△-correct decision for location and scale parameters

In selecting t out of k populations, a △-correct decision is said to be made if the smallest location parameter for the selected populations is not more than △ below the largest location parameter for the non-selected populations. (For seal? parameters there is a similar definition in terms of ratio3.) The minimum probability of △-correct decision over the entire parameter pace is shown to be equal to the minimum probability of correct selection over a preference zone determined by △.     

Applications of a probability inequality to ranking and selection and other related problems

A probability inequality of conditionally independent and identically distributed (i.i.d.) random variables obtained recently by the author is applied to ranking and selection problems. It is shown that under both the indifference-zone and the subset formulations, the probability of a correct selection (PCS) is a cumulative probability of conditionally i.i.d, random variables. Therefore bounds on both the PCS and the sample size required can be obtained from that probability inequality. Applications of that inequality to other multiple decision problems are also considered. It is illustrated that general results concerning conditionally i.i.d. random variables are applicable to many problems in multiple decision theory.     

On lower confidence bounds for pcs in truncated location parameter models

We are concerned with deriving lower confidence bounds for the probability of a correct selection in truncated location-parameter models. Two cases are considered according to whether the scale parameter is known or unknown. For each case, a lower confidence bound for the difference between the best and the second best is obtained. These lower confidence bounds are used to construct lower confidence bounds for the probability of a correct selection. The results are then applied to the problem of seleting the best exponential populationhaving the largest truncated location-parameter. Useful tables are provided for implementing the proposed methods.     

Tests for compound periodicities and estimating a non linear function

We propose two tests for testing compound periodicities which are the uniformly most powerful invariant decision procedures against simple periodicities. The second test can provide an excellent estimation of a compound periodic non linear function from observed data. These tests were compared with the tests proposed by Fisher and Siegel by Monte Carlo studies and we found that all the tests showed high power and high probability of a correct decision when all the amplitudes of underlying periods were the same. However, if there are at least several different periods with unequal amplitudes, then the second test proposed always showed high power and high probability of a correct decision, whereas the tests proposed by Fisher and Siegel gave 0 for the power and 0 for the probability of a correct decision, whatever the standard deviation of pseudo normal random numbers. Overall, the second test proposed is the best of all in view of the probability of a correct decision and power.     

Combining two classical approaches for statistical selection

This paper presents a selection procedure that combines Bechhofer's indifference zone selection and Gupta's subset selection approaches, by using a preference threshold. For normal populations with common known variance, a subset is selected of all populations that have sample sums within the distance of this threshold from the largest sample sum. We derive the minimal necessary sample size and the value for the preference threshold, in order to satisfy two probability requirements for correct selection, one related to indifference zone selection, the other to subset selection. Simulation studies are used to illustrate the method.     

Least significant spacing for ‘one versus the rest’ normal populations

Consider sample means from k(≥2) normal populations where the variances and sample sizes are equal. The problem is to find the ‘least significant difference’ or ‘spacing’ (LSS) between the two largest means, so that if an observed spacing is larger we have confidence 1 - α that the population with largest sample mean also has the largest population mean.

When the variance is known it is shown that the maximum LSS occurs when k = 2, provided a < .2723. In other words, for any value of k we may use the usual (one-tailed) least significant difference to demonstrate that one population has a population mean greater than (or equal to) the rest.

When the variance is estimated bounds are obtained for the confidence which indicate that this last result is approximately correct.     

Asymptotic properties of a generalized regression-type predictor of a finite population variance in probability sampling

A system of predictors for estimating a finite population variance is defined and shown to be asymptotically design-unbiased (ADU) and asymptotically design-consistent (ADC) under probability sampling. An asymptotic mean squared error (MSE) of a generalized regression-type predictor, generated from the system, is obtained. The suggested predictor attains the minimum expected variance of any design-unbiased estimator when the superpopulation model is correct. The generalized regression-type predictor and the predictor suggested by Mukhopadhyay (1990) are compared.     

Probability of correct selection of weibull versus gamma based on livelihood ratio

This paper proposes the use of tne likelihood ratio statistic in choosing between a Neibull or gamma model, values of the probability of correct seeiection are obtained by Monte Carlo simulation. This method provides some basis for decision even when the sample size is small. The technique is applied to four sets of data from the literature.     

Truncation of the Bechhofer-Kiefer-Sobel sequential procedure for selecting the multinomial event which has the largest probability

In this article we study the effect of truncation on the performance of an open vector-at-a-time sequential sampling procedure (P^* _B) proposed by Bechhofer, Kiefer and Sobel , for selecting the multinomial event which has the largest probability. The performance of the truncated version (P^{* B} _T) is compared to that of the original basic procedure (P^* _B). The performance characteristics studied include the probability of a correct selection, the expected number of vector-observations (n) to terminate sampling, and the variance of n. Both procedures guarantee the specified probability of a correct selection. Exact results and Monte Carlo sampling results are obtained. It is shown that P^* _{B T}is far superior to P^* _B in terms of E{n} and Var{n}, particularly when the event probabilities are equal.The performance of P^* _{B T} is also compared to that of a closed vector-at-a-time sequential sampling procedure proposed for the same problem by Ramey and Alam; this procedure has here to fore been claimed to be the best one for this problem. It is shown that p^* _{B T} is superior to the Ramey-Alam procedure for most of the specifications of practical interest.     

Selecting the Best Exponential Population When the Scale Parameters are Bounded

Consider k (≥ 2) independent exponential populations with different location and scale parameters. Call a population associated with largest of unknown location parameters as the best population. For the goal of selecting the best population, it is established that if the scale parameters are completely unknown, then the indifference-zone probability requirement can not be guaranteed by any single sample decision rule which is just and translation invariant. Under the assumption that the scale parameters are bounded above by a known constant, a single sample selection procedure is proposed for which the indifference-zone probability requirement can be guaranteed. Under the same assumption, 100P*% simultaneous upper confidence intervals for all distances from the largest location parameter are also obtained.     

On a conjecture concerning the least favorable configuration of a two-stage selection procedure

A two-stage procedure 𝓅with screening in the first stage to find the population with the largest mean out of k ≧ 2 normal populations with unknown means and a common variance is under concern. It was proposed and previousiy studied by Cohen (1959), Alam (1970) and Tamhane and Bechhofer (1977, 1979) using the indifference-zone approach. The conjecture that the least favourable parameter configuration for the probability of a correct selection is of the slippage type remained unproved for k ≧ 3. Miescke and Sehr (1980) proved the conjecture for k=3. The problem was further discussed by Gupta and Miescke (1982). A general proof for rhe conjecture will be given in this paper.     

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     

On the difference between ML and REML estimators in the modelling of multivariate longitudinal data

A random effects model is examined in the multivariate setting where more than one characteristics are measured at each time point. ML and REML estimators are obtained under the restriction that estimates of variance matrices being at least p.s.d. It is shown that REML has greater probability of giving full rank estimates of variance components matrices but as regards the efficiency in the estimation of the location parameter, correct specification of the number of random effects is needed. In general, REML provides larger estimates of variance of model parameters than ML.     

Selection of the best in 2×2 factorial designs

Selection of the cell in a 2×2 -factorial design with the greatest mean is considered. A general class of ranking and selection procedures (RSP) is constructed to include methods based on the largest marginal cell means (SP1) or on the largest cell mean (SP3). Using the preference zone approach the minimum probability of a correct solution is found, In this paper a RSP which maximizes the minimum probability of a correct solution over the preference zone is found. In this way selection of the cell with the greatest observed mean is proven to be admissible.     

On two-stage selection procedures and related probability-inequalities

In this paper we discuss a modification of the Dudewicz-Dalal procedure for the problem of selecting the population with the largest mean from k normal populations with unknown variances. We derive some inequalities and use them to lower-bound the probability of correct selection. These bounds are applied to the determination of the second-stage sample size which is required in order to achieve a prescribed probability of correct selection. We discuss the resulting procedure and compare it to that of Dudewicz and Dalai (1975).     

Convergence rates of change-point estimators and tail probabilities of the first-passage-time process

In the classical setting of the change-point problem, the maximum-likelihood estimator and the traditional confidence region for the change-point parameter are considered. It is shown that the probability of the correct decision, the coverage probability and the expected size of the confidence set converge exponentially fast as the sample size increases to infinity. For this purpose, the tail probabilities of the first passage times are studied. General inequalities are established, and exact asymptotics are obtained for the case of Bernoulli distributions. A closed asymptotic form for the expected size of the confidence set is derived for this case via the conditional distribution of the first passage times.     

随机优势在工程项目决策中的应用

随机条件下工程项目投资决策常用准则有概率最大准则、期望准则、期望方差准则，这些决策准则简单，但是由于没有考虑投资者的效用函数，决策具有一定的缺陷。因而，在工程项目投资决策者中考虑投资者效用函数的随机优势准则，利用三类效用函数和三类随机占优决策准则，通过案例演示其在工程项目投资决策中的具体应用。     

Simultaneous selection and estimation in general linear models

The problem of selecting the largest treatment parameter, and simultaneously estimating the selected treatment parameter, in a general linear model is considered in the decision theoretic Bayes approach. Both cases, where the error variance is known or unknown, are included. Bayes decision rules are derived for noninformative priors and for normal priors. The problem of finding Bayes designs, i.e. designs that have minimum Bayes risk, within a given class of designs is also discussed.     

Optimal Selection of the Most Reliable Design Based on Gamma Degradation Processes

Manufacturers are often faced with the problem of how to select the most reliable design among several competing designs in the stage of development. It becomes complicated if products are highly reliable. Under the circumstances, recent work has focused on the study with degradation data by assuming that degradation paths follow Wiener processes or random-effect models. However, it is more appropriate to use gamma processes to model degradation data with monotone-increasing pattern. This article deals with the selection problem for such processes. With a minimum probability of correct decision, optimal test plans can be obtained by minimizing the total cost.     

Indifference-zone ranking and selection: confidence intervals for true achieved P(CD)

In this paper we present procedures which attempt to allow experimenters to take at least partial advantage of more-favorable configurations of the population parameters (without, in the process, sacrificing precise probability statements about the inferences made) by obtaining confidence intervals for the true probability of a correct decision, regarded as a function of the true unknown underlying parameter values.