Showing papers in "Sequential Analysis in 1988"
TL;DR: In this article, the authors considered the problem of sequential point estimation of the autoregressive parameter in a first order auto-regressive model and proposed a sequential estimator based on the least squares estimator, which is shown to be asymtotically risk efficient as the cost of estimation error tends to infinity.
Abstract: This paper considers the problem of sequential point estimation of the autoregressive parameter in a first order autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be asymtotically risk efficient as the cost of estimation error tends to infinity, under certain regularity conditions. Furthermore, nonlinear renewal theory is used to obtain a second order approximation to the expected stopping time. The asymptotic normality and uniform integrability of the standardized stopping time are also established.
44 citations
TL;DR: Using Jensen s inequality, this article obtained a tighter bound on the probability of making a correct selection in Paulson s procedure for selecting the normal population which has the largest population mean by replacing Boole s inequality by a geometric inequality.
Abstract: Using Jensen s inequality we obtain a tighter bound on the probability of making a correct selection in Paulson s procedure for selecting the normal population which has the largest population mean by replacing Boole s inequality by a geometric inequality As a consequence we are able to use a sharper value for the constant aλ in Paulson s procedure If more than two populations are involved this should lead to an improvement in the expected number of stages to termination and expected total number of observations which is uniform in {delta;∗ P∗}. We also use the same method to improve Swanepoel and Geertsema's procedure for selecting the normal population which has the largest population mean Hoel and Mazumdar s extension of Paulson's procedure for a class of distributions in the Koopman-Darmois family and Hoel and Sobel s elimination "play the winner" procedure for selecting the binomial population with the largest probability of success
33 citations
TL;DR: In this article, a test based on Kologorov-Smirnov-type statistics is proposed for testing for changes in distribution in a nonparametric setting, both the retrospective and sequential detection settings are conside...
Abstract: Test based on Kologorov-Smirnov-type statistics are proposed for testing for changes in distribution in a nonparametric setting. Both the retrospective and sequential detection settings are conside...
12 citations
TL;DR: In this paper, the authors discuss sequential procedures for estimating the difference of means of two independent negative exponential populations with unknown locations and obtain minimum risk point estimators when the loss function is squared error plus linear cost.
Abstract: In this paper, we discuss sequential procedures for estimating the difference of means of two independent negative exponential populations with unknown locations.Up to various orders of approximation, we obtain minimum risk point estimators when the loss function is squared error plus linear cost.We address this problem separately when the scale parameters are (i) unknown but equal and (ii) unknown and unequal.In the first situation, we derive, among other things, asymptotic second order characteristics; while in the second situation, we are able to develop only the first order properties.We also give comments on truly encouraging moderate sample size performances of the proposed sampling procedures.
10 citations
TL;DR: In this article, two modifications of Sen's recursive procedure for the location model are considered and they are shown to be asymptotically equivalent to Sen's one, however easily computable.
Abstract: Sen (1984) recursive and noncursive procedures based on M–statistics to test for a possible change in the regression relationships occuring at an unknown time point. Here two modifications of Sen's recursive procedure for the location model are considered. These procedures are based on recursive M–estimators and a recursive one–step version M–estimators. They are shown to be asymptotically equivalent to Sen's one, however easily computable.
9 citations
TL;DR: In this article, the authors consider the problem of sequential estimation of a probability density function at a given point x 0, and investigate the asymptotic behaviour of the moments of the stopping rules when the length of the interval tends to zero.
Abstract: Let f be a probability density function on RP. we consider the problem of sequential estimation of f at a given point x0. For a class of recursive kernel estimators and a class of stopping rules based on the idea of fixed–width interval estimation we investigate the asymptotic behaviour of the moments of the stopping rules when the length of the interval tends to zero.
8 citations
TL;DR: In this article, a general treatment for the probrem of selecting the best (with respect to mean vector) from several multivariate normal populations is proposed. But the problem is not solved for known covariance matrices, as well as for unknown covariance matrix.
Abstract: This paper considers a general treatment for the probrem of selecting the best (with respect to mean vector) from several multivariate normal populations. The problem is solved for known covariance matrices, as well as for unknown covariance matrices. Asymptotic properties of the sample size are also discussed.
7 citations
TL;DR: In this article, a stopping time for estimating the failure rate of an exponential distribution with a smooth loss function and a gamma prior is constructed using censored data, and the optimality of this stopping time is established by using Dynkin's identity for Markov processes.
Abstract: A stopping time for estimating the failure rate of an exponential distribution with a smooth loss function and a gamma prior is constructed using censored data. The optimality of this stopping time is established by using Dynkin's identity for Markov processes. Although the results obtained are asymptotic in one sense, the optimality is exact.
5 citations
TL;DR: In each problem, a sequential procedure, based on maximum likelihood estimate, is proposed and is shown to be asymptotically risk efficient and derived as consequences of standard results in parametric counting process estimation problem.
Abstract: Time sequential and sequential point estimation problems with censored data are studied in the parametric counting process context. In each problem, a sequential procedure, based on maximum likelihood estimate, is proposed and is shown to be asymptotically risk efficient. These and other asymptotic results are derived as consequences of standard results in parametric counting process estimation problem.
4 citations
TL;DR: In this paper, a two-stage rule and a sequential rule for estimating the mean of an unspecified distribution with a given constant risk are presented. Under appropriate regularity conditions, the rules are shown to have asymptotically bounded regrets.
Abstract: This paper provides a two-stage rule and a sequential rule for estimating the mean of an unspecified distribution with a given constant risk. Under appropriate regularity conditions, the rules are shown to have asymptotically bounded regrets.
3 citations
TL;DR: In this article, a multiple response extension is proposed for the univariate calibration problem and a sequential solution is developed, which is based on the theory developed by Chow and Robbins (1965).
Abstract: A multiple response extension is proposed for the univariate calibration problem and a sequential solution is developed. The sequential sampling procedure is a generalization of existing univariate results of Perng and Tong (1974) and is based on the theory developed by Chow and Robbins (1965). It is shown that the proposed multiple response procedure is at least as good as and in many instances better than the univariate method
TL;DR: Optimal and suboptimal Bayesian sequential dsigns for the selection of the better of two Bernoulli populations are described in this article, where the objective is to minimize the number of trials on the inferior population in a fixed maximum number of iterations.
Abstract: Optimal and suboptimal Bayesian sequential dsigns for the selection of the better of two Bernoulli populations are described. The objective is to minimise the number of trials on the inferior population in a fixed maximum number of trials. Some numerical results are presented and the performance of the schemes are compared with the simple play the winner rule and with rules due to Bechhofer and Kulkarni.
TL;DR: For nonlinear costs of observation, the shape of cost functions has been shown to influence the shape and shape of optimal stopping boundaries as mentioned in this paper, and similar results are derived in the case of discrete time when the Wiener process is replaced by the sequence of partial sums arising from i.i.d. observations.
Abstract: The assumption of linear costs of observation often leads to optimal stopping boundaries which are straight lines. As for nonlinear costs of observation, the question arises how the shape of cost functions influences the shape of optimal stopping boundaries. For a certain stopping problem for the Wiener process, this was addressed by Irle (1987), for cost functions , 0 0, where Furthermore, similar results are derived in the case of discrete time, when the Wiener process is replaced by the sequence of partial sums arising from i.i.d. observations.
TL;DR: In this paper, the problem of constructing fixed-width confidence intervals for having the confidence coefficient at least or approximately α which is pre-assigned was considered, and various two-stage, sequential and three-stage sampling techniques were presented.
Abstract: We consider k (≥ 2) negative exponential populations having unknown locations θ1,...,θk, but unknown and equal scale parameter σ. We look at the problem of constructing fixed-width confidence intervals for having the confidence coefficient at least or approximately α which is preassigned. We present various two-stage, sequential and three-stage sampling techniques and discuss several exact and first- as well as second-order asymptotic properties.
TL;DR: In this paper, the authors considered three simple random sampling strategies: (i) with replacement, mean per unit estimation, (ii) without replacement and (iii) mean per distinct unit estimation and showed that the second strategy still fares better than the first, although the third strategy may not perform better than second one.
Abstract: Bounded risk estimation of the mean of a finite population is considered under three simple random sampling strategies: (i) with replacement, mean per unit estimation, (ii) with replacement, mean per distinct unit estimation, and (iii) without replacement, mean per unit estimation. It is well known that in the fixed sample size scheme, (iii) fares better than (ii) and (ii) better than (i). However, in the current context, the sample sizes are dictated by (possibly, degenerate) stopping times, and visualizing the cost (due to measurements/recording, etc.) as a function of the number of distinct units in the sample, it is shown that the second strategy still fares better than the first, although the third strategy may not perform better than the second one. Actually, in the case of known population variance, it is shown that in the light of the number of distinct units, the difference of ASN for the second and third strategies can never be greater than two or less than minus one. A similar relationship also...
TL;DR: In this paper, an extension of a theorem by Woodroofe and Yu (1981) is provided which establishes, under certain conditions, the "asymptotic second-order efficiency" (in the sense of Ghosh and Mukhopadhyay (1981)) of certain stopping rules that arise from problems in sequential estimation.
Abstract: An extension of a theorem by Woodroofe (1977) and Yu (1981) is provided which establishes, under certain conditions, the "asymptotic second-order efficiency" (in the sense of Ghosh and Mukhopadhyay (1981)) of certain stopping rules that arise from problems in sequential estimation. The outcome of a limited Monte Carlo study is also discussed.
TL;DR: In this article, a generalized recursive method was developed to obtain error probabilities and the expected sample size for a truncated SPRT with converging boundaries, and a simple relationship was established to predict the expected error probability and sample size.
Abstract: We develop a generalized recursive method to obtain error probabilities and the expected sample size for a truncated SPRT with converging boundaries. Using Wald's constant bounds, for IID normal observations, integer truncation points are obtained that guarantee actual error probabilities no worse than the desired probabilities. A simple relationship is established to predict the expected sample size. Anderson's converging boundaries are applied to a discrete observation process. It is found that some of these boundaries are conservative while others yield actual error probabilities that are higher than desired. Other converging boundaries are also considered.
TL;DR: In this article, a sequential procedure to estimate an estimable parameter of the underlying unkown distribution function is discussed, where the estimator is based on the corresponding U-statistic.
Abstract: This paper discusses a sequential procedure to estimate an estimable parameter of the underlying unkown distribution function. The estimator is based on the corresponding U-statistic. Under appropr...
TL;DR: In this paper, the estimation of the slope parameter β e R k for k ≥ 3, in a general linear model, is considered and a class of James Stein estimators is proposed and is compared with the least squares estimator under an appropriate stopping rule.
Abstract: The paper considers the estimation of the slope parameter β e R k for k ≥3, in a general linear model. A class of James Stein estimators is proposed and is compared with the least squares estimator under an appropriate stopping rule. It is shown that the sequential James Stein estimator dominates the sequential least squares estimator. Furthermore, under mild regularity conditions, a second order asymptotic risk expansion for the sequential James-Stein estimator is obtained.
Keil1
TL;DR: In this paper, the authors derived asymptotic expansions for the expected value of the first exit times T(b) = inf {n ≧ m: sn > f(b, n)}, b ≧ 0, m ≧ 1, and the associated last exit times L(b), where s denotes the n-th partial sum of a sequence of i.i.d. variables s x1,x2, with positive mean μ and finite variance σ2.
Abstract: This paper derives asymptotic expansions for the expected value of the first exit times T(b) = inf {n ≧ m: sn > f(b, n)}, b ≧ 0, m ≧ 1, and the associated last exit times L(b) = sup{n ≧ m: sn ≧ f (b, n)}, where s denotes the n-th partial sum of a sequence of i.i.d. randon variable s x1,x2, … with positive mean μ and finite variance σ2, and f(b) are twice differentiable boundaries satisfying some further regularity conditions. Two main results are presented. The first one gives expansions for ET(b) and EL(b) up to terms of order 0(1), as b→ ∞ m, without using renewal theory. These condone gives an expansion for ET(b) upto vanishing terms in the nonarithmetic case which requires to extend some results from nonlinear renewal theory, as developed by Lai and Siegmund, Woodroofe et al., to a certain parameter-dependent context.