On weighted cumulative residual entropy

In analogy with the weighted Shannon entropy proposed by Belis and Guiasu (1968 Belis, M., Guiasu, S. (1968). A quantitative-qualitative measure of information in cybernetic systems. IEEE Trans. Inf. Th. IT-4:593–594.[Crossref], [Web of Science ®] , [Google Scholar]) and Guiasu (1986 Guiasu, S. (1986). Grouping data by using the weighted entropy. J. Stat. Plann. Inference 15:63–69.[Crossref], [Web of Science ®] , [Google Scholar]), we introduce a new information measure called weighted cumulative residual entropy (WCRE). This is based on the cumulative residual entropy (CRE), which is introduced by Rao et al. (2004 Rao, M., Chen, Y., Vemuri, B.C., Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE Trans. Info. Theory 50(6):1220–1228.[Crossref], [Web of Science ®] , [Google Scholar]). This new information measure is “length-biased” shift dependent that assigns larger weights to larger values of random variable. The properties of WCRE and a formula relating WCRE and weighted Shannon entropy are given. Related studies of reliability theory is covered. Our results include inequalities and various bounds to the WCRE. Conditional WCRE and some of its properties are discussed. The empirical WCRE is proposed to estimate this new information measure. Finally, strong consistency and central limit theorem are provided.     

Finite-sample distributions of the Wald,likelihood ratio,and Lagrange multiplier test statistics in the classical linear model

In this note, it is shown that the finite-sample distributions of the Wald, likelihood ratio, and Lagrange multiplier statistics in the classical linear regression model are members of the generalized beta model introduced by McDonald and Xu (1995a McDonald, J.B., Xu, Y.J. (1995a). A generalization of the beta distribution with applications. J. Econom. 66:133–152.[Crossref], [Web of Science ®] , [Google Scholar]). This is useful for examining the properties of these test statistics. For example, this characterization makes it easy to find distribution, quantile, and density functions for each test statistic, makes it clear why Wald tests may overreject the null hypothesis using asymptotic critical values, and formalizes the fact that the Lagrange multiplier statistic follows a distribution with bounded support.     

Dynamic version of weighted cumulative residual entropy

Some extensions of Shannon entropy to the survival function have been recently proposed. Misagh et al. (2011 Misagh, F., Panahi, Y., Yari, G.H., Shahi, R. (2011, September). Weighted cumulative entropy and its estimation. In: Quality and Reliability (ICQR), 2011, IEEE International conference (pp. 477–480), IEEE.[Crossref] , [Google Scholar]) introduced weighted cumulative residual entropy (WCRE) that was studied more by Mirali et al. (2015 Mirali, M., Baratpour, S., Fakoor, V. (2015). On weighted cumulative residual entropy. Commun. Stat. Theory Methods. doi:10.1080103610926.2015.1053932.[Web of Science ®] , [Google Scholar]). In this article, the dynamic version of WCRE is proposed. Some relationships of this measure with well-known reliability measures and ageing classes are studied and some characterization results for exponential and Rayleigh distributions are provided. Also, a non parametric estimation of dynamic version of WCRE is introduced and its asymptotic behavior is investigated.     

Bimodal symmetric-asymmetric power-normal families

This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim (2005 Kim, H.J. (2005). On a class of two-piece skew-normal distributions. Statist.: J. Theoret. Appl. Statist. 39:537–553.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Arnold et al. (2009 Arnold, B.C., H.W. Gómez, and H.S. Salinas. (2009). On multiple constraint skewed models. Statist. J. Theoret. Appl. Statist. 43: 279–293.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to the exponentiated normal distribution studied in Durrans (1992 Durrans, S.R. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]), that we call the power-normal (PN) distribution. The proposed bimodal extension, the main focus of the paper, is called the bimodal power-normal model and is denoted by BPN(α) model, where α is the asymmetry parameter. The authors give some properties including moments and maximum likelihood estimation. Two important features of the model proposed is that its normalizing constant has closed and simple form and that the Fisher information matrix is nonsingular, guaranteeing large sample properties of the maximum likelihood estimators. Finally, simulation studies and real applications reveal that the proposed model can perform well in both situations.     

Comments on “On the weighted mixed Liu-type estimator under unbiased stochastic restrictions”

We make some comments about the paper of Yildiz (2017 Yildiz, N. 2017. On the weighted mixed Liu-type estimator under unbiased stochastic restrictions. Communications in Statistics Simulation and Computation 46 (9):7238–48. do?:10.1080/03610918.2016.1235189.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and correct the theorems in that paper.     

Characterization results based on dynamic Tsallis cumulative residual entropy

In this article, the concept of cumulative residual entropy (CRE) given by Rao et al. (2004 Rao, M., Chen, Y., Vemuri, B.C., Wang, F. (2004). Cumulative residual entropy: A new measure of information. IEEE Trans. Inf. Theory 50:1220–1228.[Crossref], [Web of Science ®] , [Google Scholar]) is extended to Tsallis entropy function and dynamic version, both residual and past of it. We study some properties and characterization results for these generalized measures. In addition, we provide some characterization results of the first-order statistic based on the Tsallis survival entropy.     

On extreme order statistics from heterogeneous beta distributions with applications

This paper treats the problem of stochastic comparisons for the extreme order statistics arising from heterogeneous beta distributions. Some sufficient conditions involved in majorization-type partial orders are provided for comparing the extreme order statistics in the sense of various magnitude orderings including the likelihood ratio order, the reversed hazard rate order, the usual stochastic order, and the usual multivariate stochastic order. The results established here strengthen and extend those including Kochar and Xu (2007 Kochar, S.C., Xu, M. (2007). Stochastic comparisons of parallel systems when components have proportional hazard rates. Probab. Eng. Inf. Sci. 21:597–609.[Crossref], [Web of Science ®] , [Google Scholar]), Mao and Hu (2010 Mao, T., Hu, T. (2010). Equivalent characterizations on orderings of order statistics and sample ranges. Probab. Eng. Inf. Sci. 24:245–262.[Crossref], [Web of Science ®] , [Google Scholar]), Balakrishnan et al. (2014 Balakrishnan, N., Barmalzan, G., Haidari, A. (2014). On usual multivariate stochastic ordering of order statistics from heterogeneous beta variables. J. Multivariate Anal. 127:147–150.[Crossref], [Web of Science ®] , [Google Scholar]), and Torrado (2015 Torrado, N. (2015). On magnitude orderings between smallest order statistics from heterogeneous beta distributions. J. Math. Anal. Appl. 426:824–838.[Crossref], [Web of Science ®] , [Google Scholar]). A real application in system assembly and some numerical examples are also presented to illustrate the theoretical results.     

An objective Bayesian estimation in a two-period crossover design

Two-period crossover design is one of the commonly used designs in clinical trials. But, the estimation of treatment effect is complicated by the possible presence of carryover effect. It is known that ignoring the carryover effect when it exists can lead to poor estimates of the treatment effect. The classical approach by Grizzle (1965 Grizzle, J.E. (1965). The two-period change-over design and its use in clinical trials. Biometrics 21:467–480. See Grizzle (1974) for corrections.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) consists of two stages. First, a preliminary test is conducted on carryover effect. If the carryover effect is significant, analysis is based only on data from period one; otherwise, analysis is based on data from both periods. A Bayesian approach with improper priors was proposed by Grieve (1985 Grieve, A.P. (1985). A Bayesian analysis of the two-period crossover design for clinical trials. Biometrics 41:979–990.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) which uses a mixture of two models: a model with carryover effect and another without. The indeterminacy of the Bayes factor due to the arbitrary constant in the improper prior was addressed by assigning a minimally discriminatory value to the constant. In this article, we present an objective Bayesian estimation approach to the two-period crossover design which is also based on a mixture model, but using the commonly recommended Zellner–Siow g-prior. We provide simulation studies and a real data example and compare the numerical results with Grizzle (1965 Grizzle, J.E. (1965). The two-period change-over design and its use in clinical trials. Biometrics 21:467–480. See Grizzle (1974) for corrections.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])’s and Grieve (1985 Grieve, A.P. (1985). A Bayesian analysis of the two-period crossover design for clinical trials. Biometrics 41:979–990.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])’s approaches.     

Aging and ordering properties of multivariate lifetimes with Archimedean dependence structures

This paper further studies monotone aging properties of the multivariate random lifetime. We revise the sufficient condition for the negative monotone aging property in terms of the multivariate usual stochastic order in Theorem 3.3 of Rezapour et al. (2013) Rezapour, M., Alamatsaz, M.H., Pellerey, F. (2013). Multivariate aging with Archimedean dependence structures in high dimensions. Commun. Stat. - Theory Methods 42:2056–2070.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] and derive the condition sufficient to the multivariate monotone aging properties in terms of the upper orthant order. Also we study the upper orthant order of multivariate residual lifetimes and inactivity times from populations sharing a common Archimedean survival copula and Archimedean survival copula, respectively. Two simple applications in multivariate stress-strength and frailty models are presented as well.     

Ignorability conditions for frequentist non parametric analysis of conditional distributions with incomplete data

Rubin (1976 Rubin, D.B. (1976). Inference and missing data. Biometrika 63(3):581–592.[Crossref], [Web of Science ®] , [Google Scholar]) derived general conditions under which inferences that ignore missing data are valid. These conditions are sufficient but not generally necessary, and therefore may be relaxed in some special cases. We consider here the case of frequentist estimation of a conditional cdf subject to missing outcomes. We partition a set of data into outcome, conditioning, and latent variables, all of which potentially affect the probability of a missing response. We describe sufficient conditions under which a complete-case estimate of the conditional cdf of the outcome given the conditioning variable is unbiased. We use simulations on a renal transplant data set (Dienemann et al.) to illustrate the implications of these results.     

Bootstrap methods for multivariate hypothesis testing

The nonparametric and parametric bootstrap methods for multivariate hypothesis testing are developed. They are used to approximate the null distribution of the test statistics proposed by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses—with applications. Statistics 49:475–496.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), resulting in bootstrap testing procedures. In the problem of testing for the mean vector of a multivariate distribution, the asymptotic validity of the bootstrap methods is proved. The finite sample performance of the new solutions is demonstrated by means of Monte Carlo simulation studies. They indicate that for small-sample size, the bootstrap tests provide a better finite sample properties than the asymptotic tests considered by Duchesne and Francq (2015 Duchesne, P., Francq, C. (2015). Multivariate hypothesis testing using generalized and {2}-inverses—with applications. Statistics 49:475–496.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

Shared inverse Gaussian frailty models based on additive hazards

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin, or family data), the shared frailty models were suggested. These models are based on the assumption that frailty acts multiplicatively to hazard rate. In this article, we assume that frailty acts additively to hazard rate. We introduce the shared inverse Gaussian frailty models with three different baseline distributions, namely the generalized log-logistic, the generalized Weibull, and exponential power distribution. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo technique to estimate the parameters involved in these models. We apply these models to a real-life bivariate survival dataset of McGilchrist and Aisbett (1991 McGilchrist, C.A., Aisbett, C.W. (1991). Regression with frailty in survival analysis. Biometrics 47:461–466.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) related to the kidney infection data, and a better model is suggested for the data.     

The beta skew-generalized normal distribution

In this article, we develop the skew-generalized normal distribution introduced by Arellano-Valle et al. (2004 Arellano-Valle, R.B., Gomez, H.W., Quintana, F.A. (2004). A new class of skew-normal distribution. Commun. Stat. - Theory Methods. 33(7):1465–1480.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to a new family of the Beta skew-generalized normal (BSGN) distribution . Here, we present some theorems and properties of BSGN distribution and obtain its moment-generating function.     

An optional randomized response model for estimating a rare sensitive attribute using Poisson distribution

The crux of this article is to estimate the mean of the number of persons possessing a rare sensitive attribute based on the Mangat (1991 Mangat, N.S. (1991). An optional randomized response sampling technique using non–stigmatized attribute. Statistica. 51(4):595–602. [Google Scholar]) randomization device by utilizing the Poisson distribution in simple random sampling and stratified sampling. Properties of the proposed randomized response (RR) model have been studied along with recommendations. It is also shown that the proposed model is more efficient than that of Land et al. (2011 Land, M., Singh, S., Sedory, S.A. (2011). Estimation of a rare attribute using Poisson distribution. Statistics. 46(3):351–360.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) in simple random sampling and that of Lee et al. (2013 Lee, G.S., Uhm, D., Kim, J.M. (2013). Estimation of a rare sensitive attribute in stratified sampling using Poisson distribution. Statistics. 47(3):575–589.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) in stratified random sampling when the proportion of persons possessing a rare unrelated attribute is known. Numerical illustrations are also given in support of the present study.     

MSE performance and minimax regret significance points for a HPT estimator under a multivariate t error distribution

In this article, assuming that the error terms follow a multivariate t distribution,we derive the exact formulae forthe moments of the heterogeneous preliminary test (HPT) estimator proposed by Xu (2012b Xu, H. (2012b). MSE performance and minimax regret significance points for a HPT estimator when each individual regression coefficient is estimated. Commun. Stat. Theory Methods 42:2152–2164.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We also execute the numerical evaluation to investigate the mean squared error (MSE) performance of the HPT estimator and compare it with those of the feasible ridge regression (FRR) estimator and the usual ordinary least squared (OLS) estimator. Further, we derive the optimal critical values of the preliminary F test for the HPT estimator, using the minimax regret function proposed by Sawa and Hiromatsu (1973 Sawa, T., Hiromatsu, T. (1973). Minimax regret significance points for a preliminary test in regression analysis. Econometrica 41:1093–1101.[Crossref], [Web of Science ®] , [Google Scholar]). Our results show that (1) the optimal significance level (α*) increases as the degrees of freedom of multivariate t distribution (ν₀) increases; (2) when ν₀ ? 10, the value of α* is close to that in the normal error case.     

The odd log-logistic generalized half-normal lifetime distribution: Properties and applications

Cooray and Ananda (2008 Cooray, K., Ananda, M.M.A. (2008). A Generalization of the half-normal distribution with applications to lifetime data. Commun. Stat. - Theory Methods 37:1323–1337.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) pioneered a lifetime model commonly used in reliability studies. Based on this distribution, we propose a new model called the odd log-logistic generalized half-normal distribution for describing fatigue lifetime data. Various of its structural properties are derived. We discuss the method of maximum likelihood to fit the model parameters. For different parameter settings and sample sizes, some simulation studies compare the performance of the new lifetime model. It can be very useful, and its superiority is illustrated by means of a real dataset.     

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.     

Fubini theorem for non additive measures in the framework of g-expectation

Since the seminal paper of Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), it is known that Fubini theorem for non additive measures can be available only for functions as “slice-comonotonic” in the framework of product algebra. Later, inspired by Ghirardato (1997 Ghirardato, P. 1997. On the independence for non-additive measures, with a Fubini theorem. Journal of Economic Theory 73:261–91.[Crossref], [Web of Science ®] , [Google Scholar]), Chateauneuf and Lefort (2008 Chateauneuf, A., and J. P. Lefort. 2008. Some Fubini theorems on product σ-algebras for non-additive measures. International Journal of Approximate Reasoning 48:686–96.[Crossref], [Web of Science ®] , [Google Scholar]) obtained some Fubini theorems for non additive measures in the framework of product σ-algebra. In this article, we study Fubini theorem for non additive measures in the framework of g-expectation. We give some different assumptions that provide Fubini theorem in the framework of g-expectation.     

An efficient use of moment's ratios of scrambling variables in a randomized response technique

This paper aimed at providing an efficient new unbiased estimator for estimating the proportion of a potentially sensitive attribute in survey sampling. The suggested randomization device makes use of the means, variances of scrambling variables, and the two scalars lie between “zero” and “one.” Thus, the same amount of information has been used at the estimation stage. The variance formula of the suggested estimator has been obtained. We have compared the proposed unbiased estimator with that of Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. Relevant conditions are obtained in which the proposed estimator is more efficient than Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimators. The optimum estimator (OE) in the proposed class of estimators has been identified which finally depends on moments ratios of the scrambling variables. The variance of the optimum estimator has been obtained and compared with that of the Kuk (1990 Kuk, A.Y.C. (1990). Asking sensitive questions inderectely. Biometrika 77:436–438.[Crossref], [Web of Science ®] , [Google Scholar]) and Franklin (1989 Franklin, L.A. (1989). A comparision of estimators for randomized response sampling with continuous distribution s from a dichotomous population. Commun. Stat. Theor. Methods 18:489–505.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator and Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. It is interesting to mention that the “optimum estimator” of the class of estimators due to Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) depends on the parameter π under investigation which limits the use of Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) OE in practice while the proposed OE in this paper is free from such a constraint. The proposed OE depends only on the moments ratios of scrambling variables. This is an advantage over the Singh and Chen (2009 Singh, S., Chen, C.C. (2009). Utilization of higher order moments of scrambling variables in randomized response sampling. J. Stat. Plann. Inference. 139:3377–3380.[Crossref], [Web of Science ®] , [Google Scholar]) estimator. Numerical illustrations are given in the support of the present study when the scrambling variables follow normal distribution. Theoretical and empirical results are very sound and quite illuminating in the favor of the present study.     

Pseudo-likelihood for case–cohort studies under length-biased sampling

A case–cohort design was proposed by Prentice (1986) Prentice, R.L. (1986). A case-cohort design for epidemiologic cohort studies and disease prevention trials. Biometrika 73:1–11.[Crossref], [Web of Science ®] , [Google Scholar] in order to reduce costs. It involves the collection of covariate data from all subjects who experience the event of interest, and from the members of a random subcohort. This case–cohort design has been extensively studied, but is exclusively considered for right-censored data. In this article, we propose case–cohort designs adapted to length-biased data under the proportional hazards assumption. A pseudo-likelihood procedure is described for estimating parameters and the corresponding cumulative hazard function. The large sample properties, such as consistency and weak convergence, for such pseudo-likelihood estimators are presented. We also conduct simulation studies to show that the proposed estimators are appropriate for practical use. A real Oscar Awards data is provided.