Fixed Sample Size Procedure Research Articles

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.

Multi-stage point estimation of the common location parameter of several exponential distributions

The problem of point estimation of the common location parameter of several exponential distributions is considered. The scale parameters are assumed to be unknown and unequal. Consideration is given to a family of loss functions. Non-existence of the fixed sample size procedure is established to deal with the present estimation problem. Purely sequential, accelerated sequential, three-stage and two-stage procedures are developed to handle the situation. Relative advantage and disadvantage of these estimation approaches are established.

Sequential accuracy testing plans for the applicability of existing tree volume equations

To determine whether an existing tree volume estimation equation is acceptable for application to a new species, a new region, or a local area, an accuracy test could be carried out. Three sequential accuracy testing plans (SATP) were developed, along with approximate operating characteristic and average sample number curves. These tests are extensions of fixed sample size accuracy tests using sequential probability ratio tests to obtain sequential sampling tests. Using Monte Carlo simulations with normally distributed error terms, the SATP procedures were shown to be reliable for classifying existing volume models as acceptable or not acceptable with achieved error probabilities (type I and type II error probabilities) that were lower than the nominal values specified prior to testing. Also, on average, the use of the SATP procedures resulted in a 40 to 50% reduction in the expected sample size compared with that required for an equally reliable fixed sample size procedure. A detailed example is given to illustrate the application of the SATP procedures.

Sequential interval estimation procedures for the mean exponential survival time and reliability function

The problems of confidence interval estimation are considered (a) for estimating the mean of a one-parameter exponential distribution and (b) for estimating the reliability function associated with the one-parameter exponential distribution. For the estimation problem (a), the confidence interval of ‘preassigned width and coverage probability’ is considered. For the estimation problem (b), the confidence interval of ‘fixed-ratio width and preassigned coverage probability’ is proposed. The failure of the fixed sample size procedures to deal with these estimation problems is established and sequential procedures are proposed to deal with them. The proposed sequential procedures are proved to be ‘asymptotically efficient and consistent’ in the Chow-Robbins [Chow and Robbins, Ann. Math. Statist. 36,457–462 (1965)] sense. Asymptotic distributions of the stopping times are derived and second-order approximations are obtained for the average sample numbers associated with them.

Least favorable configuration and asymptotic efficiency of multinomial selection procedures

Abstract Inverse sampling procedures to select the cell corresponding to the highest multinomial cell probability have been studied and used widely in the literature. In this paper we establish the superiority of the inverse sampling procedure to the fixed sample size procedure over all configurations of cell probabilities in the asymptotic sense. This trend is empirically shown to prevail for small sample size as well. The results have also been used to derive the least favorable configurations (LFC) for the procedures over different indifference zones and as the number of observations tends to infinity. A byproduct of this approach is simpler proofs of several useful results in the literature.

A fixed sample size selection procedure for negative binomial populations

A fixed sample size procedure for selecting the ‘best’ ofk negative binomial populations is developed. Selection is made in such a way that the probability of correct selection is at leastP* whenever the distance between the probabilities of success is at leastδ*. The exponentr is assumed to be known and the same for all populations. Extensive computer calculations* were employed to obtain the exact least favorable configuration. The smallest sample sizes needed to meet specifications (P*,δ*) are tabulated forr=1 (1)5;δ*=0.05 (0.05) 0.55 andP*=0.75, 0.80, 0.90, 0.95, 0.98, 0.99 involvingk=3 (1) 6, 8, 10 populations.

Sequential estimation of the mean of a first-order autoregressive process

This paper considers the problem of sequential point estimation, under an appropriate loss function, of the location parameter when the errors form an autoregressive process with unknown scale and autoregressive parameters, A sequential procedure is developed and an asymptotic second order expansion is provided for the difference between expected stopping time and the optimal fixed sample size procedure. Also, the asymptotic normality of the stopping time is proved. Though the procedure Is asymptotically risk efficient, it. Is not clear whether it has bounded regret.

Sequential Classification of Prey/Predator Ratios with Application to European Red Mite (Acari: Tetranychidae) and Typhlodromus pyri (Acari: Phytoseiidae) in New York Apple Orchards

A sequential sampling procedure for classifying the ratio of prey/predators with respect to a critical ratio was developed. This procedure was combined with a sequential density classification procedure for use in sampling European red mite ( Panonychus ulmi (Koch)) and a phytoseiid predator, Typhlodromus pyri (Scheuten) in New York apple orchards. Use of the sequential procedure would result in ≥40% savings in sample size for many prey and predator densities. Frequencies of erroneous classification were similar for the sequential procedure and a fixed sample size procedure that used the maximum number of samples that might be taken when using the sequential method. To use the sequential ratio classification procedure, variance-mean models for the prey and predator are required as well as knowledge of the correlation between these two populations. Sensitivity analysis showed that the procedure, as applied to European red mite and T. pyri , is robust with respect to variation in this correlation.

Closed inverse sampling procedure for selecting the largest multinomial cell probability

A class of closed inverse sampling procedures R(n,m) for selecting the multinomial cell with the largest probability is considered; here n is the maximum sample size that an experimenter can take and m is the maximum frequency that a multinomial cell can have. The proposed procedures R(n,m) achieve the same probability of a correct selection as do the corresponding fixed sample size procedures and the curtailed sequential procedures when m is at least n/2. A monotonicity property on the probability of a correct selection is proved and it is used to find the least favorable configurations and to tabulate the necessary probabilities of a correct selection and corresponding expected sample sizes

Thek-in-a-row procedure in selection theory

Ak-in-a-row procedure is proposed to select the most demanded element in a set ofn elements. We show that the least favorable configuration of the proposed procedure which always selects the element when the same element has been demanded (or observed)k times in a row has a simple form similar to those of classical selection procedures. Moreover, numerical evidences are provided to illustrate the fact thatk-in-a-row procedure is better than the usual inverse sampling procedure and fixed sample size procedure when the distance between the most demanded element and the other elements is large and when the number of elements is small.

Non—existence of fixed sample size procedures for scale families

The paper considers the existence problem of bounded risk estimators for scale families. It is shown that there does not exist such an estimator if the sample size is fixed. Two–stage procedures are considered to solve the problem.

A note on confidence interval estimation following curtailed binomial tests

Techniques of Armitage (1958) for finding confidence intervals after sequential tests (SCI) are applied to curtailed binomial test boundaries. The form of the exact randomized SCI is given. We also show that the conservative confidence interval calculated as though a fixed sample size procedure had been used (FCI) remains conservative when used after stopping on a curtailed boundary. Numerical results are obtained to assess the potential gains that may be obtained by using the conservative SCI or exact SCI over the conservative FCI for these boundaries.

Second Order Approximation to the Risk of a Sequential Procedure

Given $X_1, X_2, \cdots$, i.i.d. with mean $\mu$ and variance $\sigma^2$, suppose that at stage $n$ one wishes to estimate $\mu$ by the sample mean $\bar{X}_n$, subject to the loss function $L_n = A(\bar{X}_n - \mu)^2 + n, A > 0$. If $\sigma$ is known, the optimal fixed sample size $n_0 \approx A^{1/2}\sigma$ can be used, with corresponding risk $R_{n_0}$, but if $\sigma$ is unknown there is no fixed sample size procedure that will achieve the risk $R_{n_0}$. For the sequential estimation procedure with stopping rule $T = \inf\{n \geq n_A: n^{-1} \sum^n_1 (X_i - \bar{X}_n)^2 \leq A^{-1} n^2\}$, the second order approximation of Woodroofe (1977) to the risk $R_T$ for normal $X_i$ is extended to the distribution-free case. Specifically, if the $X_i$ have finite moments of order greater than eight and are non-lattice, under certain conditions on the delay $n_A$ it is shown that the regret $R_T - R_{n_0} = c + o(1)$ as $A \rightarrow \infty$, where $c$ depends on the first four moments of the distribution of the $X_i$. For the lattice case, bounds of the form $c_1 + o(1) \leq R_T - R_{n_0} \leq c_2 + o(1)$ are obtained, where the $c_j$ are $c \pm 3$. It follows from these approximations that the regret can take arbitrarily large negative values as the distribution of the $X_i$ varies, in contrast to previous results for normal and gamma cases.

The combined inverse binomial sampling procedure

In the Combined Inverse Binomial Procedure (CIB), independent Bernoulli trials are performed one after another until for the first time either a fixed number of successes or a fixed number of failures are obtained. The sample size is the minimum of a pair of negative binomial variables. The distribution of the sample size and its expectation as well as the problems of estimation and testing of hypotheses in CIB are dealt with. It is seen that tests based on CIB procedures can lead to savings in sample size as compared to fixed sample size procedures in certain situations. It is also briefly indicated how the theory can be extended to the case of combined inverse multinomial procedures.

On a closed sequential procedure for categorical data, and tests for equiprobable cells

Cacoullos & Sobel (1966) proposed a closed sequential procedure for identifying the cell probability in a multinomial distribution that has the largest value. They showed, inter alia, that their sequential procedure has certain advantages over the fixed sample size procedure of Bechhofer et al. (1959). This paper extends this procedure in two directions. First, moments of the probability function are written in terms of the tail probability of a negative multinomial distribution. This makes it a relatively simple matter to determine the moments in certain important special cases. Second, an exact test for equiprobable cells is proposed, and certain properties of the test are established. In particular, the test is shown to be unbiased in the sense of majorization.

Bounded Regret of a Sequential Procedure for Estimation of the Mean

Let $X_1, X_2, \cdots$ be independent observations from a population with mean $\mu$ and variance $\sigma^2$, and suppose that given a sample of size $n$ one wishes to estimate $\mu$ by $\bar{X}_n$, subject to the loss function $L_n = A\sigma^{2\beta-2}(\bar{X}_n - \mu)^2 + n, A > 0, \beta > 0$. If $\sigma$ is known, then the optimal fixed sample size $n_0$ for minimizing the risk $R_n = EL_n$ can be computed, but if $\sigma$ is unknown there is no fixed sample size procedure that will achieve the minimum risk. For the case when $\sigma$ is unknown, a number of authors have investigated the performance of sequential estimation procedures designed to come close to attaining the minimum risk $R_{n_0}$. In this paper it is shown that for the class of sequential estimation procedures with stopping rules $T_A = \inf\{n \geq n_A: n^{-1} \sum^n_{i=1} (X_i - \bar{X}_n)^2 \leq A^{-1/\beta}n^{2/\beta}\}$ the regret $R_{T_A} - R_{n_0}$ remains bounded as $A\rightarrow \infty$, under suitable assumptions on the moments of $X_1$ and the delay $n_A$, but (unlike previous results of bounded regret) without any assumption about the type of distribution of $X_1$.

Selecting the smallest normal variance through the comparisons of several likelihoods

We are dealing with the problem of selecting the normal population having the smallest variance among k normal populations. We adopt indifference zone approach with a target value of the probability of correct selection P∗. One may refer to Bechhofer et al (1968), Gibbons et al (1977) and Gupta and Panchapakesan (1979). We view this problem as a mutliple—hypothesis testing problem and we propose sequential tests which are shown to have a substantial saving in the average sample sizes compared to the corresponding well known fixed—sample size procedures. We suggest, however, two separate methods of defining the saving and work primarily with one of these notions. We consider some special cases of some or all of the population means being known. In the casesk =2 and k=3, we have presented extensive numerical results through simulations showing the merits (in almost all the simulations) of our proposed procedures. Also, for k= 2 we study various asymptotic behavior (as ) of the stopping time involved i...

Sequential medical trials

A model for sequential clinical trials is discussed. Three proposed stopping rules are studied by the Monte Carlo method for small patient horizons and mathematically for large patient horizons. They are shown to be about equally effective and asymptotically optimal from both Bayesian and frequentist points of view and are markedly superior to any fixed sample size procedure.

The Bayes Sequential Procedure for Estimating the Arrival Rate of a Poisson Process

Abstract Let Wn, n = 0, 1, …, be the time until the nth arrival of a Poisson process with rate θ. Using loss L (θ, ) = θ−2(θ − )2 and sampling costs involving cost per arrival and cost per unit time, the Bayes sequential procedure is derived. The large-sample properties of the procedure are then studied in the classical framework, and the optimal stopping time N* is shown to be asymptotically equivalent to n*, that function of θ which minimizes the expected total cost given θ. Asymptotic normality of the optimal sequential estimator is also shown. Finally, the asymptotic saving in expected total cost from using the Bayes sequential procedure instead of the Bayes fixed sample size procedure is computed.

Selection using “play-the-winner” sampling for non-negative observations

The problem of choosing the best of k populations when observations are non-negative and small observations are preferred is considered using a truncated play-the-winner sampling scheme. Upper bounds on the expected sample sizes are derived. The procedure is compared with a fixed sample size procedure on the basis of expected sample size and of risk. It is noted that the probability of correct selection (P.C.S.) is the same for the two procedures so that existing tables may be used.