populations where μ1, μ2, σ2 are unknown parameters with −∞ 0) is known! The problem is one of estimating Δ = μ1 − μ2 by some appropriately constructed fixed-width (2d) confidence interval with the confidence coefficient at least 1 − α. Here, d (>0) and 0

Consider independent observations X 1, X 2,… having a common normal probability density function with − ∞ 0). We propose to estimate f(x;σ2) by a plug-in maximum likelihood (ML) two-stage estimator under the mean integrated squared error (MISE) loss function. Our goal is to make the associated risk not to exceed a preassigned positive number c, referred to as the risk-bound. Because no fixed-sample-size methodology would handle this estimation problem, we design an appropriate two-stage estimation methodology that is shown to satisfy the asymptotic (i) first-order efficiency property (Theorem 2.1), (ii) first-order risk-efficiency property (Theorem 2.2), as well as (iii) second-order efficiency property (Theorem 2.3). The performances of the proposed methodology for small, moderate, and large sample sizes are examined with the help of simulations. We have noticed some limited robustness of the proposed methodology under mixture-normal population densities, in w...