Invariance,Optimality, and a 1-Observation Confidence Interval for a Normal Mean

In a 1965 Decision Theory course at Stanford University, Charles Stein began a digression with “an amusing problem”: is there a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown? Stein introduced the interval with endpoints ± c|X| and showed indeed that for c large enough, the minimum coverage probability (over all values for the mean and variance) could be made arbitrarily near one. While the problem and coverage calculation were in the author’s hand-written notes from the course, there was no development of any optimality result for the interval. Here, the Hunt–Stein construction plus analysis based on special features of the problem provides a “minimax” rule in the sense that it minimizes the maximum expected length among all procedures with fixed coverage (or, equivalently, maximizes the minimal coverage among all procedures with a fixed expected length). The minimax rule is a mixture of two confidence procedures that are equivariant under scale and sign changes, and are uniformly better than the classroom example or the natural interval X ± c|X|?.     

Statistical inference for the difference of two Lorenz curves

Testing the Lorenz dominance is of importance in economic and social sciences. In this article, we propose new tools to do inferences for the difference of two Lorenz curves. The asymptotic normality of the proposed smoothed nonparametric estimator is proved. We also propose a smoothed jackknife empirical likelihood (JEL) method which avoids to estimate the complicate asymptotic variance. It is proved that the proposed JEL ratio statistics converge to the standard chi-square distribution. Simulation studies and real data analysis are also conducted, and show encouraging finite-sample performance.     

Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model

The studied topic is motivated by the problem of interlaboratory comparisons. This paper focuses on the confidence interval estimation of the between group variance in the unbalanced heteroscedastic one-way random effects model. Several interval estimators are proposed and compared by means of the simulation study. The most recommended (safest) is the confidence interval based on Bonferroni's inequality.     

Bootstrapping a Universal Pivot When Nuisance Parameters are Estimated

In complete samples from a continuous cumulative distribution with unknown parameters, it is known that various pivotal functions can be constructed by appealing to the probability integral transform. A pivotal function (or simply pivot) is a function of the data and parameters that has the property that its distribution is free of any unknown parameters. Pivotal functions play a key role in constructing confidence intervals and hypothesis tests. If there are nuisance parameters in addition to a parameter of interest, and consistent estimators of the nuisance parameters are available, then substituting them into the pivot can preserve the pivot property while altering the pivot distribution, or may instead create a function that is approximately a pivot in the sense that its asymptotic distribution is free of unknown parameters. In this latter case, bootstrapping has been shown to be an effective way of estimating its distribution accurately and constructing confidence intervals that have more accurate coverage probability in finite samples than those based on the asymptotic pivot distribution. In this article, one particular pivotal function based on the probability integral transform is considered when nuisance parameters are estimated, and the estimation of its distribution using parametric bootstrapping is examined. Applications to finding confidence intervals are emphasized. This material should be of interest to instructors of upper division and beginning graduate courses in mathematical statistics who wish to integrate bootstrapping into their lessons on interval estimation and the use of pivotal functions.

[Received November 2014. Revised August 2015.]     

Inference on proportional hazard rate model parameter under Type-I progressively hybrid censoring scheme

ABSTRACT

In this paper, under Type-I progressive hybrid censoring sample, we obtain maximum likelihood estimator of unknown parameter when the parent distribution belongs to proportional hazard rate family. We derive the conditional probability density function of the maximum likelihood estimator using moment-generating function technique. The exact confidence interval is obtained and compared by conducting a Monte Carlo simulation study for burr Type XII distribution. Finally, we obtain the Bayes and posterior regret gamma minimax estimates of the parameter under a precautionary loss function with precautionary index k = 2 and compare their behavior via a Monte Carlo simulation study.     

On difference factor in assessment of dissolution similarity

In vitro dissolution similarity has been suggested as a surrogate for assessing equivalence between the pre-changed and post-changed formulations for postapproval changes of a drug. The difference factor f₁, based on the absolute mean difference, has been proposed as a criterion for evaluating similarity between dissolution profiles. Statistical properties including density function, bias, and asymptotic distribution of a consistent estimator are investigated. Due to complexity of the distribution of the estimator, we suggest the use of the confidence intervals obtained from the bootstrap method for evaluation of dissolution similarity. A simulation was conducted to examine the size and power of the proposed CI procedure. Comparisons with other criteria such as similarity factor are also provided. Numerical examples are used to illustrate the proposed CI procedure.     

Confidence regions for parameters in linear models with additional information

Using given significant additional information it is possible to improve different confidence regions for the regression parameters in a linear model. Thereby, the given informations may concern the expectation and (or) the variance of the observations, and an improvement is possible in the sense of the decrease of the confidence regions' size. In particular it is possible to improve the so called confidence ellipsoids which are often used to estimate the considered parameters.     

Some Small-Sample Properties of the Student t Confidence Intervals

For the two-sided Student t confidence interval for the mean of a normal distribution there is, for any sample size, a sufficiently large confidence level that ensures that the interval covers all the observations; there are also sufficiently small confidence levels guaranteeing, respectively, that (a) the interval does not cover all the observations and (b) the interval lies within the extreme observations. Necessary and sufficient conditions are also obtained for the width of the confidence interval to always exceed the sample range, as well as for the reverse inequality. Some implications of the results are discussed.     

Online interval scheduling: randomized and multiprocessor cases

We consider the problem of scheduling a set of equal-length intervals arriving online, where each interval is associated with a weight and the objective is to maximize the total weight of completed intervals. An optimal 4-competitive algorithm has long been known in the deterministic case, but the randomized case remains open. We give the first randomized algorithm for this problem, achieving a competitive ratio of 3.5822. We also prove a randomized lower bound of 4/3, which is an improvement over the previous 5/4 result. Then we show that the techniques can be carried to the deterministic multiprocessor case, giving a 3.5822-competitive 2-processor algorithm, and a 4/3 lower bound for any number of processors. We also give a lower bound of 2 for the case of two processors. A preliminary version of this paper appeared in the Proceedings of COCOON 2007, LNCS, vol. 4598, pp. 176–186. The work described in this paper was fully supported by a grant from City University of Hong Kong (SRG 7001969), and NSFC Grant No. 70525004 and 70702030.     

Empirical Likelihood Inference for Longitudinal Data with Missing Response Variables and Error-Prone Covariates

We consider statistical inference for longitudinal partially linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. The block empirical likelihood procedure is used to estimate the regression coefficients and residual adjusted block empirical likelihood is employed for the baseline function. This leads us to prove a nonparametric version of Wilk's theorem. Compared with methods based on normal approximations, our proposed method does not require a consistent estimators for the asymptotic variance and bias. An application to a longitudinal study is used to illustrate the procedure developed here. A simulation study is also reported.