We have designed Stein-type (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] , Annals of Mathematical Statistics) two-stage, modified two-stage (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] , Sankhya, Series A), and purely sequential strategies (Chow and Robbins, 1965 Chow, Y. S. and Robbins, H. (1965). On the Asymptotic Theory of Fixed Width Sequential Confidence Intervals for the Mean, Annals of Mathematical Statistics 36: 457–462. [Crossref] [Google Scholar] , 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 Mukhopadhyay, N. and Duggan, W. T. (1997). Can a Two-Stage Procedure Enjoy Second Order Properties? Sankhya, Series A 59: 435–448. [Google Scholar] ). In developing asymptotic second-order properties for the purely sequential methodology, however, we have heavily relied upon nonlinear renewal theory (Lai and Siegmund, 1977 Lai, T. L. and Siegmund, D. (1977). A Nonlinear Renewal Theory with Applications to Sequential Analysis I, Annals of Statistics 5: 946–954. [Crossref], [Web of Science ®] [Google Scholar] , 1979 Lai, T. L. and Siegmund, D. (1979). A Nonlinear Renewal Theory with Applications to Sequential Analysis II, Annals of Statistics 7: 60–76. [Crossref], [Web of Science ®] [Google Scholar] , Annals of Statistics; Woodroofe, 1977 Woodroofe, M. (1977). Second Order Approximation for Sequential Point and Interval Estimation, Annals of Statistics 5: 984–995. [Crossref], [Web of Science ®] [Google Scholar] , Annals of Statistics). Then, we take to extensive data analysis carried out via computer simulations when requisite sample sizes range from small to moderate to large. We find that the Stein-type two-stage estimation methodology oversamples significantly and yet the achieved risk is not close to preset goal ω. On the other hand, both modified two-stage and purely sequential estimation strategies perform remarkably well. We have validated their main theoretical first-order and second-order properties through simulated data. The latter methodologies have been illustrated and implemented using two real data sets from health studies, namely, infant mortality data and bone marrow data.

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;????) = exp(−????x−????⁻¹y), 0<x,y<∞, where ????>0 is a single unknown parameter. We develop (i) fixed-width and (ii) fixed-accuracy confidence intervals for ???? 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 P_????{X>Y}. All theoretical properties are adequately validated by large-scale simulations.

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.

Under the squared error loss plus linear cost of sampling, we revisit the minimum risk point estimation (MRPE) problem for an unknown normal mean \(\mu\) when the variance \(\sigma ^{2}\) also remains unknown. We begin by defining a new class of purely sequential MRPE methodologies based on a general estimator \(W_{n}\) for \(\sigma\) satisfying a set of conditions in proposing the requisite stopping boundary. Under such appropriate set of sufficient conditions on \(W_{n}\) and a properly constructed associated stopping variable, we show that (i) the normalized stopping time converges in law to a normal distribution (Theorem 3.3 ), and (ii) the square of such a normalized stopping time is uniformly integrable (Theorem 3.4 ). These results subsequently lead to an asymptotic second-order expansion of the associated regret function in general (Theorem 4.1 ). After such general considerations, we include a number of substantial illustrations where we respectively substitute appropriate multiples of Gini’s mean difference and the mean absolute deviation in the place of the general estimator \(W_{n}\). These illustrations show a number of desirable asymptotic first-order and second-order properties under the resulting purely sequential MRPE strategies. We end this discourse by highlighting selected summaries obtained via simulations.