Showing papers in "Sequential Analysis in 1994"
TL;DR: In this article, the boundary crossing probabilities of generalized likelihood ratio statistics in multiparameter exponential families were investigated and shown to be asymptotically optimal from both frequentist and Bayesian points of view.
Abstract: By developing and applying certain results on boundary crossing probabilities of generalized likelihood ratio statistics in multiparameter exponential families, we show that Lai's (1988a) modification of Schwarz's (1962) sequential likelihood ratio tests of one-sided hypotheses in the one-parameter case can be extended to general hypotheses in the multiparameter case. Such tests are shown to be asymptotically optimal, from both frequentist and Bayesian points of view, and numerical results are also given to demonstrate their performance.
30 citations
TL;DR: In this paper, the problem of sequential point estimation and fixed accuracy confidence set procedures of autoregressive parameters in a ρ-th order stationary auto-regression model is considered.
Abstract: This paper considers the problem of sequential point estimation and fixed accuracy confidence set procedures of autoregressive parameters in a ρ-th order stationary autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be risk efficient as the cost of estimation error tends to infinity. Furthermore, the proposed procedure for fixed-width confidence set is shown to be both asymptotically consistent and asymptotically efficient as the width approaches zero.
21 citations
TL;DR: In this article, a new approach for specifying the alternative distribution was proposed and a new eliminating procedure for selecting the best one of k experimental categories when the measurements from each category have a one-dimensional Koopman-Darmois distribution.
Abstract: By using a new approach for specifying the alternative distribution, we obtain a new eliminating procedure for selecting the best one of k experimental categories when the measurements from each category have a one-dimensional Koopman-Darmois distribution. When the k populations are normal with a common known variance, a modification of this approach results in a new closed eliminating procedure for selecting the population with the greatest mean. The Monte Carlo results summarized in Tables I and II indicate the new procedures lead to a reduction in the average total sample size when compared to the other available sequential procedures. We also obtain a sequential procedure for the case when the "best" experimental category is only preferred when it is better than a specified standard.
19 citations
TL;DR: In this article, exact formulae for the operating characteristic as well as for the average sample number of SPRT's are derived using solutions of special integral equations, and exact results for SPRT operating characteristics are derived.
Abstract: Let X1, X2,… be independent distributed random variables where is unknown. Using solutions of special integral equations, exact formulae for the operating characteristic as well as for the average sample number of SPRT’s are derived
10 citations
TL;DR: In this article, a general framework for optiinal stopping problems associated with multivariate point processes is presented and the optimal stopping problem is solved in a finite and an infinite horizon setting, and various special models are coilsidered.
Abstract: A general framework for optiinal stopping problems associated with multivariate point processes is presented. Such problems arise. for example. in finding optimal selection/detection strategies in discovering new species, error detection and job search problems. The optimal stopping problem is solved in a finite and an infinite horizon setting. and various special models are coilsidered. It turns out that in regenerative type of modlels where the interarrival time distribtutions have properties known from reliability theory. such as NBUE, IFR and DFR, the optimal stopping times have appealirig properties.
7 citations
TL;DR: In this article, the optimal length for each treatment at each stage is to be decided in a multistage clinical trial, where the total number of patients to be involved in the trial is N, which is random.
Abstract: K treatments that yield dichotomous responses are considered in a multistage clinical trial. The total number of patients to be involved in the trial is N, which is random. The optimal length for each treatment at each stage is to be decided. The objective is to maximize the expected total number of successes of the trial. Two cases are considered: (1) Two stages. It is shown that the rate of the optimal length of the first stage is no greater than √E(N) as E(N) goes to infinity, when the distribution of N, Q, yields a regular discount sequence. The rate may be smaller than √E(N) which depends on the distribution of N and the prior distributions on the probabilities of success of the treatments. Under certain conditions, the rate is exactly √E(N). Nevertheless, the rate may be greater than E(N) without the regularity of Q. (2) r stages. It is shown that M∗/E(N) converges to zero in probability as E(N) goes to infinity when Q has the regularity, where M∗ is the sum of the optimal lengths for the first r - ...
6 citations
TL;DR: In this paper, approximate probabilities of error for Armitage's (1947) test are derived and a method of adjusting the error rates used to establish the decision boundaries in order to attain the nominal error rates is developed.
Abstract: Abraham Wald developed the Sequential Probability Ratio Test in the 1940's to perform simple vs. simple hypothesis tests that would control both Type I and Type II error rates. Some applications require a test of three hypotheses. In addition, to perform a simple vs. composite two-sided test, a three-hypothesis test with all hypotheses simple has been suggested. Methods have been proposed that will test three hypotheses sequentially. They range widely in simplicity andaccuracy. In this paper,approximate probabilities of error for Armitage's (1947) test arederived. A method of adjusting the error rates used to establish the decision boundaries in order to attain the nominal error rates is developed.The procedure is compared to existing ones by Monte-Carlo simulation
5 citations
TL;DR: In this paper, the excess over the boundaries used in the test is approximated as a simple function of the parameter to be tested by using the condition of the test statistic immediately before the stopping time in normal and exponential cases.
Abstract: Since Wald developed the sequential probability ratio test, many studies have been done to approximate the characteristics of the test. One of the major efforts among them is to approximate the excess over the boundaries used in the test. In this paper the excess is approximated as a simple function of the parameter to be tested by using the condition of the test statistic immediately before the stopping time in normal and exponential cases. The use of the estimated excess shows good performances in estimating the operating characteristic function, the average sample number, and the probability mass function of the sarnple number. It also make it possible to determine the boundary values which can give the error probabilities close to the desired ones.
5 citations
TL;DR: In this article, the bias and risk of sequential estimators are compared in terms of the bias bias and the risk for estimating the variance of a normal population in certain class of estimators when the loss function is squared error plus sampling cost.
Abstract: We consider the sequential procedures for estimating the variance of a normal population in certain class of estimators when the loss function is squared error plus sampling cost. In this paper some sequential estimators are applied and compared in terms of the bias and the risk.
4 citations
TL;DR: This work extends the kernel-type recursive estimators of a density and its derivatives, based on a fixed number of observations, to the case where the sample size is random.
Abstract: Menon et al. (1984) proposed the kernel-type recursive estimators of a density and its derivatives, based on a fixed number of observations. We extend the estimators to the case where the sample size is random. Convergence rates of the normal approximation for the kernel-type sequential estimators are studied.
4 citations
TL;DR: In this paper, the authors compared the average time to termination of stopping rules for deciding when all the individuals in the population have been seen and found that the best performance is provided by two of the stopping rules based on the cumulative waiting time to sight a given number of unmarked individuals.
Abstract: Sightings of any member of a population of unknown size N occur according to a homogeneous Poisson process. The processes are independent and have a common rate. Individuals are given a distinctive tag at the time they are first sighted. Various stopping rules have appeared in different areas of the literature for deciding when all the individuals in the population have been seen. The average time to termination of these rules is compared. Computational results suggest that the best performances are provided by two of the rules based on the cumulative waiting time to sight a given number of unmarked individuals
TL;DR: In this paper, the authors examined the role and applicability of the theory of general multistage piecewise sampling methodologies in the area of estimation as well as selection and ranking.
Abstract: A piecewise sequential methodology and its merits have been recently discussed in Mukhopadhyay and Sen (1993), in the context of certain estimation problems. In the present paper, we examine the role and applicability of the theory of general multistage piecewise sampling methodologies in the area of estimation as well as selection and ranking. In each problem, second-order asymptotic characteristics of piecewise multistage methodologies are indicated.
TL;DR: In this paper, the authors consider a process X whose paths are right-continuous and have limits from the left, and denote by P its law, and show that under some conditions, Wald's sequential procedure for testing H0 : P = P0 versus H1 : P=P1 is optimal among a certain class of decision rules, in the sense that it minimizes the Kullback informations under both probability measures.
Abstract: We consider a process X whose paths are right-continuous and have limits from the left, and denote by P its law. If P0 and P1 are two given probability measures, we show that under some conditions, Wald's sequential procedure for testing H0 : P = P0 versus H1 : P = P1 is optimal among a certain class of decision rules, in the sense that it minimizes the Kullback informations under both probability measures. As a conclusion,we give some applications of this result.
TL;DR: In this article, a repeated significance test on regression coefficients in a linear regression model is proposed for the sequential comparison of two medical treatments whose effectiveness is influenced by prognostoc factors.
Abstract: In clinical trials we often need a sequential testing procedure for a difference between two medical treatments whose effectiveness is influenced by prognostoc factors. This article considers a repeated significance test on regression coefficients in a linear regression model. We first derive approximations for the overall significance level and power of the test and compare our test with a fixed sample test. We then discuss applications of these results to the sequential comparison of two treatments and also discuss the effect of allocation rules on the behavior of the test statistics.
TL;DR: In this article, the average run length properties of some control procedures for monitoring the mean and the variance of a process were evaluated, and it was shown that the Extreme-Value chart is the most efficient of the three charts when it is desired to detect the presence of mixture alternatives.
Abstract: This paper evaluates the average run length properties of some control procedures for monitoring the mean and the variance of a process. In particular, a comparison is made between the and Extreme—Value charts when the underlying distribution of the control chart statistic is not normal but is a mixture of normal distributions. It is shown that the Extreme—Value chart is the most efficient of the three charts when it is desired to detect the presence of mixture alternatives. Furthermore, the power of the Extreme—Value chart is close to the power of the likelihood ratio test corresponding to the underlying test problem
TL;DR: In this article, the problem of finding the cell with the smallest cell probability is revisited in a multinomial setting with a fixed number k of cells and an inverse sampling procedure is used, unlike past work on this problem.
Abstract: In a multinomial setting with a fixed number k of cells. the problem of screening out cells to find the "best" cell, i.e., the one with the smallest cell probability, or looking for a (small) subset of cells containing the best cell is revisited. An inverse sampling procedure is used, unlike past work on this problem ([l], [2], [3], and [4]). Finding the cell with the smallest cell probability is clearly more difficult than finding the one with the argest cell probability. The proposed procedure takes one observation at a time (as usual) and igns a zero to all those (and only those) k - 1 cells into which the observation does not fall Sampling continues sequentially and stops as soon as any one cell has accumulated r zeros. For any given integer c (with 0 ≤ c < r), we put into the selected subset (SS) all those cells with at least r - c zeros and assert that this selected subset contains the best cell. It is important to note that for the slippage configuration (SC) we can attain any specified lower bound...
TL;DR: In this article, the transition probabilities of finite Markov chains are studied by means of non-linear Markov renewal theorems containing the first-passage times and excess, and expressions for asymptotic values of the powers of tests are obtained.
Abstract: Repeated likelihood ratio tests for the transition probabilities of finite Markov chains are studied By means of non-linear Markov renewal theorems containing the first-passage timesand excess,expressions for asymptotic values of the powers of tests are obtained.
TL;DR: In this paper, it is shown that the bootstrap can be used at any point during a sequential analysis to predict the final values of important quantities, which may lead to modification of the experiment with a view to obtain improved final performance.
Abstract: Application of the bootstrap to problems in sequential analysis is discussed It is shown that the bootstrap can be used at any point during a sequential analysis to predict the final values of important quantities This may lead to modification of the experiment with a view to obtain improved final performance with regard to the accuracy or other aspects of the experiment required
TL;DR: In this article, a stopping time for the price search is constructed for such situations, where the buyer must update his/her beliefs in a Bayesian manner, but the prior distribution is not completely known to the buyer.
Abstract: We consider the problem of a consumer desiring to buy an item at as low a price as possible based on a finite sequence of price quotations obtained sequentially from various sellers. This is a version of the so-called best-choice problem. It is assumed that the optimal decision is concerned with the probability-maximizing approach. When the distribution of price quotations is completely known, the optimal buying policy is myopic. Many authors have shown that the myopic policy is still optimal in some cases where the price distribution has unknown parameter(s) and the buyer's prior on this parameter undergoes Bayesian updating as successive prices are received. In this article, we examine the case in which the buyer must update his/her beliefs in a Bayesian manner, but the prior distribution is not completely known to the buyer. We assume, however, that some auxiliary information is available to the buyer. Using empirical Bayes techniques, a stopping time for the price search is constructed for such situations
TL;DR: In this article, the authors propose to implement the sequential procedure of Bose and Boukai (1993) in smaller "pieces" along the lines of Mukhopadhyay and Sen (1993).
Abstract: In a certain class of two—parameter exponential distributions, we consider minimum risk point estimation problems for one of the parameters. We propose to implement the sequential procedure of Bose and Boukai (1993) in smaller “pieces” along the lines of Mukhopadhyay and Sen (1993). Unlike the fully sequential procedure, one can obtain an unbiased estimator for the variance of the storpping number in a piecewise methodology. On top of this, the asymptotic second—order expansions of the regret functions for both the piecewise and fully sequential estimators are same up to o(1) term and the piecewise sampling scheme is operationally more covenient. The effect on cost due to parallel processing is also discussed
TL;DR: In this article, under a very general dependence set-up, a rate of convergence of the randomly stopped normallocated log-I lkellihoods to standard normal varlables was shown.
Abstract: In this paper we,under a very general dependence set-up,obtaln a rate of convergence of the randomly stopped normallzed log-I lkel lhood ratlo statlstlc to standard normal varlables.Two examples are taken Into account.Non-unlform rates of convergence and bounds for Lp norms are also obtained under some qulte general condltlons.Martlngale techniques have been exploited throughout.
TL;DR: For special pairs of parameters there is no excess over the boundary in the case of one-sided SPRT's on Poisson, Bernoulli, and geometric parameters.
Abstract: For special pairs of parameters there is no excess over the boundary in the case ofone-sided SPRT's on Poisson, Bernoulli, and geometric parameters. Using the hitting timetheorem we find exact formulas for the distributions of the sample sizes and errorprobabilities in these cases. Applications are also given to queueing theory.
TL;DR: In this paper, a stopping rule is proposed for the purpose of bounding the mean integrated squared error of nonparametric regression estimates on an interval, which is designed to achieve sufficiently good fit in estimating the regression function, where the limit is taken as one requires increasingly better fit.
Abstract: A stopping rule is proposed for the purpose of bounding the mean integrated squared error of nonparametric regression estimates on an interval. The rale is designed to achieve sufficiently good fit in estimating the regression function, It is shown that the stopping rale is asymptotically efficient both in terms of sample size and fit, where the limit is taken as one requires increasingly better fit