Bayesian Interval Estimates Research Articles

Bayesian estimates for failure-free operation based on trials using measurement of definitive parameters

We obtain relations for Bayesian interval estimates for parameters of failure-free operation for components; the estimates are obtained from the results of trials that measure definitive parameters on the basis of dynamic parametric models of operability.

Bayes Estimators for the Continuous Uniform Distribution

Classical estimators for the parameter of a uniform distribution on the interval (0, θ) are often discussed in mathematical statistics courses, but students are frequently left wondering how to distinguish which among the variety of classical estimators are better than the others. We show how classical estimators can be derived as Bayes estimators from a family of improper prior distributions. We believe that linking the estimation criteria in a Bayesian framework is of value to students in a mathematical statistics course, and we believe that the students benefit from the exposure to Bayesian methods. In addition, we compare classical and Bayesian interval estimators for the parameter θ and illustrate the Bayesian analysis with an example.

Interval Estimation of Bivariate Correlations With Missing Data on Both Variables: A Bayesian Approach

The posterior distribution of the bivariate correlation ( ρxy) is analytically derived given a data set consisting N1 cases measured on both x and y, N2 cases measured only on x, and N3 cases measured only on y. The posterior distribution is shown to be a function of the subsample sizes, the sample correlation ( rxy) computed from the N1 complete cases, a set of four statistics which measure the extent to which the missing data are not missing completely at random, and the specified prior distribution for ρxy. A sampling study suggests that in small ( N = 20) and moderate ( N = 50) sized samples, posterior Bayesian interval estimates will dominate maximum likelihood based estimates in terms of coverage probability and expected interval widths when the prior distribution for ρxy is simply uniform on (0, 1). The advantage of the Bayesian method when more informative priors based on beta densities are employed is not as consistent.

Bayesian HPM intervals for the number of nonconforming items in a lot

The incorporation of prior information or personal beliefs into a statistical procedure is the basis for the Bayesian approach to statistical inference. We discuss an information-theoretic approach which minimizes the amount of extraneous information embedded in a prior distribution yet leaves intact the prior beliefs. The procedure is used to select a Polya prior distribution for the number of nonconforming items in a lot. Theoretically speaking, the Polya distribution is the appropriate distribution for describing how the number of nonconforming items vary from lot to lot when the process fraction nonconforming varies according to a beta distribution. Assuming hypergeometric sampling from a lot, we present tables which give Bayesian interval estimates of the number of nonconforming items in a lot based on the number of nonconforming items in the sample. Two cases are considered: prior information as to the average process/lot fraction nonconforming is either (a) not available or (b) believed to be a specific value.

Bayesian analysis of a multiple-recapture model

In the Bayesian analysis of a multiple-recapture census, different diffuse prior distributions can lead to markedly different inferences about the population size N. Through consideration of the Fisher information matrix it is shown that the number of captures in each sample typically provides little information about N. This suggests that if there is no prior information about capture probabilities, then knowledge of just the sample sizes and not the number of recaptures should leave the distribution of Nunchanged. A prior model that has this property is identified and the posterior distribution is examined. In particular, asymptotic estimates of the posterior mean and variance are derived. Differences between Bayesian and classical point and interval estimators are illustrated through examples.

Small-sample properties of finite population ratios

Employing Bayesian finite population resampling techniques [referred to as finite population Bayesian bootstrap, FPBB, by Lo 1988], asymptotic posterior distributions of a linear combination of functions of stratum ratios and the corresponding confidence interval estimates are approximated. A straight forward algorithm is provided for the evaluation of the performance of the FPBB approximations to the asymptotic posterior distributions. Monte Carlo simulation is used to construct confidence interval estimates based on the FPBB method and to (graphically) demonstrate the performance of the FPBB method inapproximating the asymptotic posterior distributions of functions of ratios using Efron's (1982) Law School data. The results of the Monte Carlo investigations indicate that the FPBB based interval estimates compare favorably with the usual large-sample Bayesian interval estimates even for moderate bootstrap samples

正規母集団からのファジィ区間データに基づく近似的ベイズ区間推定

正規母集団からのファジィ区間データに基づく近似的ベイズ区間推定

Conditional properties of Bayesian interval estimates

Consider the construction of an interval estimate for a scalar parameter of interest in the presence of orthogonal nuisance parameters. A conditional prior density on the parameter of interest that is proportional to the square root of its information element, generates one-sided Bayes intervals that are approximately confidence intervals as well, having coverage error of orderO(1/n), wheren is the sample size. We show that the frequency property of these intervals also holds conditionally on a locally ancillary statistic near the true parameter value.

Bayesian Interval Estimates Which are Also Confidence Intervals

SUMMARY Let Y 1, . . ., Yn denote independent observations each distributed according to a distribution depending on a scalar parameter θ suppose that we are interested in constructing an interval estimate for θ. One approach is to use Bayesian inference. For a given prior density, we can construct an interval such that the posterior probability that θ lies in the interval is some specified value. In this paper, a method is given for constructing a Bayesian interval estimate such that the coverage probability of the interval is approximately equal to the posterior probability of the interval.

An alternative decision rule for quality control in sequential sampling plans

The conventional rule for acceptance sampling based on sequential samples is based on the rationale of hypothesis testing developed by Wald (1947). This type of decision rule tests the hypothesis of P=p1 versus P=p2 as a proxy for determining whether P>d or P<d with P1<d<p2. It requires a zone of indifference between the rejection and acceptance levels p2 and p1. In this note, we propose an alternative rule for making the decision based on the confidence level of a one-sided Bayesian interval estimate of the parameter. This method results in direct determination of whether the proportion of defects P in the population is greater or less than a prespecified level d, rather than test two points as proxy for the decision. We present a numerical illustration of the rule and an example of determining rejection and acceptance numbers. Ue also compare the results with two conventional rules.

Bayesian bootstrap lower confidence interval estimation of the reliability and failure rate

The bootstrap, the Bayesian bootstrap and the Bayesian bootstrap with a Dirichlet process prior estimation of the reliability and failure rate functions is studied by using Monte Carlo techniques. These concepts are applied to real life data.

Correct Bayesian and frequentist intervals are similar

This paper argues that Bayesians and frequentists will normally reach numerically similar conclusions, when dealing with vague data or sparse data. It is shown that both statistical methodologies can deal reasonably with vague data. With sparse data, in many important practical cases Bayesian interval estimates and frequentist confidence intervals are approximately equal, although with discrete data the frequentist intervals are somewhat longer. This is not to say that the two methodologies are equally easy to use: The construction of a frequentist confidence interval may require new theoretical development. Bayesian methods typically require numerical integration, perhaps over many variables. Also, Bayesians can easily fall into the trap of over-optimism about their amount of prior knowledge. But in cases where both intervals are found correctly, the two intervals are usually not very different.

Neyman-Pearson and Bayes Interval Estimates for Sampling by Atiributes

Confidence intervals can provide important information about statistical estimates of EMP protection parameters during test programs which are often constrained by budgetary or other considerations. Under some circumstances Bayesian interval estimates are contained within Neyman-Pearson estimates. For example, this is true in the single stage case with uniform prior distribution under the constraint of equation (15). Differences between the Neynlan-Pearson and Bayesian intervals (with uniform prior distribution) rapidly decrease as the sample size increases. For sample sizes in excess of roughly 40 these differences are usually of little practical importance. Multistage sampling is usually employed to reduce the experimental program with a potential penalty in the width of the confidence interval. As demonstrated above, two stage confidence intervals contain corresponding intervals computed from a fixed sample size when the outcome is near acceptance-rejection boundaries and the level of confidence exceeds about 90%. When all or nearly all samples fail at some later stage, confidence bounds computed from multistage sampling are noticeably different from those computed with fixed sample size. In this circumstance the mathematical decision is to reject the sample. However, the mathematical decision is often only part of the overall operational decision. This mathematically unlikely occurrence can serve as a warning that unanticipated change or error has occurred during the test. Confidence limits computed from multistage sampling must then be employed with the usual caution which surrounds an unanticipated result.

HP-41C Programs for Bayesian Binomial and Exponential Interval Estimation with a Uniform Prior on the Reliability

HP-41C handheld calculator programs are presented for Bayes interval estimation of: 1) the reliability R of a component for binomial data; 2) both R and the MTBF for exponential data. In both cases, the ``ignorance'' prior is employed: uniform on R over the range 0 to 1. The input data are numbers of trials and successes for the binomial case; and test time, mission time and failures for the exponential. This paper describes the models and includes the programs.