Confidence Interval Estimation in Ultrastructural Model

The present article proposes a methodology to construct confidence interval for slope parameter and joint confidence region for intercept term and slope parameter in an ultra-structural model with equation error and correlated measurement errors.     

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.     

Minimum-area confidence sets for a normal distribution

When making inference on a normal distribution, one often seeks either a joint confidence region for the two parameters or a confidence band for the cumulative distribution function. A number of methods for constructing such confidence sets are available, but none of these methods guarantees a minimum-area confidence set. In this paper, we derive both a minimum-area joint confidence region for the two parameters and a minimum-area confidence band for the cumulative distribution function. The minimum-area joint confidence region is asymptotically equivalent to other confidence regions in the literature, but the minimum-area confidence band improves on existing confidence bands even asymptotically.     

Statistical inference for two exponential populations under joint progressive type-I censored scheme

In this article, we introduce a new scheme called joint progressive type-I (JPC-I) censored and as a special case, joint type-I censored scheme. Bayesian and non Bayesian estimators have been obtained for two exponential populations under both JPC-I censored scheme and joint type-I censored. The maximum likelihood estimators of the parameters, the asymptotic variance covariance matrix, have been obtained. Bayes estimators have been developed under squared error loss function using independent gamma prior distributions. Moreover, approximate confidence region based on the asymptotic normality of the maximum likelihood estimators and credible confidence region from a Bayesian viewpoint are also discussed and compared with two Bootstrap confidence regions. A numerical illustration for these new results is given.     

Empirical Likelihood Confidence Region for the Parameter in a Partially Linear Errors-in-Variables Model

In this article, a partially linear errors-in-variables model is considered, and empirical log-likelihood ratio statistic for the unknown parameter in the model is suggested. It is proved that the proposed statistic is asymptotically standard chi-square distribution under some suitable conditions, and hence it can be used to construct the confidence region of the parameter. A simulation study indicates that, in terms of coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the least-squares method.     

Empirical Likelihood Confidence Region for Parameters in Semi-linear Errors-in-Variables Models

Abstract.  This paper proposes a constrained empirical likelihood confidence region for a parameter in the semi-linear errors-in-variables model. The confidence region is constructed by combining the score function corresponding to the squared orthogonal distance with a constraint on the parameter, and it overcomes that the solution of limiting mean estimation equations is not unique. It is shown that the empirical log likelihood ratio at the true parameter converges to the standard chi-square distribution. Simulations show that the proposed confidence region has coverage probability which is closer to the nominal level, as well as narrower than those of normal approximation of generalized least squares estimator in most cases. A real data example is given.     

Empirical Likelihood Inference for the Parameter in Additive Partially Linear EV Models

In this article, we consider empirical likelihood inference for the parameter in the additive partially linear models when the linear covariate is measured with error. By correcting for attenuation, a corrected-attenuation empirical log-likelihood ratio statistic for the unknown parameter β, which is of primary interest, is suggested. We show that the proposed statistic is asymptotically standard chi-square distribution without requiring the undersmoothing of the nonparametric components, and hence it can be directly used to construct the confidence region for the parameter β. Some simulations indicate that, in terms of comparison between coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the profile-based least-squares method. We also give the maximum empirical likelihood estimator (MELE) for the unknown parameter β, and prove the MELE is asymptotically normal under some mild conditions.     

Estimation of the two-parameter bathtub-shaped lifetime distribution with progressive censoring

In this paper, we investigate the estimation problem concerning a progressively type-II censored sample from the two-parameter bathtub-shaped lifetime distribution. We use the maximum likelihood method to obtain the point estimators of the parameters. We also provide a method for constructing an exact confidence interval and an exact joint confidence region for the parameters. Two numerical examples are presented to illustrate the method of inference developed here. Finally, Monte Carlo simulation studies are used to assess the performance of our proposed method.     

Simulation-based consistent inference for biased working model of non-sparse high-dimensional linear regression

Variable selection in regression analysis is of importance because it can simplify model and enhance predictability. After variable selection, however, the resulting working model may be biased when it does not contain all of significant variables. As a result, the commonly used parameter estimation is either inconsistent or needs estimating high-dimensional nuisance parameter with very strong assumptions for consistency, and the corresponding confidence region is invalid when the bias is relatively large. We in this paper introduce a simulation-based procedure to reformulate a new model so as to reduce the bias of the working model, with no need to estimate high-dimensional nuisance parameter. The resulting estimators of the parameters in the working model are asymptotic normally distributed whether the bias is small or large. Furthermore, together with the empirical likelihood, we build simulation-based confidence regions for the parameters in the working model. The newly proposed estimators and confidence regions outperform existing ones in the sense of consistency.     

Empirical Likelihood for Semiparametric Varying-Coefficient Heteroscedastic Partially Linear Errors-in-Variables Models

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in semiparametric varying-coefficient heteroscedastic partially linear errors-in-variables models. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.     

Empirical Likelihood for a Heteroscedastic Partial Linear Errors-in-Variables Model

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in heteroscedastic partially linear errors-in-variables model with martingale difference errors. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.     

Hybrid regions for post-change mean in a sequence of normal variables

The hybrid bootstrap uses resampling ideas to extend the duality approach to the interval estimation for a parameter of interest when there are nuisance parameters. The confidence region constructed by the hybrid bootstrap may perform much better than the ordinary bootstrap region in a situation where the data provide substantial information about the nuisance parameter, but limited information about the parameter of interest. We apply this method to estimate the post-change mean after a change is detected by a stopping procedure in a sequence of independent normal variables. Since distribution theory in change point problems is generally a challenge, we use bootstrap simulation to find empirical distributions of test statistics and calculate critical thresholds. Both likelihood ratio and Bayesian test statistics are considered to set confidence regions for post-change means in the normal model. In the simulation studies, the performance of hybrid regions are compared with that of ordinary bootstrap regions in terms of the widths and coverage probabilities of confidence intervals.     

A new confidence region for the multinomial distribution

A simple confidence region is proposed for the multinomial parameter. It is designed for situations having zero cell counts. Simulation studies as well as a real data application show that it performs at least as well as than at least two of the most common confidence regions.     

Confidence Intervals in Regression That Utilize Uncertain Prior Information About a Vector Parameter

Consider a linear regression model with independent normally distributed errors. Suppose that the scalar parameter of interest is a specified linear combination of the components of the regression parameter vector. Also suppose that we have uncertain prior information that a parameter vector, consisting of specified distinct linear combinations of these components, takes a given value. Part of our evaluation of a frequentist confidence interval for the parameter of interest is the scaled expected length, defined to be the expected length of this confidence interval divided by the expected length of the standard confidence interval for this parameter, with the same confidence coefficient. We say that a confidence interval for the parameter of interest utilizes this uncertain prior information if (a) the scaled expected length of this interval is substantially less than one when the prior information is correct, (b) the maximum value of the scaled expected length is not too large and (c) this confidence interval reverts to the standard confidence interval, with the same confidence coefficient, when the data happen to strongly contradict the prior information. We present a new confidence interval for a scalar parameter of interest, with specified confidence coefficient, that utilizes this uncertain prior information. A factorial experiment with one replicate is used to illustrate the application of this new confidence interval.     

Minimum volume confidence sets for parameters of normal distributions

AStA Advances in Statistical Analysis - Under a proper restriction, we establish the minimum volume confidence set (interval and region) for parameter of any normal distribution. Compared with...     

Confidence sets based on the positive part James–Stein estimator with the asymptotically constant coverage probability

The asymptotic expansions for the coverage probability of a confidence set centred at the James–Stein estimator presented in our previous publications show that this probability depends on the non-centrality parameter τ² (the sum of the squares of the means of normal distributions). In this paper we establish how these expansions can be used for a construction of confidence region with constant confidence level, which is asymptotically (the same formula for both case τ→0 and τ→∞) equal to some fixed value 1?α. We establish the shrinkage rate for the confidence region according to the growth of the dimension p and also the value of τ for which we observe quick decreasing of the coverage probability to the nominal level 1?α. When p→∞ this value of τ increases as O(p^1/4). The accuracy of the results obtained is shown by the Monte-Carlo statistical simulations.     

Exact likelihood inference for two exponential populations based on a joint generalized Type-I hybrid censored sample

Following the work of Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring. Comm Statist Theory Methods. 1988;17:1857–1870], several results have been developed regarding the exact likelihood inference of exponential parameters based on different forms of censored samples. In this paper, the conditional maximum likelihood estimators (MLEs) of two exponential mean parameters are derived under joint generalized Type-I hybrid censoring on the two samples. The moment generating functions (MGFs) and the exact densities of the conditional MLEs are obtained, using which exact confidence intervals are then developed for the model parameters. We also derive the means, variances, and mean squared errors of these estimates. An efficient computational method is developed based on the joint MGF. Finally, an example is presented to illustrate the methods of inference developed here.     

Exact tests and confidence regions in nonlinear regression

Nonlinear regression models with spherically symmetric error vectors and a single nonlinear parameter are considered. On the basis of a new geometric approach, exact one- and two-sided tests and confidence regions for the nonlinear parameter are derived in the cases of known and unknown error variances. A geometric measure representation formula is used to determine the power functions of the tests if the error variance is known and to derive different lower bounds for the power function of a one-sided test in the case of an unknown error variance. The latter can be done quite effectively by constructing and measuring several balls inside the critical region. A numerical study compares the results for different density generating functions of the error distribution.     

Joint confidence region estimation on predictive values

For evaluating diagnostic accuracy of inherently continuous diagnostic tests/biomarkers, sensitivity and specificity are well-known measures both of which depend on a diagnostic cut-off, which is usually estimated. Sensitivity (specificity) is the conditional probability of testing positive (negative) given the true disease status. However, a more relevant question is “what is the probability of having (not having) a disease if a test is positive (negative)?”. Such post-test probabilities are denoted as positive predictive value (PPV) and negative predictive value (NPV). The PPV and NPV at the same estimated cut-off are correlated, hence it is desirable to make the joint inference on PPV and NPV to account for such correlation. Existing inference methods for PPV and NPV focus on the individual confidence intervals and they were developed under binomial distribution assuming binary instead of continuous test results. Several approaches are proposed to estimate the joint confidence region as well as the individual confidence intervals of PPV and NPV. Simulation results indicate the proposed approaches perform well with satisfactory coverage probabilities for normal and non-normal data and, additionally, outperform existing methods with improved coverage as well as narrower confidence intervals for PPV and NPV. The Alzheimer's Disease Neuroimaging Initiative (ADNI) data set is used to illustrate the proposed approaches and compare them with the existing methods.