Extended exponential distribution based on order statistics

The extended exponential distribution due to Nadarajah and Haghighi (2011 Nadarajah, S., Haghighi, F. (2011). An extension of the exponential distribution. Statistics 45:543–558.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) is an alternative to and always provides better fits than the gamma, Weibull, and the exponentiated exponential distributions whenever the data contain zero values. We establish recurrence relations for the single and product moments of order statistics from the extended exponential distribution. These recurrence relations enable computation of the means, variances, and covariances of all order statistics for all sample sizes in a simple and efficient manner. By using these relations, we tabulate the means, variances, and covariances of order statistics and derive best linear unbiased estimates of the extended exponential distribution. Finally, a data application is provided.     

A bivariate generalized geometric distribution with applications

This paper proposes a bivariate version of the univariate discrete generalized geometric distribution considered by Gómez–Déniz (2010 Gómez–Déniz, E. (2010). Another generalization of the geometric distribution. Test 19:399–415.[Crossref], [Web of Science ®] , [Google Scholar]). The proposed bivariate distribution can have a positive or negative correlation coefficient which can be used for modeling bivariate-dependent count data. After discussing some of its properties, maximum likelihood estimation is discussed. Two illustrative examples are given for fitting and demonstrating the usefulness of the new bivariate distribution proposed here.     

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.     

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.     

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.     

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.     

Estimation for exponential distribution based on competing risk middle censored data

Abstract

For a general censoring scheme called “middle censoring” scheme which was proposed by Jammalamadaka and Mangalam (2003 Jammalamadaka, S.R., Mangalam, V. (2003). Nonparametric estimation for middle censored data. J. Nonparametric Statist. 15:253–265.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in nonparametric set up. In this article, point and interval estimation problems are considered for the exponential distribution when the failure data is middle censored with two independent competing failure risks. Different methods are introduced to estimate the unknown model parameters such as maximum likelihood estimation, midpoint approximation, equivalent quantities estimation. The Bayesian estimation is also considered with gamma priors. Two numerical examples are analyzed to show the performance of the proposed methods.     

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.     

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.     

On a stochastic process with a heavy-tailed distributed component describing inventory model type of (s,S)

In the present study, the stochastic process X(t) describing inventory model type of (s, S) with a heavy-tailed distributed demands is considered. The asymptotic expansions at sufficiently large values of parameter β = S ? s for the ergodic distribution and nth-order moment of the process X(t) based on the main results of the studies Teugels (1968 Teugels, J.L. (1968). Renewal theorems when the first or the second moment is infinite. Ann. Math. Stat. 39(4):1210–1219.[Crossref] , [Google Scholar]) and Geluk and Frenk (2011 Geluk, J.L., Frenk, J.B.G. (2011). Renewal theory for random variables with a heavy tailed distribution and finite variance. Stat. Probab. Lett. 81:77–82.[Crossref], [Web of Science ®] , [Google Scholar]) are obtained.     

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.     

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 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.     

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]).     

A revisit of the distribution of linear combinations of Dirichlet components

Provost and Cheong (2000 Provost, S. B., and Y.-H. Cheong. 2000. On the distribution of linear combinations of the components of a Dirichlet random vector. The Canadian Journal of Statistics 28 (2):417–25.[Crossref], [Web of Science ®] , [Google Scholar]) show the importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts. The c-characteristic function has been shown to be very useful and more practical in some distributions that are hard to manage with the traditional characteristic functions. The importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts, is well known. We first provide its inversion formula which is practical in determining the distribution function of a random variable when its c-characteristic function is known. We then use this inversion formula to find an expression of probability density function of linear combinations of components of any Dirichlet vector. This would generalize the currently well known results.     

The Marshall–Olkin extended generalized Rayleigh distribution: Properties and applications

In this paper, a new extension for the generalized Rayleigh distribution is introduced. The proposed model, called Marshall–Olkin extended generalized Rayleigh distribution, arises based on the scheme introduced by Marshall and Olkin (1997) Marshall, A.W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika 84:641–652.[Crossref], [Web of Science ®] , [Google Scholar]. A comprehensive account of the mathematical properties of the new distribution is provided. We discuss about the estimation of the model parameters based on two estimation methods. Empirical applications of the new model to real data are presented for illustrative purposes.     

Credibility premium for rate-making systems

Credibility formula has been developed in many fields of actuarial sciences. Based upon Payandeh (2010 Payandeh, A.T. (2010). A new approach to the credibility formula. Insur.: Math. Econ. 46(2):334–338.[Crossref], [Web of Science ®] , [Google Scholar]), this article extends concept of credibility formula to relatively premium of a given rate-making system. More precisely, it calculates Payandeh’s (2010 Payandeh, A.T. (2010). A new approach to the credibility formula. Insur.: Math. Econ. 46(2):334–338.[Crossref], [Web of Science ®] , [Google Scholar]) credibility factor for zero-inflated Poisson gamma distributions with respect to several loss functions. A comparison study has been given.     

Test of Independence for Baker’s Bivariate Distributions

Baker (2008 Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. Journal of Multivariate Analysis 99: 2312–2327.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new method for constructing multivariate distributions with given marginals based on order statistics. In this paper, we provide a test of independence for a pair of absolutely continuous random variables (X, Y) jointly distributed according to Baker’s bivariate distributions. Our purpose is to test the hypothesis that X and Y are independent versus the alternative that X and Y are positively (negatively) quadrant dependent. The asymptotic distribution of the proposed test statistic is investigated. Also, the powers of the proposed test and the class of distribution-free tests proposed by Kochar and Gupta (1987 Kochar, S. G., Gupta, R. P. (1987). Competitors of Kendall-tau test for testing independence against positive quadrant dependence. Biometrika 74(3): 664–666.[Crossref], [Web of Science ®] , [Google Scholar]) are compared empirically via a simulation 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.     

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.