Efficient inverse probability weighting method for quantile regression with nonignorable missing data

Quantitle regression (QR) is a popular approach to estimate functional relations between variables for all portions of a probability distribution. Parameter estimation in QR with missing data is one of the most challenging issues in statistics. Regression quantiles can be substantially biased when observations are subject to missingness. We study several inverse probability weighting (IPW) estimators for parameters in QR when covariates or responses are subject to missing not at random. Maximum likelihood and semiparametric likelihood methods are employed to estimate the respondent probability function. To achieve nice efficiency properties, we develop an empirical likelihood (EL) approach to QR with the auxiliary information from the calibration constraints. The proposed methods are less sensitive to misspecified missing mechanisms. Asymptotic properties of the proposed IPW estimators are shown under general settings. The efficiency gain of EL-based IPW estimator is quantified theoretically. Simulation studies and a data set on the work limitation of injured workers from Canada are used to illustrated our proposed methodologies.     

Bivariate discrete generalized exponential distribution

In this paper, we develop a bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distribution as proposed by Nekoukhou, Alamatsaz and Bidram [Discrete generalized exponential distribution of a second type. Statistics. 2013;47:876–887]. It is observed that the proposed bivariate distribution is a very flexible distribution and the bivariate geometric distribution can be obtained as a special case of this distribution. The proposed distribution can be seen as a natural discrete analogue of the bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivariate Anal. 2009;100:581–593]. We study different properties of this distribution and explore its dependence structures. We propose a new EM algorithm to compute the maximum-likelihood estimators of the unknown parameters which can be implemented very efficiently, and discuss some inferential issues also. The analysis of one data set has been performed to show the effectiveness of the proposed model. Finally, we propose some open problems and conclude the paper.     

Using orthogonal array for constructing three-level search designs

We consider the problem of constructing search designs for 3^m factorial designs. By using projection properties of some three-level orthogonal arrays, some search designs are obtained for 3 ? m ? 11. The new obtained orthogonal search designs are capable of searching and identifying up to four two-factor interactions and estimating them along with the general mean and main effects. The resulted designs have very high searching probabilities; it means that besides the well-known orthogonal structure, they have high ability in searching the true effects.     

Inference for probability of selection with dependently truncated data using a Cox model

A truncated sample consists of realizations of two variables L and T subject to the constraint L < T. One simple solution to dependently truncated data is to take L as a covariate of T in the Cox model. We aimed at studying the probability of selection, P(L < T), in this framework. We proposed the point estimator and derived its asymptotic distribution. Both truncated-only data and censored and truncated data were generated in the simulation study. The proposed point and variance estimators showed good performance in various simulated settings. The bone marrow transplant registry data were analyzed as the illustrative example.     

Phase-I monitoring of general linear profiles in multistage processes

In this article, the general linear profile-monitoring problem in multistage processes is addressed. An approach based on the U statistic is first proposed to remove the effect of the cascade property in multistage processes. Then, the T² chart and a likelihood ratio test (LRT)-based scheme on the adjusted parameters are constructed for Phase-I monitoring of the parameters of general linear profiles in each stage. Using simulation experiments, the performance of the proposed methods is evaluated and compared in terms of the signal probability for both weak and strong autocorrelations, for processes with two and three stages, as well as for two sample sizes. According to the results, the effect of the cascade property is effectively removed and hence each stage can be monitored independently. In addition, the result shows that the LRT approach provides significantly better results than the T² method and outperforms it under different shift and autocorrelation scenarios. Moreover, the proposed methods perform better when larger sample sizes are used in the process. Two illustrative examples, including a real case and a simulated example, are used to show the applicability of the proposed methods.     

Comparison of confidence intervals for the Behrens-Fisher problem

The Behrens–Fisher problem concerns the inferences for the difference between means of two independent normal populations without the assumption of equality of variances. In this article, we compare three approximate confidence intervals and a generalized confidence interval for the Behrens–Fisher problem. We also show how to obtain simultaneous confidence intervals for the three population case (analysis of variance, ANOVA) by the Bonferroni correction factor. We conduct an extensive simulation study to evaluate these methods in respect to their type I error rate, power, expected confidence interval width, and coverage probability. Finally, the considered methods are applied to two real dataset.     

Stress−strength reliability with general form distributions

In this article, we present characterizations, associated with the stress?strength reliability, of distributions with some general exponential and general inverse exponential forms. We obtain a general form for stress?strength reliability, when the distributions of the stress and the strength are non identical and independently distributed with either form. We obtain different point and interval estimators of stress?strength reliability. The theoretical results obtained can be directly applied whenever the distributions of the stress and the strength possess the forms discussed. A simulation study is carried out showing satisfactory performance of the estimators obtained. Moreover, a real data example is presented.     

On the comparison of Warner's and unrelated question randomized response plans for estimating sensitive finite population proportions

We consider the problem of estimation of a finite population proportion (P) related to a sensitive attribute under Warner's (1965 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) randomized response plan and the unrelated question plan due to Horvitz et al. (1967 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and prove that for a given probability sampling design, given any linear unbiased estimator (LUE) of P based on Warner's (1965 Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60:63–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) plan with any given value of the plan parameter, there exists an LUE of P based on the unrelated question plan with a uniformly smaller variance for suitable choices of the plan parameters. Assuming that only the attribute is sensitive but its complement is innocuous, the same is also shown to be true when the plan parameters for the two plans are so chosen so that both offer the same specified level of privacy.     

On the comparison of Horvitz–Thompson and Murthy's sampling strategies for estimating finite population proportions in direct and randomized response surveys

We consider the problem of unbiased estimation of a finite population proportion and compare the relative efficiency of the unequal probability sampling strategies due to Horvitz and Thompson (1952 Horvitz, D.G., Thompson, D.J. (1952). A generalization of sampling without replacement. J Am Stat Assoc. 47:663–685.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Murthy (1957 Murthy, M.N. (1957). Ordered and unordered estimators in sampling without replacement. Sankhya 18:379–390. [Google Scholar]) under a super-population model. It is shown that the model expected variance is smaller for the Murthy's (1957 Murthy, M.N. (1957). Ordered and unordered estimators in sampling without replacement. Sankhya 18:379–390. [Google Scholar]) strategy both when these two sampling strategies are based on data obtained from (i) a direct survey, and (ii) a randomized response (RR) survey employing some RR technique following a general RR model.     

Exact expressions for the weights used in least-squares regression estimation for the log-logistic and Weibull distribution

Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting, and this technique outperforms the maximum likelihood (ML) estimation often in small samples. This leads to a highly heteroskedastic regression problem. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples, the ML performs best, but it is shown that in smaller samples, the weighted regression outperforms the ML estimation with respect to bias and mean square error.