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.     

On dynamic weighted survival entropy of order α

Recently, Abbasnejad et al. (2010 Abbasnejad, M., Arghami, N.R., Morgenthaler, S., Mohtashami Borzadaran, G.R. (2010). On the dynamic survival entropy. Stat. Probab. Lett. 80:1962–1971.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a measure of uncertainty based on survival function, called the survival entropy of order α. A dynamic form of the survival entropy of order α is also proposed by them. In this paper, we derive the weighted form of these measures. The properties of the new measures are also discussed.     

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.     

On generalized exponential chain ratio estimators under stratified two-phase random sampling

Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) have suggested generalized exponential chain ratio estimators under stratified two-phase sampling scheme for estimating the finite population mean. However, the bias and mean square error (MSE) expressions presented in that work need some corrections, and consequently the study based on efficiency comparison also requires corrections. In this article, we revisit Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]) estimator and provide the correct bias and MSE expressions of their estimator. We also propose an estimator which is more efficient than several competing estimators including the classes of estimators in Sanaullah et al. (2014 Sanaullah, A., Ali, H.M., Noor ul Amin, M., Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Appl. Math. Comput. 226:541–547.[Crossref], [Web of Science ®] , [Google Scholar]). Three real datasets are used for efficiency comparisons.     

Semi-parametric Efficient Inference for Heteroscedastic Semivarying-coefficient Models

Semivarying-coefficient models with heteroscedastic errors are frequently used in statistical modeling. When the error is conditional heteroskedastic, Ahmad, et al. (2005 Ahmad, I., Leelahanon, S., Li, Q. (2005). Efficient estimation of a semiparametric partially linear varying coefficient model. Ann. Statist. 33(1):258–283.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a general series method to obtain an efficient estimation. In this article we study the heteroscedastic semi-varying coefficient models with a nonparametric variance function, not only use the semi-parametric efficient normal approximation method to derive a family of semi-parametric efficient estimator, but also use the semi-parametric efficient empirical likelihood method to construct the efficient empirical likelihood confidence regions. The proposed estimators retain the double robustness feature of semi-parametric efficient estimator.     

A Flexible Skew-Generalized Normal Distribution

Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we will introduce a uni/bimodal generalization of the Azzalini's skew-normal distribution which is indeed an extension of the skew-generalized normal distribution obtained by Arellano-Valle et al. (2004 Arellano-Valle , R. B. , Gómez , H. W. , Quintana , F. A. ( 2004 ). A new class of Skew-normal distributions . Commun. Statist. 33 : 1465 – 1480 .[Taylor & Francis Online] , [Google Scholar]). Our new distribution contains more parameters and thus it is more flexible in data modeling. Indeed, certain univariate case of the so called flexible skew-symmetric distribution of Ma and Genton (2004 Ma , Y. , Genton , M. G. ( 2004 ). Flexible class of skew-symmetric distributions . Scan. J. Statist. 31 : 459 – 468 .[Crossref], [Web of Science ®] , [Google Scholar]) is also a particular case of our proposed model. We will first study some basic distributional properties of the new extension, such as its distribution function, limiting behavior and moments. Then, we will investigate some useful results regarding its relation with other known distributions, such as student's t and skew-Cauchy distributions. In addition, we will present certain methods to generate the new distribution and, finally, we shall apply the model to a real data set to illustrate its behavior comparing to some rival models.     

Performance of some ridge regression estimators for the multinomial logit model

This article considers several estimators for estimating the ridge parameter k for multinomial logit model based on the work of Khalaf and Shukur (2005 Khalaf, G., and G. Shukur. 2005. Choosing ridge parameters for regression problems. Commun. Statist. Theor. Meth., 34:1177–1182.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Alkhamisi et al. (2006 Alkhamisi, M., G. Khalaf, and G. Shukur. 2006. Some modifications for choosing ridge parameters. Commun. Statist. Theor. Meth. 35:2005–2020.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), and Muniz et al. (2012 Muniz, G., B. M. G. Kibria, K. Månsson, and G. Shukur. 2012. On developing ridge regression parameters: A graphical investigation. in SORT. 36: 115–138.[Web of Science ®] , [Google Scholar]). The mean square error (MSE) is considered as the performance criterion. A simulation study has been conducted to compare the performance of the estimators. Based on the simulation study we found that increasing the correlation between the independent variables and the number of regressors has negative effect on the MSE. However, when the sample size increases the MSE decreases even when the correlation between the independent variables is large. Based on the minimum MSE criterion some useful estimators for estimating the ridge parameter k are recommended for the practitioners.     

Improved Exponential Type Estimators of the Mean of a Sensitive Variable in the Presence of Nonsensitive Auxiliary Information

Recently, Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) proposed an exponential type estimator to improve the efficiency of mean estimator based on randomized response technique. In this article, we propose an improved exponential type estimator which is more efficient than the Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) estimator, which in turn was shown to be more efficient than the usual mean estimator, ratio estimator, regression estimator, and the Gupta et al. (2012 Gupta, S., Shabbir, J., Sousa, R., Corte-Real, P. (2012). Regression estimation of the mean of a sensitive variable in the presence of auxiliary information. Communications in Statistics – Theory and Methods 41:2394–2404.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) estimator. Under simple random sampling without replacement (SRSWOR) scheme, bias and mean square error expressions for the proposed estimator are obtained up to first order of approximation and comparisons are made with the Koyuncu et al. (2013 Koyuncu, N., Gupta, S., Sousa, R. (2014). Exponential type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics- Simulation and Computation[PubMed], [Web of Science ®] , [Google Scholar]) estimator. A simulation study is used to observe the performances of these two estimators. Theoretical findings are also supported by a numerical example with real data. We also show how to, extend the proposed estimator to the case when more than one auxiliary variable is available.     

The flexible skew Laplace distribution

ABSTRACT

Skew-symmetric distributions have been discussed by several research-ers. In this article we construct a skew-symmetric Laplace distribution, which is the generalization of distribution given by Ali et al. (2009 Ali, M., Pal, M., Woo, J. (2009). Skewed reflected distributions generated by the Laplace kernel. Aust. J. Statist. 38:45–58. [Google Scholar]) and Nekoukhou and Alamatsaz (2012 Nekoukhou, V., Alamatsaz, M.H. (2012). A family of skew-symmetric-Laplace distributions. Statist. Papers. 53(3):685–696.[Crossref], [Web of Science ®] , [Google Scholar]). This new distribution contains more parameters, and this induces flexibility properties, such as unimodality or bimodality. We study on some properties of this distribution. In the last section we also provide an application with a real data. Concerning example has recently been discussed by Nekoukhou et al. (2013 Nekoukhou, V., Alamatsaz, M.H., Aghajani, A.H. (2013). A flexible skew-generalized normal distribution. Commun. Statist. Theory Methods. 42(13):2324–2334.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) to apply to their model. We compare the behavior of our distribution to their distribution on this example.     

Buffered Autoregressive Models With Conditional Heteroscedasticity: An Application to Exchange Rates

This article introduces a new model called the buffered autoregressive model with generalized autoregressive conditional heteroscedasticity (BAR-GARCH). The proposed model, as an extension of the BAR model in Li et al. (2015 Li, G.D., Guan, B., Li, W.K., and Yu, P. L.H. (2015), “Hysteretic Autoregressive Time Series Models,” Biometrika, 102, 717–723.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), can capture the buffering phenomena of time series in both the conditional mean and variance. Thus, it provides us a new way to study the nonlinearity of time series. Compared with the existing AR-GARCH and threshold AR-GARCH models, an application to several exchange rates highlights the importance of the BAR-GARCH model.     

On the optimal allocation of active redundancies in series system

This paper studies the allocations of two non identical active redundancies in series systems in terms of the reversed hazard rate order and hazard rate order, which generalizes some results built in Valdés and Zequeira (2003 Valdés, J. E., and R. I. Zequeira 2003. On the optimal allocation of an active redundancy in a two-component series system. Stat. Probab. Lett. 63:325–32.[Crossref], [Web of Science ®] , [Google Scholar], 2006 Valdés, J. E., and R. I. Zequeira 2006. On the optimal allocation of two active redundancies in a two-component series system. Oper. Res. Lett. 34:49–52.[Crossref], [Web of Science ®] , [Google Scholar]).     

Optimality of ratio type estimation methods for population mean in the presence of missing data

This article proposes various Searls-type ratio imputation methods (STRIM) on the lines of Ahmed et al. (2006 Ahmed, M. S., O. Al-Titi, Z. Al-Rawi, and W. Abu-Dayyeh. 2006. Estimation of a population mean using different imputation methods. Stat. Trans. 7 (6):1247–1264. [Google Scholar]). It is a well-known fact that the optimal ratio type estimator attains the MSE of regression estimator (or optimal difference estimator) but while using Searls-type transformation (STT) (Searls (1964 Searls, D. T. 1964. The utilization of a known coefficient of variation in the estimation procedure. J. Am. Stat. Assoc. 59:1225–1226.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])) this may not always happen. These STRIM are shown to perform better than the imputation procedures of Ahmed et al. (2006 Ahmed, M. S., O. Al-Titi, Z. Al-Rawi, and W. Abu-Dayyeh. 2006. Estimation of a population mean using different imputation methods. Stat. Trans. 7 (6):1247–1264. [Google Scholar]). The STRIM may even outperform the Searls type difference imputation methods (STDIM) proposed by us in our earlier work, Bhushan and Pandey (2016 Bhushan, S., and A. P. Pandey. 2016. Optimal imputation of the missing data for estimation of population mean. Journal of Statistics and Management System 19 (6):755–69.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). This study is concluded with the numerical study along with the theoretical comparison.     

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.     

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.     

Multiple dependent state repetitive sampling plans based on one-sided process capability indices

In this research, multiple dependent state and repetitive group sampling are used to design a variable sampling plan based on one-sided process capability indices, which consider the quality of the current lot as well as the quality of the preceding lots. The sample size and critical values of the proposed plan are determined by minimizing the average sample number while satisfying the producer's risk and consumer's risk at corresponding quality levels. In addition, comparisons are made with the existing sampling plans [Pearn and Wu (2006a Pearn, W. L., and C. W. Wu. 2006a. Critical acceptance values and sample sizes of a variables sampling plan for very low fraction of defectives. Omega: International Journal of Management Science 34 (1):90–101.[Crossref], [Web of Science ®] , [Google Scholar]), Yen et al. (2015 Yen, C. H., C. H. Chang, and M. Aslam. 2015. Repetitive variable acceptance sampling plan for one-sided specification. Journal of Statistical Computation and Simulation 85 (6):1102–16.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar])] in terms of average sample number and operating characteristic curve. Finally, an example is provided to illustrate the proposed plan.     

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.     

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.     

The Combined Model: A Tool for Simulating Correlated Counts with Overdispersion

The combined model as introduced by Molenberghs et al. (2007 Molenberghs, G., Verbeke, G., Demétrio, C. (2007). An extended random-effects approach to modeling repeated, overdispersed count data. Lifetime Data Analysis 13:513–531.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], 2010 Molenberghs, G., Verbeke, G., Demétrio, C., Vieira, A. (2010). A family of generalized linear models for repeated measures with normal and conjugate random effects. Statistical Science 25:325–347.[Crossref], [Web of Science ®] , [Google Scholar]) has been shown to be an appealing tool for modeling not only correlated or overdispersed data but also for data that exhibit both these features. Unlike techniques available in the literature prior to the combined model, which use a single random-effects vector to capture correlation and/or overdispersion, the combined model allows for the correlation and overdispersion features to be modeled by two sets of random effects. In the context of count data, for example, the combined model naturally reduces to the Poisson-normal model, an instance of the generalized linear mixed model in the absence of overdispersion and it also reduces to the negative-binomial model in the absence of correlation. Here, a Poisson model is specified as the parent distribution of the data conditional on a normally distributed random effect at the subject or cluster level and/or a gamma distribution at observation level. Importantly, the development of the combined model and surrounding derivations have relevance well beyond mere data analysis. It so happens that the combined model can also be used to simulate correlated data. If a researcher is interested in comparing marginal models via Monte Carlo simulations, a necessity to generate suitable correlated count data arises. One option is to induce correlation via random effects but calculation of such quantities as the bias is then not straightforward. Since overdispersion and correlation are simultaneous features of longitudinal count data, the combined model presents an appealing framework for generating data to evaluate statistical properties, through a pre-specification of the desired marginal mean (possibly in terms of the covariates and marginal parameters) and a marginal variance-covariance structure. By comparing the marginal mean and variance of the combined model to the desired or pre-specified marginal mean and variance, respectively, the implied hierarchical parameters and the variance-covariance matrices of the normal and Gamma random effects are then derived from which correlated Poisson data are generated. We explore data generation when a random intercept or random intercept and slope model is specified to induce correlation. The data generator, however, allows for any dimension of the random effects although an increase in the random-effects dimension increases the sensitivity of the derived random effects variance-covariance matrix to deviations from positive-definiteness. A simulation study is carried out for the random-intercept model and for the random intercept and slope model, with or without the normal and Gamma random effects. We also pay specific attention to the case of serial correlation.     

An Approximation Model of the Collective Risk Model with INAR(1) Claim Process

Cossette et al. (2010 Cossette, H., Marceau, E., Maume-Deschamps, V. (2010). Discerte-time risk models based on time series for count random variables. ASTIN Bull. 40:123–150.[Crossref], [Web of Science ®] , [Google Scholar], 2011 Cossette, H., Marceau, E., Toureille, F. (2011). risk models based on time series for count random variables. Insur. Math. Econ. 48:19–28.[Crossref], [Web of Science ®] , [Google Scholar]) gave a novel collective risk model where the total numbers of claims satisfy the first-order integer-valued autoregressive process. For a risk model, it is interesting to investigate the upper bound of ruin probability. However, the loss increments of the above model are dependent; it is difficult to derive the upper bound of ruin probability. In this article, we propose an approximation model with stationary independent increments. The upper bound of ruin probability and the adjustment coefficient are derived. The approximation model is illustrated via four simulated examples. Results show that the gap of the approximation model and dependent model can be ignored by adjusting values of parameters.     

Efficient Mixture Design Fitting Quadratic Surface with Quantile Responses Using First-degree Polynomial

This study considers efficient mixture designs for the approximation of the response surface of a quantile regression model, which is a second degree polynomial, by a first degree polynomial in the proportions of q components. Instead of least squares estimation in the traditional regression analysis, the objective function in quantile regression models is a weighted sum of absolute deviations and the least absolute deviations (LAD) estimation technique should be used (Bassett and Koenker, 1982 Bassett, G., Koenker, R. (1982). An empirical quantile function for linear models with i.i.d. errors. Journal of the American Statistical Association 77:407–415.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]; Koenker and Bassett, 1978 Koenker, R., Bassett, G. (1978). Regression quantiles. Econometrica 46(1):33–50.[Crossref], [Web of Science ®] , [Google Scholar]). Therefore, the standard optimal mixture designs like the D-optimal or A-optimal mixture designs for the least squared estimation are not appropriate. This study explores mixture designs that minimize the bias between the approximated 1st-degree polynomial and a 2nd-degree polynomial response surfaces by the LAD estimation. In contrast to the standard optimal mixture designs for the least squared estimation, the efficient designs might contain elementary centroid design points of degrees higher than two. An example of a portfolio with five assets is given to illustrate the proposed efficient mixture designs in determining the marginal contribution of risks by individual assets in the portfolio.