Generalized Kerridge’s inaccuracy measure for conditionally specified models

Recently, conditional Renyi’s divergence of order α and Kerridge’s inaccuracy measures are studied by Navarro et al. (2014 Navarro, J., Sunoj, S.M., Linu, M.N. (2014). Characterizations of bivariate models using some dynamic conditional information divergence measures. Commun. Stat. Theory Methods 43:1939–1948.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). In the present article, a generalized dynamic conditional Kerridge’s inaccuracy measure is introduced, which can be represented as the sum of conditional Renyi’s divergence and Renyi’s entropy. Some useful bounds are obtained using the concept of likelihood ratio order. The results are extended to weighted distributions. Sufficient conditions are obtained for the monotonicity of the proposed measure. Characterizations for bivariate exponential conditional distribution are presented based on the proposed measure.     

New classes of discrete bivariate distributions with application to football data

Motivated by Lee and Cha [The American Statistician 69 (2015) 221–230], we introduce 18 new families of discrete bivariate distributions. We give explicit expressions for their probability mass functions, cumulative distribution functions, product moments, and moment-generating functions. We reanalyze the football data in Lee and Cha (2015 Lee, H., Cha, J.H. (2015). On two general classes of discrete bivariate distributions. Am. Stat. 69:221–230.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and show that some of the newly proposed distributions provide better fits than the two families proposed by Lee and Cha.     

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.     

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.     

A Bayesian control chart for a common coefficient of variation

By using the medical data analyzed by Kang et al. (2007 Kang, C.W., Lee, M.S., Seong, Y.J., Hawkins, D.M. (2007). A control chart for the coefficient of variation. J. Qual. Technol. 39(2):151–158.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), a Bayesian procedure is applied to obtain control limits for the coefficient of variation. Reference and probability matching priors are derived for a common coefficient of variation across the range of sample values. By simulating the posterior predictive density function of a future coefficient of variation, it is shown that the control limits are effectively identical to those obtained by Kang et al. (2007 Kang, C.W., Lee, M.S., Seong, Y.J., Hawkins, D.M. (2007). A control chart for the coefficient of variation. J. Qual. Technol. 39(2):151–158.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for the specific dataset they used. This article illustrates the flexibility and unique features of the Bayesian simulation method for obtaining posterior distributions, predictive intervals, and run-lengths in the case of the coefficient of variation. A simulation study shows that the 95% Bayesian confidence intervals for the coefficient of variation have the correct frequentist coverage.     

Frailty models power variance function with cure fraction and latent risk factors negative binomial

In this article, we propose a flexible cure rate model, which is an extension of Cancho et al. (2011 Cancho, V.G., Rodrigues, J., de Castro, M. (2011). A flexible model for survival data with a cure rate: A Bayesian approach. J. Appl. Stat. 38:57–70.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) model, by incorporating a power variance function (PVF) frailty term in latent risk. The model is more flexible in terms of dispersion and it also quantifies the unobservable heterogeneity. The parameter estimation is reached by maximum likelihood estimation procedure and Monte Carlo simulation studies are considered to evaluate the proposed model performance. The practical relevance of the model is illustrated in a real data set of preventing cancer recurrence.     

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.     

Bayesian Estimation for Linear Functions of Poisson Rates

The probability matching prior for linear functions of Poisson parameters is derived. A comparison is made between the confidence intervals obtained by Stamey and Hamilton (2006 Stamey, J., Hamilton, C. (2006). A note on confidence intervals for a linear function of Poisson rates. Commun. Statist. Simul. &; Computat. 35(4):849–856.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and the intervals derived by us when using the Jeffreys’ and probability matching priors. The intervals obtained from the Jeffreys’ prior are in some cases fiducial intervals (Krishnamoorthy and Lee, 2010 Krishnamoorthy, K., Lee, M. (2010). Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases. J. Statist. Plann. Infere. 140(5):1182–1192.[Crossref], [Web of Science ®] , [Google Scholar]). A weighted Monte Carlo method is used for the probability matching prior. The power and size of the test, using Bayesian methods, is compared to tests used by Krishnamoorthy and Thomson (2004 Krishnamoorthy, K., Thomson, J. (2004). A more powerful test for comparing two Poisson means. J. Statist. Plann. Infere. 119(1):23–35.[Crossref], [Web of Science ®] , [Google Scholar]). The Jeffreys’, probability matching and two other priors are used.     

Linear transformation models for survival analysis with tumor growth information in cancer screening study

The complication in analyzing tumor data is that the tumors detected in a screening program tend to be slowly progressive tumors, which is the so-called left-truncated sampling that is inherent in screening studies. Under the assumption that all subjects have the same tumor growth function, Ghosh (2008 Ghosh, D. (2008). Proportional hazards regression for cancer studies. Biometrics 64:141–148.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) developed estimation procedures for the Cox proportional hazards model. Shen (2011a Shen, P.-S. (2011a). Proportional hazards regression for cancer screening data. J. Stat. Comput. Simul. 18:367–377.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) demonstrated that Ghosh (2008 Ghosh, D. (2008). Proportional hazards regression for cancer studies. Biometrics 64:141–148.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar])'s approach can be extended to the case when each subject has a specific growth function. In this article, under linear transformation model, we present a general framework to the analysis of data from cancer screening studies. We developed estimation procedures under linear transformation model, which includes Cox's model as a special case. A simulation study is conducted to demonstrate the potential usefulness of the proposed estimators.     

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.     

A new efficient unrelated randomized response model

The present paper suggests an interesting and useful ramification of the unrelated randomized response model due to Pal and Singh (2012 Pal, S., and S. Singh. 2012. A new unrelated question randomized response model. Statistics 46 (1):99–109.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) [A new unrelated question randomized response model. Statistics 46 (1), 99–109] that can be used for any sampling scheme. We have shown theoretically and numerically that the proposed model is more efficient than Pal and Singh (2012 Pal, S., and S. Singh. 2012. A new unrelated question randomized response model. Statistics 46 (1):99–109.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) model.     

A Modification of a Percentile Estimation Procedure Based on Generalized Polya Urns

Adaptive designs find an important application in the estimation of unknown percentiles for an underlying dose-response curve. A nonparametric adaptive design was suggested by Mugno et al. (2004 Mugno, R.A., Zhus, W., Rosenberger, W.F. (2004). Adaptive urn designs for estimating several percentiles of a dose-response curve. Statist. Med. 23(13):2137–2150.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to simultaneously estimate multiple percentiles of an unknown dose-response curve via generalized Polya urns. In this article, we examine the properties of the design proposed by Mugno et al. (2004 Mugno, R.A., Zhus, W., Rosenberger, W.F. (2004). Adaptive urn designs for estimating several percentiles of a dose-response curve. Statist. Med. 23(13):2137–2150.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) when delays in observing responses are encountered. Using simulations, we evaluate a modification of the design under varying group sizes. Our results demonstrate unbiased estimation with minimal loss in efficiency when compared to the original compound urn design.     

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.     

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]).     

Matrix mean squared error comparisons of some biased estimators with two biasing parameters

To deal with multicollinearity problem, the biased estimators with two biasing parameters have recently attracted much research interest. The aim of this article is to compare one of the last proposals given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) with Liu-type estimator (Liu 2003 Liu, K. 2003. Using Liu-type estimator to combat collinearity. Communications in Statistics: Theory and Methods 32 (5):1009–20.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and k ? d class estimator (Sakallioglu and Kaciranlar 2008 Sakallioglu, S., and S. Kaciranlar. 2008. A new biased estimator based on ridge estimation. Statistical Papers 49:669–89.[Crossref], [Web of Science ®] , [Google Scholar]) under the matrix mean squared error criterion. As well as giving these comparisons theoretically, we support the results with the extended simulation studies and real data example, which show the advantages of the proposal given by Yang and Chang (2010 Yang, H., and X. Chang. 2010. A new two-parameter estimator in linear regression. Communications in Statistics: Theory and Methods 39 (6):923–34.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) over the other proposals with increasing multicollinearity level.     

Multivariate Bayesian U-type asymmetric designs for non parametric response surface prediction under correlated errors

The objective of this paper is to study U-type designs for Bayesian non parametric response surface prediction under correlated errors. The asymptotic Bayes criterion is developed in terms of the asymptotic approach of Mitchell et al. (1994 Mitchell, T., Sacks, J., Ylvisaker, D. (1994). Asymptotic Bayes criteria for nonparametric response surface design. Ann. Stat. 22:634–651.[Crossref], [Web of Science ®] , [Google Scholar]) for a more general covariance kernel proposed by Chatterjee and Qin (2011 Chatterjee, K., Qin, H. (2011). Generalized discrete discrepancy and its applications in experimental designs. J. Stat. Plann. Inference 141:951–960.[Crossref], [Web of Science ®] , [Google Scholar]). A relationship between the asymptotic Bayes criterion and other criteria, such as orthogonality and aberration, is then developed. A lower bound for the criterion is also obtained, and numerical results show that this lower bound is tight. The established results generalize those of Yue et al. (2011 Yue, R.X., Qin, H., Chatterjee, K. (2011). Optimal U-type design for Bayesian nonparametric multiresponse prediction. J. Stat. Plann. Inference 141:2472–2479.[Crossref], [Web of Science ®] , [Google Scholar]) from symmetrical case to asymmetrical U-type designs.     

General mathematical properties,regression and applications of the log-gamma-generated family

The construction of some wider families of continuous distributions obtained recently has attracted applied statisticians due to the analytical facilities available for easy computation of special functions in programming software. We study some general mathematical properties of the log-gamma-generated (LGG) family defined by Amini, MirMostafaee, and Ahmadi (2014 Amini, M., S. M. T. K. MirMostafaee, and J. Ahmadi. 2014. Log-gamma-generated families of distributions. Statistics 48:913–32.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). It generalizes the gamma-generated class pioneered by Risti? and Balakrishnan (2012 Risti?, M. M., and N. Balakrishnan. 2012. The gamma exponentiated exponential distribution. Journal of Statistical Computation and Simulation 82:1191–206.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We present some of its special models and derive explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves, Shannon entropy, Rényi entropy, reliability, and order statistics. Models in this family are compared with nested and non nested models. Further, we propose and study a new LGG family regression model. We demonstrate that the new regression model can be applied to censored data since it represents a parametric family of models and therefore can be used more effectively in the analysis of survival data. We prove that the proposed models can provide consistently better fits in some applications to real data sets.     

Optimal allocations of coverage limits for two independent random losses of insurance policy

This article recasts the optimal allocations of coverage limits for two independent random losses. Under some regularity conditions on the two concerned probability density functions, we build the sufficient and necessary condition for the existence of the optimal allocation of coverage limits, and derive the optimal allocation whenever they do exist. The results supplement Lu and Meng (2011 Lu, Z.Y., Meng, L.L. (2011). Stochastic comparisons for allocations of upper limits and deductibles with applications. Insur.: Math. Econ. 48:338–343.[Crossref], [Web of Science ®] , [Google Scholar], Proposition 5.2) and Hu and Wang (2014 Hu, S., Wang, R. (2014). Stochastic comparisons and optimal allocation for policy limits and deductibles. Commun. Stat. – Theory Methods 43:151–164.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.1).     

Some extensions of circular balanced and circular strongly balanced repeated measurements designs

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this paper the works of Iqbal and Tahir (2009 Iqbal, I., and M. H. Tahir. 2009. Circular strongly balanced repeated measurements designs. Communications in Statistics—Theory and Methods 38:3686–96.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Iqbal, Tahir, and Ghazali (2010 Iqbal, I., M. H. Tahir, and S. S. A. Ghazali. 2010. Circular first- and second-order balanced repeated measurements designs. Communications in Statistics—Theory and Methods 39:228–40.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) are generalized for the construction of circular-balanced and circular strongly balanced repeated measurements designs through the method of cyclic shifts for three periods.