Asymptotic and Resampling-Based Confidence Intervals for P(X < Y)

ABSTRACT

We consider asymptotic and resampling-based interval estimation procedures for the stress-strength reliability P(X < Y). We developed and studied several types of intervals. Their performances are investigated using simulation techniques and compared in terms of attainment of the nominal confidence level, symmetry of lower and upper error rates, and expected length. Recommendations concerning their use are given.     

Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Small Sample Confidence Intervals for the Odds Ratio

Abstract

Numerous methods—based on exact and asymptotic distributions—can be used to obtain confidence intervals for the odds ratio in 2 × 2 tables. We examine ten methods for generating these intervals based on coverage probability, closeness of coverage probability to target, and length of confidence intervals. Based on these criteria, Cornfield’s method, without the continuity correction, performed the best of the methods examined here. A drawback to use of this method is the significant possibility that the attained coverage probability will not meet the nominal confidence level. Use of a mid-P value greatly improves methods based on the “exact” distribution. When combined with the Wilson rule for selection of a rejection set, the resulting method is a procedure that performed very well. Crow’s method, with use of a mid-P, performed well, although it was only a slight improvement over the Wilson mid-P method. Its cumbersome calculations preclude its general acceptance. Woolf's (logit) method—with the Haldane–Anscombe correction— performed well, especially with regard to length of confidence intervals, and is recommended based on ease of computation.     

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.     

Asymptotic Confidence Limits for Performance Measures of a Repairable System with Imperfect Service Station

This article studies the asymptotic confidence limits for the steady-state availability, failure frequency, and mean time to failure of a repairable K-out-of-(M + S) system with M operating devices, S spares, and an imperfect service station that may be interrupted by a breakdown when it is repairing for the failed devices.     

A General Model for Repeated Audit Controls Using Monotone Subsampling

Abstract

In categorical repeated audit controls, fallible auditors classify sample elements in order to estimate the population fraction of elements in certain categories. To take possible misclassifications into account, subsequent checks are performed with a decreasing number of observations. In this paper a model is presented for a general repeated audit control system, where k subsequent auditors classify elements into r categories. Two different subsampling procedures will be discussed, named “stratified” and “random” sampling. Although these two sampling methods lead to different probability distributions, it is shown that the likelihood inferences are identical. The MLE are derived and the situations with undefined MLE are examined in detail; it is shown that an unbiased MLE can be obtained by stratified sampling. Three different methods for constructing confidence upper limits are discussed; the Bayesian upper limit seems to be the most satisfactory. Our theoretical results are applied to two cases with r = 2 and k = 2 or 3, respectively.     

On a Confidence Interval for the Mean of an Asymmetric Distribution

A clarification is given of the main result (1.1) in Communications in Statistics: Theory and Methods 34:753–766. The term {1 + 6a(r ? a)}^1/3 is to be understood as sgn(1 + 6a(r ? a)) | 1 + 6a(r ? a)|^1/3. The result is expressed in a more user-friendly form. An issue is raised regarding the common usage of the expression x ^1/n when n is even.     

Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models

We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n. In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known-variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically, the intervals based on the thresholding estimators are larger even by an order of magnitude when the estimators are tuned to perform consistent variable selection. For the unknown-variance case, we provide nontrivial lower bounds and a small numerical study for the coverage probabilities in finite samples. We also conduct an asymptotic analysis where the results from the known-variance case can be shown to carry over asymptotically if the number of degrees of freedom n ? k tends to infinity fast enough in relation to the thresholding parameter.     

Usual and Shortest Confidence Intervals on Odds Ratios from Logistic Regression

Based on the large-sample normal distribution of the sample log odds ratio and its asymptotic variance from maximum likelihood logistic regression, shortest 95% confidence intervals for the odds ratio are developed. Although the usual confidence interval on the odds ratio is unbiased, the shortest interval is not. That is, while covering the true odds ratio with the stated probability, the shortest interval covers some values below the true odds ratio with higher probability. The upper and lower limits of the shortest interval are shifted to the left of those of the usual interval, with greater shifts in the upper limits. With the log odds model γ + Xβ, in which X is binary, simulation studies showed that the approximate average percent difference in length is 7.4% for n (sample size) = 100, and 3.8% for n = 200. Precise estimates of the covering probabilities of the two types of intervals were obtained from simulation studies, and are compared graphically. For odds ratio estimates greater (less) than one, shortest intervals are more (less) likely to include one than are the usual intervals. The usual intervals are likelihood-based and the shortest intervals are not. The usual intervals have minimum expected length among the class of unbiased intervals. Shortest intervals do not provide important advantages over the usual intervals, which we recommend for practical use.     

Some Properties of Order Statistics When the Sample Size Is Random

ABSTRACT

The study of r-out-of-n systems is of utmost importance in reliability theory. In this note, we study closure of different partial orders under the formation of r-out-of-N and (N ? s)-out-of-N systems when the number of components N, forming the system, is a random variable having support {k, k + 1,…}, where k is a fixed positive integer, r ∈ {1,…, k} and s ∈ {0, 1,…, k ? 1}. This generalizes quite a few results already known in the literature. We also study the closure of different partial orders when two systems are formed out of different random number of components.     

Frequentist and Bayesian Interval Estimators for the Normal Mean

It is well known that a Bayesian credible interval for a parameter of interest is derived from a prior distribution that appropriately describes the prior information. However, it is less well known that there exists a frequentist approach developed by Pratt (1961 Pratt , J. W. ( 1961 ). Length of confidence intervals . J. Amer. Statist. Assoc. 56 : 549 – 657 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) that also utilizes prior information in the construction of frequentist confidence intervals. This frequentist approach produces confidence intervals that have minimum weighted average expected length, averaged according to some weight function that appropriately describes the prior information. We begin with a simple model as a starting point in comparing these two distinct procedures in interval estimation. Consider X ₁,…, X _n that are independent and identically N(μ, σ²) distributed random variables, where σ² is known, and the parameter of interest is μ. Suppose also that previous experience with similar data sets and/or specific background and expert opinion suggest that μ = 0. Our aim is to: (a) develop two types of Bayesian 1 ? α credible intervals for μ, derived from an appropriate prior cumulative distribution function F(μ) more importantly; (b) compare these Bayesian 1 ? α credible intervals for μ to the frequentist 1 ? α confidence interval for μ derived from Pratt's frequentist approach, in which the weight function corresponds to the prior cumulative distribution function F(μ). We show that the endpoints of the Bayesian 1 ? α credible intervals for μ are very different to the endpoints of the frequentist 1 ? α confidence interval for μ, when the prior information strongly suggests that μ = 0 and the data supports the uncertain prior information about μ. In addition, we assess the performance of these intervals by analyzing their coverage probability properties and expected lengths.     

Exact confidence limits for the probability of response in two-stage designs

In addition to point estimate for the probability of response in a two-stage design (e.g. Simon's two-stage design for binary endpoints), confidence limits should be computed and reported. The current method of inverting the p-value function to compute the confidence interval does not guarantee coverage probability in a two-stage setting. The existing exact approach to calculate one-sided limits is based on the overall number of responses to order the sample space. This approach could be conservative because many sample points have the same limits. We propose a new exact one-sided interval based on p-value for the sample space ordering. Exact intervals are computed by using binomial distributions directly, instead of a normal approximation. Both exact intervals preserve the nominal confidence level. The proposed exact interval based on the p-value generally performs better than the other exact interval with regard to expected length and simple average length of confidence intervals.     

The Variable Sampling Interval (VSI) in an Analysis of Taguchi's On-line Quality Monitoring Procedure for Variables

The procedure of on-line process control for variables proposed by Taguchi consists of inspecting the mth item (a single item) of every m items produced and deciding, at each inspection, whether the mean value is increased or not. If the value of the monitored statistic is outside of the control limits, one decides the process is out-of-control and the production is stopped for adjustment; otherwise, it continues. In this article, a variable sampling interval (with a longer L and a shorter m ≤ L) chart with two set of limits is used. These limits are the warning (±W) and the control (±C), where W ≤ C. The process is stopped for adjustment when an observation falls outside of the control limits or a sequence of h observations falls between the warning limits and the control limits. The longer sample interval is used after an adjustment or when an observation falls inside the warning limits; otherwise, the short sampling interval is used. The properties of an ergodic Markov chain are used to evaluate the time (in units) that the process remains in-control and out-of-control, with the aim of building an economic–statistical model. The parameters (the sampling intervals m and L, the control limits W and C and the length of run h) are optimized by minimizing the cost function with constraints on the average run lengths (ARLs) and the conformity fraction. The performance of the current proposal is more economical than the decision taken based on a sequence of length h = 1, L = m, and W = C, which is the model employed in earlier studies. A numerical example illustrates the proposed procedure.     

An optimal sign test for one-sample bivariate location model using an alternative bivariate ranked-set sample

The aim of this paper is to find an optimal alternative bivariate ranked-set sample for one-sample location model bivariate sign test. Our numerical and theoretical results indicated that the optimal designs for the bivariate sign test are the alternative designs with quantifying order statistics with labels {((r+1)/2, (r+1)/2)}, when the set size r is odd and {(r/2+1, r/2), (r/2, r/2+1)} when the set size r is even. The asymptotic distribution and Pitman efficiencies of these designs are derived. A simulation study is conducted to investigate the power of the proposed optimal designs. Illustration using real data with the Bootstrap algorithm for P-value estimation is used.     

On the Validity of the Bootstrap in Non‐Parametric Functional Regression

Abstract. We consider the functional non‐parametric regression model Y= r( χ )+?, where the response Y is univariate, χ is a functional covariate (i.e. valued in some infinite‐dimensional space), and the error ? satisfies E(? | χ ) = 0. For this model, the pointwise asymptotic normality of a kernel estimator of r (·) has been proved in the literature. To use this result for building pointwise confidence intervals for r (·), the asymptotic variance and bias of need to be estimated. However, the functional covariate setting makes this task very hard. To circumvent the estimation of these quantities, we propose to use a bootstrap procedure to approximate the distribution of . Both a naive and a wild bootstrap procedure are studied, and their asymptotic validity is proved. The obtained consistency results are discussed from a practical point of view via a simulation study. Finally, the wild bootstrap procedure is applied to a food industry quality problem to compute pointwise confidence intervals.     

Small-sample intervals for regression

For the general linear regression model Y = Xη + e, we construct small-sample exponentially tilted empirical confidence intervals for a linear parameter 6 = a^Tη and for nonlinear functions of η. The coverage error for the intervals is O_p(1/n), as shown in Tingley and Field (1990). The technique, though sample-based, does not require bootstrap resampling. The first step is calculation of an estimate for η. We have used a Mallows estimate. The algorithm applies whenever η is estimated as the solution of a system of equations having expected value 0. We include calculations of the relative efficiency of the estimator (compared with the classical least-squares estimate). The intervals are compared with asymptotic intervals as found, for example, in Hampel et at. (1986). We demonstrate that the procedure gives sensible intervals for small samples.     

Estimation of the Parameters of a Two-phase Tandem Queue with a Second Optional Service

In this article, we consider a two-phase tandem queueing model with a second optional service. In this model, the service is done by two phases. The first phase of service is essential for all customers and after the completion of the first phase of service, any customer receives the second phase of service with probability α, or leaves the system with probability 1 ? α. Also, there are two heterogeneous servers which work independently, one of them providing the first phase of service and the other a second phase of service. In this model, our main purpose is to estimate the parameters of the model, traffic intensity, and mean system size, in the steady state, via maximum likelihood and Bayesian methods. Furthermore, we find asymptotic confidence intervals for mean system size. Finally, by a simulation study, we compute the confidence levels and mean length for asymptotic confidence intervals of mean system size with a nominal level 0.95.