BAYES SEQUENTIAL ESTIMATION OF POISSON MEAN UNDER A LINEX LOSS FUNCTION

This paper considers Bayes sequential estimation of the mean of a Poisson distribution using a LINEX loss function and a cost c > 0 for each observation. Under a Gamma prior distribution, it is shown that an asymptotically pointwise optimal rule is asymptotically non-deficient in the sense of Woodroofe (1981).     

Optimal Sequential Tests for Two Simple Hypotheses

Abstract

A general problem of testing two simple hypotheses about the distribution of a discrete-time stochastic process is considered. The main goal is to minimize an average sample number over all sequential tests whose error probabilities do not exceed some prescribed levels. As a criterion of minimization, the average sample number under a third hypothesis is used (modified Kiefer–Weiss problem). For a class of sequential testing problems, the structure of optimal sequential tests is characterized. An application to the Kiefer–Weiss problem for discrete-time stochastic processes is proposed. As another application, the structure of Bayes sequential tests for two composite hypotheses, with a fixed cost per observation, is given. The results are also applied for finding optimal sequential tests for discrete-time Markov processes. In a particular case of testing two simple hypotheses about a location parameter of an autoregressive process of order 1, it is shown that the sequential probability ratio test has the Wald–Wolfowitz optimality property.     

A sequential sampling plan for exponential distribution

In this article, a sequential variable sampling plan is studied. Suppose that the quality of an item in a batch is measured by a random variable with exponential distribution; its parameter is unknown having a gamma prior distribution. Then by using Bayesian approach and considering a Markov decision process, the optimality equations for the minimum total expected cost are formulated. We show that an optimal decision rule will have a control limit structure and monotonicity. A backward induction method is suggested that is a finite algorithm for the numerical solution of the sequential sampling plan.     

Minimum risk sequential point estimation of the stress-strength reliability parameter for exponential distribution

Abstract

In this article, using purely and two-stage sequential procedures, the problem of minimum risk point estimation of the reliability parameter (R) under the stress–strength model, in case the loss function is squared error plus sampling cost, is considered when the random stress (X) and the random strength (Y) are independent and both have exponential distributions with different scale parameters. The exact distribution of the total sample size and explicit formulas for the expected value and mean squared error of the maximum likelihood estimator of the reliability parameter under the stress–strength model are provided under the two-stage sequential procedure. Using the law of large numbers and Monte Carlo integration, the exact distribution of the stopping rule under the purely sequential procedure is approximated. Moreover, it is shown that both proposed sequential procedures are finite and for special cases the exact distribution of stopping times has a degenerate distribution at the initial sample size. The performances of the proposed methodologies are investigated with the help of simulations. Finally, using a real data set, the procedures are clearly illustrated.     

LERCHE'S SEQUENTIAL TEST FOR THE DRIFT OF A BROWNIAN MOTION WITH A SMOOTH PRIOR

Lerche (1986) studies a specific sequential problem concerned with testing the sign of the drift (say θ) of a Brownian motion. He shows that a repeated significance test boundary based on frequentist principles is also an optimal Bayesian boundary with 0–1 decision loss and sampling costs cθ² per unit time, where c > 0. In this paper, we re-investigate Lerche's problem for a more general smooth prior. Using the approach developed by Simons et al. (1989), we derive an expansion of the optimal stopping boundary by solving a free boundary problem for the heat equation. The first term of the solution of the free boundary problem is exactly the same as for Lerche's optimal stopping boundary, while the first term of the corresponding Bayes risk is slightly different from Lerche's Bayes risk. An alternative approach to approximating the boundary can be obtained by using a tangent approximation for two-sided Brownian exit densities. Doing so yields results that are close to the results obtained by solving the free problem.     

Determination of multiple dependent state repetitive group sampling plan based on the process capability index

Abstract

In this article, we propose a sampling plan called a multiple dependent state repetitive group sampling plan for variable inspection based on the process capability index C_pk. The proposed sampling plan is applicable for the inspection of normally distributed quality characteristics when both mean and variance are assumed to be unknown. This new plan under variable inspection will be very useful particularly in compliance testing. Tables are also constructed for the determination of optimal parameters for easy selection and implementation of the plan. The optimal parameters can be determined by using the approach of two points on the operating characteristic curve. Symmetric and asymmetric cases based on the fraction nonconforming by the lower and the upper specification limits are also considered. Advantages of the proposed plan are also discussed. It is also shown that the proposed plan outperforms other existing sampling plans under variable inspection.     

Sequential minimum risk point estimation (MRPE) methodology for a normal mean under Linex loss plus sampling cost: First-order and second-order asymptotics

Abstract

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 linear cost of sampling. A number of important asymptotic first-order and asymptotic second-order properties' characteristics have been developed and proved thoroughly. Extensive sets of simulations tend to validate nearly all of these asymptotic properties for small to medium to large optimal fixed sample sizes.     

Truncated Sequential Procedures

Abstract

Sequential planning with the restriction that the number of groups is absolutely bounded by some fixed number K is considered. These truncated sequential plans will stop sampling after most K groups of observations are collected. Practically, it reflects the following situation. There are only K days available to complete a certain study (e.g., clinical trials, to get FDA approval or investigate conditions of the field before it is time for farming). Every day, a sample of the desired size is selected. Based on the interim analysis performed, an inference is made or another batch of observations is collected as long as total number of groups does not exceed K. Truncated sequentially planned probability ratio tests (TSPPRTs) on a lattice are considered in detail. Construction and methods of evaluation of TSPPRTs are proposed.     

Empirical bayes sequential estimation of the mean

We consider the empirical Bayes problem where the component problem is the sequential estimation of the mean of a distribution with squared error decision loss plus a sampling cost. An empirical Bayes sequential estimation procedure is exhibited which is asymptotically optimal. Asymptotic efficiency of the empirical Bayes stopping time sequence is also established. The performance of the proposed empirical Bayes procedure is studied with the help of a Monte Carlo study.     

An Asymptotic Second-Order Lower Bound for the Bayes Risk of a Sequential Procedure

Abstract

Consider the problem of estimating the difference of the means of two populations, where each population distribution is a member of the one-parameter exponential family of probability distributions. A Bayesian approach is adopted in which the mean difference is estimated under the squared error loss and the prior distributions are of the form proposed by Diaconis and Ylivisaker [Diaconis, P.; Ylivisaker, D. Conjugate priors for exponential families. Ann. of Statist. 1979, 6, 269–281]. The main result determines an asymptotic second-order lower bound for the Bayes risk of a sequential procedure that takes N observations from one population and t ? N from the other population, and estimates the mean difference by the Bayes estimator, where N is determined according to a sequential design and t denotes the total number of observations sampled from both populations.     

A Linear Programming Approach to Sequential Hypothesis Testing

Abstract

Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. This result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function, with respect to these multipliers, coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is jointly linear in the cost function and the Lagrangian multipliers and can be solved for both with off-the-shelf algorithms. To illustrate the procedure, optimal sequential tests for Gaussian random sequences with different dependency structures are derived, including the Gaussian AR(1) process.     

Nonparametric Sequential Minimax Estimation of the Drift Coefficient in Diffusion Processes

Abstract

A sequential procedure for estimating the drift function of a diffusion process is constructed. The asymptotic properties are established, such as the optimal covergence rate for a global risk and the asymptotic efficiency for a local risk of the procedure as well as the asymptotic normality of observation time.

     

Two-Stage Procedures for High-Dimensional Data

Abstract

In this article, we consider a variety of inference problems for high-dimensional data. The purpose of this article is to suggest directions for future research and possible solutions about p ? n problems by using new types of two-stage estimation methodologies. This is the first attempt to apply sequential analysis to high-dimensional statistical inference ensuring prespecified accuracy. We offer the sample size determination for inference problems by creating new types of multivariate two-stage procedures. To develop theory and methodologies, the most important and basic idea is the asymptotic normality when p → ∞. By developing asymptotic normality when p → ∞, we first give (a) a given-bandwidth confidence region for the square loss. In addition, we give (b) a two-sample test to assure prespecified size and power simultaneously together with (c) an equality-test procedure for two covariance matrices. We also give (d) a two-stage discriminant procedure that controls misclassification rates being no more than a prespecified value. Moreover, we propose (e) a two-stage variable selection procedure that provides screening of variables in the first stage and selects a significant set of associated variables from among a set of candidate variables in the second stage. Following the variable selection procedure, we consider (f) variable selection for high-dimensional regression to compare favorably with the lasso in terms of the assurance of accuracy and the computational cost. Further, we consider variable selection for classification and propose (g) a two-stage discriminant procedure after screening some variables. Finally, we consider (h) pathway analysis for high-dimensional data by constructing a multiple test of correlation coefficients.     

Interval Estimation Approach to Counting by Weighing: A Sequential Scheme

Abstract

Considerable time and energy involved in complete counting of large, but pre-specified, batch of N _s items could be saved by using weights of items. This article considers the case when the underlying distribution of weights, and its mean and variance are unknown. The problem is then reduced to that of finding an estimator of the mean and an optimal (small) sample size based on which a number [Ncirc] _n of items in the batch can be determined. Using a fixed-width interval criterion, Nickerson [Nickerson, D.M. Another look at counting by weighing. Commun. Statist. Simula. 1993, 22 (2), 323–343] derived an optimal sample size, but it depends on the unknown coefficient of variation. For this case, we propose a batch-type sequential sampling scheme which requires substantially fewer sampling operations and no prior knowledge of the coefficient of variation, but performs as well as Nickerson's and other available procedures in the literature. This shows that a little bit of sampling using substantially fewer sampling operations can significantly reduce the effort of complete counting.     

Sequentially estimating the required optimal observed number of tagged items with bounded risk in the recapture phase under inverse binomial sampling

Abstract

Estimation of a closed population size (N) under inverse binomial sampling consists of four basic steps: First, one captures t items, then tag these t items, followed by releasing the t tagged items back to the population. Then, one draws items from the population one by one until s tagged items are recaptured where s is fixed in advance. In the recapturing stage (fourth step), items are normally drawn with replacement. But, without replacement, sampling will not be impacted much if N is large. Under squared error loss (SEL) as well as weighted SEL, we propose sequential methodologies to come up with bounded risk point estimators of an optimal choice of s, leading to an appropriate sequential estimator of N: The sequential estimation methodologies are supplemented with appropriate first-order asymptotic properties, followed by extensive data analyses.     

A heuristic approach for near optimal truncated sequential test of exponential distribution

Abstract

Optimal truncated sequential test of the exponential distribution is studied in this paper. Definitions and relative concepts of optimal truncated sequential test are discussed. A heuristic approach, sample space sorting method (SSSM) and the procedures are established to solve a near optimal truncated sequential test. Comparing with the test plans provided by international standard IEC 61124 (2012 IEC 61124 (2012). Reliability Testing - Compliance Tests for Constant Failure Rate and Constant Failure Intensity, International Electrotechnical Commission, Geneva. [Google Scholar]) and Russian national standard GOST R 27.402 (1995 GOST R 27.402 (1995). Dependability Techniques-Compliance Test Plans for Mean Operating Time to Failure or between Failures-Part 1: Exponential Case, Russian National Standard, Moscow (in Russian). [Google Scholar]), the results show that the near optimal tests proposed in this paper can keep the error probabilities almost equivalence to the nominal error levels and save more synthetical expected test times and maximum expected test times simultaneously.     

Asymptotic Non-Deficiency of Some Procedures for a Particular Exponential Family Under Relative LINEX Loss

Abstract

The problem of Bayes sequential estimation of the unknown parameter in a particular exponential family of distributions with relative LINEX loss and fixed cost for each observation is considered in this article. Optimal, nearly optimal, and asymptotically pointwise optimal procedures with deterministic stopping rules are derived, and the approximate optimal procedures are shown to be asymptotically nondeficient in the sense of Woodroofe (1981 Woodroofe , M. ( 1981 ). A.P.O. Rules are Asymptotically Non-Deficient for Estimation with Squared Error Loss , in Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 58 : 331 – 341 .[Crossref], [Web of Science ®] , [Google Scholar]). In addition, a robust procedure with a deterministic stopping rule, which does not depend on the parameters of the prior distribution, is proposed, and the asymptotic second-order expansion of the corresponding Bayes risk is obtained.     

Recovery of Platinum and Palladium from Spent Automobile Catalyst by Solvent-Impregnated Resins

ABSTRACT

Separation of platinum and palladium, using solvent-impregnated resin (SIR) impregnating di-hexylamine (DHA), di-n-octylamine (DOA), or di-hexyl sulfide (DHS), coated by polyvinyl alcohol crosslinked by glutaraldehyde, was investigated. Coating of the SIR successfully suppressed leakage of the extractant during adsorption. The coated SIR was applied to chromatographic recovery of the two metals and quantitative adsorption–elution was performed via the frontal analysis mode. The adsorption of Pt and Pd was achieved, while adsorption of the co-existing metals in the leaching solution was suppressed. Separation of Pt and Pd was achieved by employing sequential chromatography system together with the quantitative elution of Pt from DHA-SIR with the mixture of 0.5 mol/L thiourea and 0.5 mol/L HCl and that of Pd from DHS-SIR with the mixture of 0.1 mol/L thiourea and 0.5 mol/L HCl solution. Finally, elementary Pt was then obtained from the eluent of sequential chromatography system by simultaneous precipitation and reduction of Pt with sodium borohydride with 99.2% purity.     

Some remarks on minimax sequential tests for stationary ornstein–uhilenbeck processes

In this paper sequential tests of one sided hypotheses H₁ : μ ≤ μ₀ and H₂ : μ > μ₀ for the velocity parameter of a stationary Ornstein-Uhlenbeck process(OUP) are discussed. When the variance parameter of the OUP is known the results obtained here complement a recent work of Roters (1991) by proving the existence of minimax sequential tests under concave and symmetric (about μ₀) loss l(μ)with l(μ₀) > 0 and linear costs of observation. For unkown variance parameter minimax sequential tests are derived for all loss functions discussed here and in the above mentioned paper.