Strawderman-type estimators for a scale parameter with application to the exponential distribution

It is shown that Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique for estimating the variance of a normal distribution can be extended to estimating a general scale parameter in the presence of a nuisance parameter. Employing standard monotone likelihood ratio-type conditions, a new class of improved estimators for this scale parameter is derived under quadratic loss. By imposing an additional condition, a broader class of improved estimators is obtained. The dominating procedures are in form analogous to those in Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]. Application of the general results to the exponential distribution yields new sufficient conditions, other than those of Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38] and Kubokawa [1994. A unified approach to improving equivariant estimators. Ann. Statist. 22, 290–299], for improving the best affine equivariant estimator of the scale parameter. A class of estimators satisfying the new conditions is constructed. The results shed new light on Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique.     

Inferences on the intraclass correlation coefficients in the unbalanced two-way random effects model with interaction

In this paper, the hypothesis testing and interval estimation for the intraclass correlation coefficients are considered in a two-way random effects model with interaction. Two particular intraclass correlation coefficients are described in a reliability study. The tests and confidence intervals for the intraclass correlation coefficients are developed when the data are unbalanced. One approach is based on the generalized p-value and generalized confidence interval, the other is based on the modified large-sample idea. These two approaches simplify to the ones in Gilder et al. [2007. Confidence intervals on intraclass correlation coefficients in a balanced two-factor random design. J. Statist. Plann. Inference 137, 1199–1212] when the data are balanced. Furthermore, some statistical properties of the generalized confidence intervals are investigated. Finally, some simulation results to compare the performance of the modified large-sample approach with that of the generalized approach are reported. The simulation results indicate that the modified large-sample approach performs better than the generalized approach in the coverage probability and expected length of the confidence interval.     

Improved estimation of the smallest scale parameter of gamma distributions

In this work improved point and interval estimation of the smallest scale parameter of independent gamma distributions with known shape parameters are studied in an integrated fashion. The approach followed is based on formulating the model in such a way that enables us to treat the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The class of improved point and interval estimators is enriched. Within this class, a subclass of generalized Bayes estimators of a simple form is identified.     

Inference of non-centrality parameter of a truncated non-central chi-squared distribution

Non-central chi-squared distribution plays a vital role in statistical testing procedures. Estimation of the non-centrality parameter provides valuable information for the power calculation of the associated test. We are interested in the statistical inference property of the non-centrality parameter estimate based on one observation (usually a summary statistic) from a truncated chi-squared distribution. This work is motivated by the application of the flexible two-stage design in case–control studies, where the sample size needed for the second stage of a two-stage study can be determined adaptively by the results of the first stage. We first study the moment estimate for the truncated distribution and prove its existence, uniqueness, and inadmissibility and convergence properties. We then define a new class of estimates that includes the moment estimate as a special case. Among this class of estimates, we recommend to use one member that outperforms the moment estimate in a wide range of scenarios. We also present two methods for constructing confidence intervals. Simulation studies are conducted to evaluate the performance of the proposed point and interval estimates.     

Estimation of the smallest scale parameter of two-parameter exponential distributions

Improved point and interval estimation of the smallest scale parameter of n independent populations following two-parameter exponential distributions are studied. The model is formulated in such a way that allows for treating the estimation of the smallest scale parameter as a problem of estimating an unrestricted scale parameter in the presence of a nuisance parameter. The classes of improved point and interval estimators are enriched with Stein-type, Brewster and Zidek-type, Maruyama-type and Strawderman-type improved estimators under both quadratic and entropy losses, whereas using as a criterion the coverage probability, with Stein-type, Brewster and Zidek-type, and Maruyama-type improved intervals. The sampling framework considered incorporates important life-testing schemes such as i.i.d. sampling, type-II censoring, progressive type-II censoring, adaptive progressive type-II censoring, and record values.     

Inference on the Common Variance of Correlated Normal Random Variables

ABSTRACT

Scale equivariant estimators of the common variance σ², of correlated normal random variables, have mean squared errors (MSE) which depend on the unknown correlations. For this reason, a scale equivariant estimator of σ² which uniformly minimizes the MSE does not exist. For the equi-correlated case, we have developed three equivariant estimators of σ²: a Bayesian estimator for invariant prior as well as two non-Bayesian estimators. We then generalized these three estimators for the case of several variables with multiple unknown correlations. In addition, we developed a system of confidence intervals which produce the desired coverage probability while being efficient in terms of expected length.     

SHRINKAGE PREDICTION IN THE EXPONENTIAL DISTRIBUTION WITH A PRIOR INTERVAL FOR THE SCALE PARAMETER

The author proposes the best shrinkage predictor of a preassigned dominance level for a future order statistic of an exponential distribution, assuming a prior estimate of the scale parameter is distributed over an interval according to an arbitrary distribution with known mean. Based on a Type II censored sample from this distribution, we predict the future order statistic in another independent sample from the same distribution. The predictor is constructed by incorporating a preliminary confidence interval for the scale parameter and a class of shrinkage predictors constructed here. It improves considerably classical predictors for all values of the scale parameter within its dominance interval containing the confidence interval of a preassigned level.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

Better approximate confidence intervals for a binomial parameter

This paper discusses five methods for constructing approximate confidence intervals for the binomial parameter Θ, based on Y successes in n Bernoulli trials. In a recent paper, Chen (1990) discusses various approximate methods and suggests a new method based on a Bayes argument, which we call method I here. Methods II and III are based on the normal approximation without and with continuity correction. Method IV uses the Poisson approximation of the binomial distribution and then exploits the fact that the exact confidence limits for the parameter of the Poisson distribution can be found through the x² distribution. The confidence limits of method IV are then provided by the Wilson-Hilferty approximation of the x². Similarly, the exact confidence limits for the binomial parameter can be expressed through the F distribution. Method V approximates these limits through a suitable version of the Wilson-Hilferty approximation. We undertake a comparison of the five methods in respect to coverage probability and expected length. The results indicate that method V has an advantage over Chen's Bayes method as well as over the other three methods.     

Smallest confidence intervals for one binomial proportion

We specify three classes of one-sided and two-sided $1-\alpha$ confidence intervals with certain monotonicity and symmetry on the confidence limits for the probability of success, the parameter in a binomial distribution. For each class of one-sided confidence intervals the smallest interval, in the sense of the set inclusion, is obtained based on the direct analysis of coverage probability functions. A simple sufficient and necessary condition for the existence of the smallest two-sided confidence interval is provided and the smallest interval is derived if it exists. Thus the proposed intervals are uniformly most accurate, and have the uniformly minimum expected length as well.     

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 the lengths of t-based confidence intervals

Confidence interval is a basic type of interval estimation in statistics. When dealing with samples from a normal population with the unknown mean and the variance, the traditional method to construct t-based confidence intervals for the mean parameter is to treat the n sampled units as n groups and build the intervals. Here we propose a generalized method. We first divide them into several equal-sized groups and then calculate the confidence intervals with the mean values of these groups. If we define “better” in terms of the expected length of the confidence interval, then the first method is better because the expected length of the confidence interval obtained from the first method is shorter. We prove this intuition theoretically. We also specify when the elements in each group are correlated, the first method is invalid, while the second can give us correct results in terms of the coverage probability. We illustrate this with analytical expressions. In practice, when the data set is extremely large and distributed in several data centers, the second method is a good tool to get confidence intervals, in both independent and correlated cases. Some simulations and real data analyses are presented to verify our theoretical results.     

Large sample interval mapping method for genetic trait loci in finite regression mixture models

This article investigates the large sample interval mapping method for genetic trait loci (GTL) in a finite non-linear regression mixture model. The general model includes most commonly used kernel functions, such as exponential family mixture, logistic regression mixture and generalized linear mixture models, as special cases. The populations derived from either the backcross or intercross design are considered. In particular, unlike all existing results in the literature in the finite mixture models, the large sample results presented in this paper do not require the boundness condition on the parametric space. Therefore, the large sample theory presented in this article possesses general applicability to the interval mapping method of GTL in genetic research. The limiting null distribution of the likelihood ratio test statistics can be utilized easily to determine the threshold values or p-values required in the interval mapping. The limiting distribution is proved to be free of the parameter values of null model and free of the choice of a kernel function. Extension to the multiple marker interval GTL detection is also discussed. Simulation study results show favorable performance of the asymptotic procedure when sample sizes are moderate.     

Approximate confidence intervals for a linear combination of binomial proportions: A new variant

We propose a new adjustment for constructing an improved version of the Wald interval for linear combinations of binomial proportions, which addresses the presence of extremal samples. A comparative simulation study was carried out to investigate the performance of this new variant with respect to the exact coverage probability, expected interval length, and mesial and distal noncoverage probabilities. Additionally, we discuss the application of a criterion for interpreting interval location in the case of small samples and/or in situations in which extremal observations exist. The confidence intervals obtained from the new variant performed better for some evaluation measures.     

Estimating the common hazard rate of several exponential distributions

Abstract

In the present communication, we consider the estimation of the common hazard rate of several exponential distributions with unknown and unequal location parameters with a common scale parameter under a general class of bowl-shaped scale invariant loss functions. We have shown that the best affine equivariant estimator (BAEE) is inadmissible by deriving a non smooth improved estimator. Further, we have obtained a smooth estimator which improves upon the BAEE. As an application, we have obtained explicit expressions of improved estimators for special loss functions. Finally, a simulation study is carried out for numerically comparing the risk performance of various estimators.     

A new confidence interval in measurement error model with the reliability ratio known

For the slope parameter of the measurement error model with the reliability ratio known, this article constructs a fiducial generalized confidence interval (FGCI) which is proved to have correct asymptotic coverage. Simulation results demonstrate that the FGCI often outperforms the existing intervals in terms of empirical coverage probability, average interval length, and false parameter coverage rate. Two examples are also provided to illustrate our approach.     

Gamma-minimax results for the class of unimodal priors

Olman and Shmundak proved 1985 that in estimating a bounded normal mean under squared error loss the Bayes estimator with respect to the uniform distribution on the parameter interval is gamma-minimax when the parameter interval is sufficiently small and the class of priors consists of all symmetric and unimodal distributions. Recently, one of the authors showed that this result remains valid for quite general families of distributions which satisfy some regularity conditions. In the present paper a generalization to the class of unimodal priors with fixed mode is derived. It is proved that the Bayes estimator with respect to a suitable mixture of two uniform distributions is gamma-minimax for sufficiently small parameter intervals. To that end appropriate characterizations of a saddle point in the corresponding statistical games are established. Some results of a numerical study are presented.     

Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

A note on the equivariant estimation of an exponential scale using progressively censored data

We consider the problem of estimating the scale parameter θ$\theta$ of the shifted exponential distribution with unknown location based on a type II progressively censored sample. Under a large class of bowl-shaped loss functions, a smooth estimator, that dominates the minimum risk equivariant estimator of θ$\theta$, is proposed. A numerical study is performed and shows that the improved estimator yields significant risk reduction over the MRE.