Nitis Mukhopadhyay

Sequential methods and their applications

Preface Objectives, Coverage, and Hopes Introduction Back to the Origin Recent Upturn and Positive Feelings The Objectives The Coverage Aims and Scope Final Thoughts Why Sequential? Introduction Tests of Hypotheses Estimation Problems Selection and Ranking Problems Computer Programs Sequential Probability Ratio Test Introduction Termination and Determination of A and B ASN Function and OC Function Examples and Implementation Auxiliary Results Sequential Tests for Composite Hypotheses Introduction Test for the Variance Test for the Mean Test for the Correlation Coefficient Test for the Gamma Shape Parameter Two-Sample Problem: Comparing the Means Auxiliary Results Sequential Nonparametric Tests Introduction A Test for the Mean: Known Variance A Test for the Mean: Unknown Variance A Test for the Percentile A Sign Test Data Analyses and Conclusions Estimation of the Mean of a Normal Population Introduction Fixed-Width Confidence Intervals Bounded Risk Point Estimation Minimum Risk Point Estimation Some Selected Derivations Location Estimation: Negative Exponential Distribution Introduction Fixed-Width Confidence Intervals Minimum Risk Point Estimation Selected Derivations Point Estimation of the Mean of an Exponential Population Introduction Minimum Risk Estimation Bounded Risk Estimation Data Analyses and Conclusions Other Selected Multistage Procedures Some Selected Derivations Fixed-Width Intervals from MLEs Introduction General Sequential Approach General Accelerated Sequential Approach Examples Data Analyses and Conclusions Some Selected Derivations Distribution-Free Methods in Estimation Introduction Fixed-Width Confidence Intervals for the Mean Minimum Risk Point Estimation for the Mean Bounded Length Confidence Interval for the Median Data Analyses and Conclusions Other Selected Multistage Procedures Some Selected Derivations Multivariate Normal Mean Vector Estimation Introduction Fixed-Size Confidence Region: SIGMA = sigma2H Fixed-Size Confidence Region: Unknown Dispersion Matrix Minimum Risk Point Estimation: Unknown Dispersion Matrix Data Analyses and Conclusions Other Selected Multistage Procedures Some Selected Derivations Estimation in a Linear Model Introduction Fixed-Size Confidence Region Minimum Risk Point Estimation Data Analyses and Conclusions Other Selected Multistage Procedures Some Selected Derivations Estimating the Difference of Two Normal Means Introduction Fixed-Width Confidence Intervals Minimum Risk Point Estimation Other Selected Multistage Procedures Some Selected Derivations Selecting the Best Normal Population Introduction Indifference Zone Formulation Two-Stage Procedure Sequential Procedure Data Analyses and Conclusions Other Selected Multistage Procedures Some Selected Derivations Sequential Bayesian Estimation Introduction Selected Fixed Sample Size Concepts Elementary Sequential Concepts Data Analysis Selected Applications Introduction Clinical Trials Integrated Pest Management Experimental Psychology: Cognition of Distance A Problem from Horticulture Other Contemporary Areas of Applications Appendix: Selected Reviews, Tables, and Other Items Introduction Big O(.) and Little o(.) Some Probabilistic Notions and Results A Glimpse at Nonlinear Renewal Theory Abbreviations and Notation Statistical Tables References Index Exercises appear at the end of each chapter.

Sequential estimation problems for negative exponential populations

The literature on sequential estimation problems for negative exponential populations has been reviewed here, We attempt to bring in all the published and unpublished materials known to us in a fairly coherent fashion. Both the concepts and theoretical findings are discussed.

Multistage selection and ranking procedures : second-order asymptotics

Theory of sequential and multistage procedures selecting the best normal population selecting the best negative exponential population estimation of ordered parameters selecting the best component in a multivariate normal population estimation after selection and ranking additional topics notation and abbreviations.

TWO-STAGE ESTIMATION OF A LINEAR FUNCTION OF NORMAL MEANS WITH SECOND-ORDER APPROXIMATIONS

ABSTRACT When a confidence interval tends to be too wide, its effectiveness in bolstering any inferential statements becomes limited. Hence, an experimenter may opt to construct a confidence interval with some preassigned “small” width and preassigned “large” confidence coefficient so that any inferences drawn from this can be of some value in practice. We consider k(≥2) independent normal populations with unknown means and unknown and unequal variances. We discuss the estimation problem for a linear function of the population means with a fixed-width (=2d) confidence interval having the preassigned confidence coefficient (≥1 − α), d > 0, 0 < α < 1. But, the goal of having such a confidence interval with both preassigned width and confidence coefficient is not attainable when the sample sizes are held fixed in advance [Dantzig (1940)[8], Ghosh et al. (1997, Sec. 3.7)[13]]. Chapman (1950)[4] first gave a Stein-type two-stage procedure for the problem on hand when k = 2. It is known that in a k-sample problem, the analogous two-stage procedure requires the upper percentage points of the distribution of the sum of k independent Students t variates. First, a Cornish-Fisher expansion of such a percentage point is derived (Theorem 2.1) in general, followed by the Tables 1 and 2 of these (approximate) percentage points which are constructed by using this expansion, with the pilot sample sizes not necessarily all equal, when k = 2, α = .05, .01 and k = 3, α = .05. Next, under the limited additional assumption that each unknown population variance has a known positive lower bound, the Chapman type two-stage estimation procedure is modified along the lines of the one-sample considerations of Mukhopadhyay and Duggan (1997)[23]. For this modified two-stage procedure, various second-order expansions for both the lower and upper bounds of the average sample sizes (Theorem 2.2) and the associated confidence coefficient (Theorem 2.3) are obtained. We may remark that the second-order expansions are meant to provide faster rates of convergence for useful approximations. Then, through extensive sets of simulations we show that the extent of over-sampling experienced by the Chapman-type procedure is significantly reduced under the new modification when k = 2(1)5, α = .05. We include examples and data to illustrate usefulness of the modified k-sample two-stage estimation technique when k = 2, 3. Additionally, the importance of the asymptotic second-order terms is highlighted with the help of data analysis (Examples 1 and 2, Sec. 3).

Sequential estimation of a linear function of mean vectors

Let π1,...,πk be k independent populations where we assume that the ith population distribution is The parameters and σi e (0,∞) for i = l,...,k are all assumed unknown, but H1einf:,...,Hk are known positive definite pzp matrices. We estimate parameters of the form where cis are known nonzero constants for i = 1,...,k, by means of ellipsoidal confidence regions. Various two-stage and sequential procedures are proposed and some of their exact and asymptotic properties are studied. Statistical methods and some of their characteristics are discussed both when H1,...,Hk are simultaneously diagonalizable as well as when they are not.

Exact Risks of Sequential Point Estimators of the Exponential Parameter

Abstract Under purely sequential sampling schemes, a theory is developed for the exact determination of the distributions of two classes of stopping variables (rules) in order to handle point estimation problems for the parametric functionals in an exponential distribution. Explicit formulae are derived for the expected value and risks of sequential estimators of the mean, failure rate, and reliability function of an exponential distribution. These are utilized to compare performances of several competing estimators of the mean and the failure rate. Recommended by Ben Boukai

Nearly Optimal Change-Point Detection with an Application to Cybersecurity

Abstract We address the sequential change-point detection problem for the Gaussian model where baseline distribution is Gaussian with variance σ2 and mean μ such that σ2 = aμ, where a > 0 is a known constant; the change is in μ from one known value to another. First, we carry out a comparative performance analysis of four detection procedures: the Cumulative Sum (CUSUM) procedure, the Shiryaev–Roberts (SR) procedure, and two its modifications—the Shiryaev–Roberts–Pollak and Shiryaev–Roberts–r procedures. The performance is benchmarked via Pollaks maximal average delay to detection and Shiryaevs stationary average delay to detection, each subject to a fixed average run length to false alarm. The analysis shows that in practically interesting cases the accuracy of asymptotic approximations is “reasonable” to “excellent”. We also consider an application of change-point detection to cybersecurity for rapid anomaly detection in computer networks. Using real network data we show that statistically traffics intensity can be well described by the proposed Gaussian model with σ2 = aμ instead of the traditional Poisson model, which requires σ2 = μ. By successively devising the SR and CUSUM procedures to “catch” a low-contrast network anomaly (caused by an Internet Control Message Protocol reflector attack), we then show that the SR rule is quicker. We conclude that the SR procedure is a better cyber “watch dog” than the popular CUSUM procedure.

An alternative formulation of accelerated sequential procedures with applications to parametric and nonparametric estimation

A new type of accelerated sequential procedure is developed and its associated second-order characteristics are investigated. In the case of a few known sequential estimation problems, it is shown that the new methodology (i) is flexible in terms of what one plugs in as the estimator of the nuisance parameter in the boundary condition, and (ii) can economize at the same time on the starting sample size condition to provide second-order properties similar to those already available for the existing methodologies. The power and novelty of the newly constructed sampling techniques are also substantiated by the simplicity of derivation of the second-order expansions for the regret function associated with the estimator of the scale parameter of an exponential distribution, as well as in the context of one-sample and two-sample problems for estimating means when the distributions are unspecified.

Triple stage point estimation for the exponential location parameter

This paper deals with the problem of estimating the minimum lifetime (guarantee time) of the two parameter exponential distribution through a three-stage sampling procedure. Several forms of loss functions are considered. The regret associated with each loss function is determined. The results in this paper generalize the basic results of Hall (1981, Ann. Statist., 9, 1229–1238).

Three-Stage point Estimation Procedures For a Normal Mean

Three stage sampling procedures have been discussed with the goal of achieving (i) the minimum risk and (ii) the bounded risk separately for point estimation of the mean of a normal population whose variance is completely unknown. The remainder term in the second-order asymptotic expansion of the risk function is shown to have the exact same order as that obtained in the case of the corresponding purely sequential procedure for both these problems. The results obtained for the second problem also form major extensions of the properties derived in Mukhopadhyay (1985, Sequential Analysis, 4, 311-319)