Fixed Sample Size Procedure Research Articles

On Fixed-Accuracy Confidence Intervals for the Parameters of Lindley Distribution and Its Extensions

The purpose of the present paper is to deal with sequential estimation of the parameter θ in a Lindley distribution. A fixed-accuracy confidence interval for θ with a preassigned confidence coefficient is developed. It is established that, no fixed sample size procedure can solve the estimation problem and hence a purely sequential methodology is proposed to deal with the situation. The first-order asymptotic efficiency and consistency properties associated with our purely sequential strategy are derived. Similar estimation strategies are also outlined for a few other extensions of the Lindley distribution. Extensive simulation analysis is carried out to validate the theoretical findings. We also provide a real data example, where we estimate the parameter related to the “initial mass function" for a particular cluster of stars.

Purely Sequential and k-Stage Procedures for Estimating the Mean of an Inverse Gaussian Distribution

In the first part of this paper, we propose purely sequential and k-stage (k ≥ 3) procedures for estimation of the mean μ of an inverse Gaussian distribution having prescribed ‘proportional closeness’. The problem is constructed in such a manner that the boundedness of the expected loss is equivalent to the estimation of parameter with given ‘proportional closeness’. We obtain the associated second-order approximations for both the procedures. Second part of this paper deals with developing the minimum risk and bounded risk point estimation problems for estimating the mean μ of an inverse Gaussian distribution having unknown scale parameter λ. We propose an useful family of loss functions for both the problems and our aim is to control the associated risk functions. Moreover, we establish the failure of fixed sample size procedures to deal with these problems and hence propose purely sequential and k-stage (k ≥ 3) procedures to estimate the mean μ. We also obtain the second-order approximations associated with our sequential procedures. Further, we provide extensive sets of simulation studies and real data analysis to show the performances of our proposed procedures.

Sequential tests of multiple hypotheses controlling false discovery and nondiscovery rates

We propose a general and flexible procedure for testing multiple hypotheses about sequential (or streaming) data that simultaneously controls both the false discovery rate (FDR) and false nondiscovery rate (FNR) under minimal assumptions about the data streams, which may differ in distribution and dimension and be dependent. All that is needed is a test statistic for each data stream that controls its conventional type I and II error probabilities, and no information or assumptions are required about the joint distribution of the statistics or data streams. The procedure can be used with sequential, group sequential, truncated, or other sampling schemes. The procedure is a natural extension of Benjamini and Hochberg’s (1995) widely used fixed sample size procedure to the domain of sequential data, with the added benefit of simultaneous FDR and FNR control that sequential sampling affords. We prove the procedure’s error control and give some tips for implementation in commonly encountered testing situations.

Sequential point estimation procedures for the parameter of a family of distributions

For the family of distributions considered by Chaturvedi and Alam, the problems of minimum risk point estimation and bounded risk point estimation of the parameter are handled. Consideration is given to squared-error loss function. The failure of fixed sample size procedure is established. Sequential procedures are proposed to deal with the situation. For both the estimation problems,the uniformly minimum variance unbiased estimator/maximum likelihood estimator and minimum mean square estimator are utilized to develop the fixed sample size procedures. Second-order approximations are derived for the sequential procedures and ‘improved’ estimators are proposed. First-order properties are also discussed. The negative ‘regret’ [see Starr and Woodroofe] is achieved.

Selection among Bernoulli populations in comparison with a standard

In this article we consider the problem of comparing several Bernoulli populations in order to identify the population producing the largest success probability that is also larger than a given standard. We propose a fixed sample size procedure for which we develop the probability of a correct selection and the least favorable configuration. We also propose a curtailed version of the fixed sample size procedure and show that the curtailed procedure reaches the same probability of a correct selection as the fixed sample size procedure, while using fewer observations. We provide tables to implement the procedures and illustrate them via an example. We use simulations to estimate the savings by the curtailment procedure.

Two-stage fixed-width and bounded-width confidence interval estimation methodologies for the common correlation in an equi-correlated multivariate normal distribution

In this article, two-stage and fixed sample size procedures are developed for constructing confidence intervals for the common correlation ρ of an equi-correlated multivariate normal distribution. Two different approaches for estimation are considered, namely, fixed-width and bounded-width interval estimation. In the fixed-width confidence interval estimation problem, a two-stage procedure is developed and the exact distribution of the corresponding stopping variable and the exact coverage probability of the interval estimator are derived. Asymptotic optimality properties such as asymptotic first-order efficiency and asymptotic consistency properties of the two-stage procedure are established. Bounded-width confidence intervals are obtained by applying fixed-accuracy estimation methodologies of Mukhopadhyay and Banerjee (2014, 2015a,b) and Banerjee and Mukhopadhyay (2016) using different transformations on ρ. Both fixed sample size and two-stage sampling methodologies for bounded-width confidence interval estimation of are developed incorporating such transformations. Finally, performances of all the procedures are compared via extensive sets of simulation studies.

Sequential Minimum Risk Point Estimation of the Parameters of an Inverse Gaussian Distribution

SYNOPTIC ABSTRACTIn the first part of this article, a minimum risk estimation procedure is developed for estimating the mean μ of an inverse Gaussian distribution having an unknown scale parameter λ. A weighted squared-error loss function is assumed, and we aim at controlling the associated risk function. First and second-order asymptotic properties are also established for our stopping rule. The second part deals with developing a minimum risk estimation procedure for estimating the scale parameter λ of an inverse Gaussian distribution. We make use of a squared-error loss function here. The failure of a fixed sample size procedure is established and, hence, some sequential procedures are proposed to deal with this situation. For this estimation problem, we make use of the uniformly minimum variance unbiased estimator (UMVUE) and the minimum mean square estimator (MMSE) of the associated parameters. Second-order approximations are derived for the sequential procedures and improved estimators are proposed.

Risk-efficient sequential estimation of multivariate random coefficient autoregressive process

A vector-valued autoregressive time series model is considered. The autoregressive coefficients of the model are random with possible dependencies among them. Estimation of the large number of parameters in such models becomes costly with an increase in dimension. A sequential procedure is proposed that promises a significant gain in the sample size thus reduction in the cost of implementation. The procedure is also risk efficient in the sense that as the cost of sampling becomes negligible the asymptotic predictive risk of the proposed procedure reaches the oracle predictive risk corresponding to the best fixed sample size procedure that assumes the values of the nuisance parameters to be known. Extensive simulation results are presented to illustrate the properties of the proposed procedure in a finite sample.

Adaptive sequential selection procedures with random subset sizes

ABSTRACTWe introduce a new family of sequential selection procedures wherein the subsets selected have random sizes. In comparison to subset selection procedures that select subsets of fixed size, the new procedures alleviate the need to specify the subset size prior to the experiment. We discuss the application of such procedures in the context of early phase clinical trials. The new procedures retain the adaptive features of the Levin-Robbins-Leu family of sequential subset selection procedures for selecting subsets of fixed size, namely, sequential elimination of inferior treatments and sequential recruitment of superior treatments. These two adaptive features respectively address ethical concerns that diminish interest in nonadaptive procedures and also allow promising treatments to be brought forward for further testing without having to wait until the end of the trial. The new procedures differ from the classical subset selection procedures of Shanti S. Gupta in terms of their respective goals and operating characteristics and we compare the two approaches in a simulation study. The findings suggest that whereas Gupta’s procedure achieves its goal of including the single best treatment in the final selected subset with high probability, it does so by virtue of a nonadaptive, fixed sample size procedure that lacks necessary flexibility in the context of clinical research. By contrast, the new procedures aim to select treatment subsets that satisfy a different criterion, that of acceptable subset selection with high probability, while allowing adaptive elimination and recruitment and other flexibilities which we discuss to fit the practical needs of selection methods in clinical research.

Multiple Hypothesis Tests Controlling Generalized Error Rates for Sequential Data

The $\gamma$-FDP and $k$-FWER multiple testing error metrics, which are tail probabilities of the respective error statistics, have become popular recently as less-stringent alternatives to the FDR and FWER. We propose general and flexible stepup and stepdown procedures for testing multiple hypotheses about sequential (or streaming) data that simultaneously control both the type I and II versions of $\gamma$-FDP, or $k$-FWER. The error control holds regardless of the dependence between data streams, which may be of arbitrary size and shape. All that is needed is a test statistic for each data stream that controls the conventional type I and II error probabilities, and no information or assumptions are required about the joint distribution of the statistics or data streams. The procedures can be used with sequential, group sequential, truncated, or other sampling schemes. We give recommendations for the procedures' implementation including closed-form expressions for the needed critical values in some commonly-encountered testing situations. The proposed sequential procedures are compared with each other and with comparable fixed sample size procedures in the context of strongly positively correlated Gaussian data streams. For this setting we conclude that both the stepup and stepdown sequential procedures provide substantial savings over the fixed sample procedures in terms of expected sample size, and the stepup procedure performs slightly but consistently better than the stepdown for $\gamma$-FDP control, with the relationship reversed for $k$-FWER control.

Sequential Point Estimation of a Function of the Exponential Scale Parameter

We consider sequential point estimation of a function of the scale parameter of an exponential distribution subject to the loss function given as a sum of the squared error and a linear cost. For a fully sequential sampling scheme, we present a sufficient condition to get a second order approximation to the risk of the sequential procedure as the cost per observation tends to zero. In estimating the mean, our result coincides with that of Woodroofe (1977). Further, in estimating the hazard rate for example, it is shown thatour sequential procedure attains the minimum risk associated with the best fixed sample size procedure up to the order term.

Sequential Testing for Sparse Recovery

This paper studies sequential methods for recovery of sparse signals in high dimensions. When compared with fixed sample size procedures, in the sparse setting, sequential methods can result in a large reduction in the number of samples needed for reliable signal support recovery. Starting with a lower bound, we show any coordinate-wise sequential sampling procedure fails in the high dimensional limit provided the average number of measurements per dimension is less then log(s)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), where s is the level of sparsity and D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ) is the Kullback-Leibler divergence between the underlying distributions. A series of sequential probability ratio tests, which require complete knowledge of the underlying distributions is shown to achieve this bound. Motivated by real-world experiments and recent work in adaptive sensing, we introduce a simple procedure termed sequential thresholding, which can be implemented when the underlying testing problem satisfies a monotone likelihood ratio assumption. Sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than log(s)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ), achieving the lower bound. For comparison, we show any nonsequential procedure fails provided the number of measurements grows at a rate less than log(n)/D(P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ||P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ), where n is the total dimension of the problem.

Bayesian Fixed Sample Size Procedure for Selecting the Better of Two Poisson Populations With General Loss Function

In this paper an optimal (Bayesian) fixed sample size procedure for selecting the better of two Poisson populations is proposed and studied . Bayesian decision-theoretic approach with general loss function and Gamma priors are used to construct this procedure .The numerical result of this procedure are given with different loss functions constant , linear and quadratic , in one equation we can obtain the Bayes risk for the three types of the loss functions : constant , linear and quadratic . in this paper the numerical results are given by using Math Works Matlab ver. 7.10.0 .

An Accelerated Sequential Class to Minimize Combined Risk for Simultaneous Estimation of Parameters of Several

The fixed sample size procedure &nbsp;sometimes fail to deal with the estimation problem in which it&nbsp; is desired to control the combined &nbsp;risk associated with the estimation of parameters of various probability distributions simultaneously. In this paper for a general probabilistic model is proposed an &ldquo;accelerated&rdquo; sequential class to minimize the combined risk for simultaneous estimation of parameters of several probability distributions. The proposed class is shown to provide solutions for many estimation problems under different probabilistic setups for &ldquo; given precision &rdquo; problems. &nbsp;Besides, positive and negative moments for the stopping times are obtained and they are used for deriving the asymptotic expression for the &ldquo;regret&rdquo;&nbsp; associated with&nbsp; the class of &ldquo;accelerated&rdquo; sequential estimation procedure.

Impact of controlling the sum of error probability in the sequential probability ratio test

A generalized modified method is proposed to control the sum of error probabilities in sequential probability ratio test to minimize the weighted average of the two average sample numbers under a simple null hypothesis and a simple alternative hypothesis with the restriction that the sum of error probabilities is a pre-assigned constant to find the optimal sample size and finally a comparison is done with the optimal sample size found from fixed sample size procedure. The results are applied to the cases when the random variate follows a normal law as well as Bernoullian law.

A two-stage procedure for estimation of the common mean of several normal populations having unequal and unknown variances

The problem of fixed-width confidence interval estimation of the common mean of several normal populations having unknown and unequal variances is considered. A two-stage procedure is developed to tackle the problem and its nice asymptotic properties are studied. Zacks (9) considered sequential procedures to construct fixed-width confidence interval for the common mean of two normal populations assuming one population's variance to be known. Mukhopadhyay and Narayan (6) generalized the results of Zacks (9) when both the variances were unknown. They also considered the problem of constructing fixed-width confidence interval for the common location parameter of two exponential populations having unequal and unknown scale parameters. Costanza, Hamdy and Son (4) dealt with the problem of constructing a confidence interval of pre-assigned width and coverage probability for the common location parameter of several exponential populations, assuming their scale parameters to be unknown and unequal. They established the non- existence of the fixed sample size procedure to deal with this estimation problem. In order to tackle with the problem, along the lines of Stein (8), they developed a 'routine-type' two-stage procedure. For the same estimation problem Chaturvedi, Tiwari and Pandey (2) developed three-stage, accelerated sequential and purely sequential procedures and obtained the associated second-order approximations. In a later communication, Chaturvedi, Tiwari and Pandey (3) considered the problem of bounded risk point estimation of the common location parameter of several exponential distributions and proposed some multistage procedures and derived the second-order approximations. In the present note, we consider the problem of fixed-width confidence interval estimation of the common mean of several normal populations having unknown and possibly unequal variances. In Section 2, we consider the set-up of the estimation problem and establish the failure of the fixed sample size procedure to deal with it. In Section 3, a two-stage procedure is developed for the fixed-width confidence interval estimation problem and its asymptotic properties are obtained.

Group Testing with a Goal in Estimating the Number of Defects Under Imperfect Environmental Stress Screen Levels

ABSTRACT A fixed sample size procedure is proposed to obtain initial and then refined estimates of defect density for Environmental Stress Screening (ESS) plans. Ideas of group testing, i.e., testing units in batches instead of individually, where each test only indicates whether the test batch contains only good units or whether it contains at least one defective, is applied to the problem of estimating the probability π of an arbitrary unit being defective. Five estimators are given with their root mean square errors or asymptotic variances. An example and tables of procedure parameters are given to illustrate the proposed method.

Efficient cycle time-throughput curve generation using a fixed sample size procedure

The cycle time-throughput curve is one of the most important analytical tools used to assess operating policies in manufacturing systems. Unfortunately, the generation of this curve is complicated and time consuming when generation is based upon extensive simulation analysis. This paper presents a simulation-based, fixed sample size strategy for generating a cycle time-throughput curve with minimal mean square error that mitigates the typical problems associated with a simulation-based cycle time-throughput curve. The strategy comprises two components, the sampling method and sampling weights. A queuing network of five workstations in series is used for validation of the approach. Results indicate that the sampling method using antithetic variates is effective in reducing the variance as well as bias of a cycle time-throughput curve. Furthermore, this method is robust to the sample size. Given a sufficiently large sample, the combination of common random numbers and antithetic variates is preferred. A reduction in the sample size and complexity of the system increases the significance of the sampling weights.

Optimal adjustment strategies for a process with run-to-run variation and 0–1 quality loss

As short run manufacturing becomes more prevalent, run-to-run components of variation, such as that contributed by set-up error, have greater potential to crucially affect product quality. While efforts should be made to eliminate such between-run variance contributing factors, some will always be present. Here, we assume there is one such factor which we envisage as set-up error that, unless the process is adjusted, remains fixed throughout a run. We develop a single adjustment strategy based on taking a sample of fixed size from the process. If a significant set-up error is indicated, a single compensatory adjustment, equal to the predicted process offset, is executed. The actual procedure depends on process parameters, including adjustment error, run size, and adjustment and sampling costs. The procedure not only specifies the adjustment amount, if any, but the time during the run at which to adjust. The procedure is, optimal among all fixed sample size procedures for the chosen cost function. Besides incorporating adjustment and sampling costs, the cost function is based on a 0–1 loss criterion, where the loss is 0 (1) units per item produced if the process offset caused by set-up error is less than or equal to (greater than) a specified amount. Tables are provided, with examples, illustrating the procedure for representative values of process parameters, costs, and run sizes.

A two-stage procedure for selecting the largest multinomial cell probability when nuisance cell is present

This paper considers the problem of selecting the cell with the largest cell probability of multinomial distribution when one of the cells is considered as nuisance cell. A multinomial distribution is supposed to have three cells for the first attempt to this problem. A Stein-type two-stage procedure is proposed to select the best cell under the two types of indifference-zone formulation in ranking and selection theory. We first identify the configuration at which the probability of correct selection is minimum under the fixed-sample size procedure when the nuisance cell probability is fixed. Then we propose a two-stage procedure to (1) estimate the nuisance cell probability in the first stage, and (2) select the best cell in the second stage. It is shown that the proposed two-stage procedure is asymptotically efficient. The tables of the procedure parameters are provided and a Monte Carlo simulation is given to illustrate the efficiency of the procedure.