A generalized binomial exponential 2 distribution: modeling and applications to hydrologic events

Developing statistical methods to model hydrologic events is always interesting for both statisticians and hydrologists, because of its importance in hydraulic structures design and water resource planning. Because of this, a flexible 3-parameter generalization of the exponential distribution is introduced based on the binomial exponential 2 (BE2) distribution [2 H.S. Bakouch, M. Aghababaei Jazi, S. Nadarajah, A. Dolati, and R. Roozegar, A lifetime model with increasing failure rate, Appl. Math. Model. 38 (2014), pp. 5392–5406. doi: 10.1016/j.apm.2014.04.028[Crossref], [Web of Science ®] , [Google Scholar]]. The proposed distribution involving the exponential, gamma and BE2 distributions as submodels; and it exhibits decreasing, increasing and bathtub-shaped hazard rates, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties, parameters estimation and information matrix of the distribution are investigated. The proposed distribution, Gumbel, generalized Logistic and other distributions are utilized to model and fit two hydrologic data sets. The distribution is shown to be more appropriate to the data than the compared distributions using the selection criteria: average scaled absolute error, Akaike information criterion, Bayesian information criterion and Kolmogorov–Smirnov statistics. As a result, some hydrologic parameters of the data are obtained such as return level, conditional mean, mean deviation about the return level and the rth moments of order statistics.     

Multivariate Generalized Distributions of Order k

Abstract

The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1988. Multivariate distributions of order k. Statistics and Probability Letters, 7(3): 207–216.  [Google Scholar]). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990 Philippou, A. N., Antzoulakos, D. L. and Tripsiannis, G. A. 1990. Multivariate distributions of order k, part II. Statistics and Probability Letters, 10(1): 29–35.  [Google Scholar]). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996 Sen, K. and Jain, R. 1996. “Cluster generalized negative binomial distribution”. In Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday Edited by: Borthakur, A. C. 227–241. New Delhi: New Age International Publishers.  [Google Scholar]). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997 Sen, K. and Jain, R. 1997. A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics—Theory and Methods, 26: 871–884. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated.     

Recurrence relations for single and product moments of generalized order statistics from Marshall–Olkin extended Pareto distributions

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(3):641–652.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new method of adding parameter to expand a family of distributions. Using this concept, in this article, the Marshall–Olkin extended Pareto distribution is introduced and some recurrence relations for single and product moments of generalized order statistics are studied. Also the results are deduced for record values and order statistics.     

THE MEDIAN ESTIMATE OF THE AUTOREGRESSIVE LOCATION PARAMETER

ABSTRACT

The concept of generalized order statistics was introduced by Kamps (1995 Kamps , U. ( 1995 ) A Concept of Generalized Order Statistics . Germany : B. G. Teubner Stuttgart [Crossref] , [Google Scholar]) to unify several concepts that have been used in statistics such as order statistics, record values, and sequential order statistics. Estimation of the parameters of the Burr type XII distribution are obtained based on generalized order statistics. The maximum likelihood and Bayes methods of estimation are used for this purposes. The Bayes estimates are derived by using the approximation form of Lindley (1980 Lindley , D. V. ( 1980 ). Approximate Bayesian methods . J. Trabajos de Estadistica 31 : 223 – 237 .[Crossref] , [Google Scholar]). Estimation based on upper records from the Burr model is obtained and compared by using Monte Carlo simulation study. Our results are specialized to the results of AL-Hussaini and Jaheen (1992 AL-Hussaini , E. K. , Jaheen , Z. F. ( 1992 ). Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model . J. Statist. Comput. Simul. 41 : 31 – 40 .[Taylor &; Francis Online] , [Google Scholar]) which are based on ordinary order statistics.     

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.     

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.     

Aspects of Dependence in Generalized Farlie-Gumbel-Morgenstern Distributions

In this article, we obtained a dependence measure for generalized Farlie-Gumbel-Morgenstern (FGM) family in view of Kochar and Gupta (1987 Kochar , S. G. , Gupta , R. P. ( 1987 ). Competitors of Kendall-tau test for testing independence against PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar]) and then compared this measure with Spearman's rho and Kendall's tau in FGM family. Moreover, we evaluated the empirical power of 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 PQD . Biometrika 74 ( 3 ): 664 – 669 .[Crossref], [Web of Science ®] , [Google Scholar], 1990 Kochar , S. G. , Gupta , R. P. ( 1990 ). Distribution-free tests based on sub-sample extrema for testing against positive dependence . Australian Journal of Statistics 32 : 45 – 51 .[Crossref] , [Google Scholar]) based on exact distribution of a U-statistics. This is derived via a simulation study for sample of sizes n = 6, 8, 10, 12, 16, and 20. Also, we compared our simulation results with those achieved by Amini et al. (2010 Amini , M. , Jabbari , H. , Mohtashami Borzadaran , G. R. , Azadbakhsh , M. ( 2010 ). Power comparison of independence test for the Farlie-Gumbel-Moregenstern family . Communications of the Korean Statistical Society 17 ( 4 ): 493 – 505 .[Crossref] , [Google Scholar]) and Güven and Kotz (2008 Güven , B. , Kotz , S. ( 2008 ). Test of independence for generalized Farlie-Gumbel-Morgenstern distributions . Journal of Computational and Applied Mathematics 212 : 102 – 111 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Bias Correction in Generalized Nonlinear Models with Dispersion Covariates

We derive general formulae for the second-order biases of maximum likelihood estimates of the parameters in generalized nonlinear models with dispersion covariates. This result generalizes previous work by Botter and Cordeiro (1998 Botter , D. A. , Cordeiro , G. M. ( 1998 ). Improved estimates for generalized linear models with dispersion covariates . J. Statist. Comput. Simul. 62 : 91 – 104 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Cordeiro and McCullagh (1991 Cordeiro , G. M. , McCullagh , P. ( 1991 ). Bias correction in generalized linear models . J. Roy Statist. Soc. B 53 : 629 – 643 . [Google Scholar]). The practical use of such bias corrections is illustrated in a simulation study.     

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]).     

Stochastic Comparison on General Inactivity Time and General Residual Life of k-Out-of-n Systems

This paper conducts stochastic comparison on general residual life and general inactivity time of (n ? k + 1)-out-of-n systems and investigates the stochastic behavior of the general inactivity time of a system with units having decreasing reversed hazard rate. These results strengthen some conclusions in both Khaledi and Shaked (2006 Khaledi , B. , Shaked , M. ( 2006 ). Ordering conditional residual lifetimes of coherent systems . Journal of Statistical Planning and Inference 137 : 1173 – 1184 .[Crossref], [Web of Science ®] , [Google Scholar]) and Hu et al. (2007 Hu , T. , Jin , W. , Khaledi , M. ( 2007 ). Ordering conditional distributions of generalized order statistics . Probability in the Engineering and Informational Sciences 21 : 401 – 417 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Random Effects,Fixed Effects and Hausman's Test for the Generalized Mixed Regressive Spatial Autoregressive Panel Data Model

This article suggests random and fixed effects spatial two-stage least squares estimators for the generalized mixed regressive spatial autoregressive panel data model. This extends the generalized spatial panel model of Baltagi et al. (2013 Baltagi, B. H., Egger, P., Pfaffermayr, M. (2013). A generalized spatial panel data model with random effects. Econometric Reviews 32:650–685.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) by the inclusion of a spatial lag term. The estimation method utilizes the Generalized Moments method suggested by Kapoor et al. (2007 Kapoor, M., Kelejian, H. H., Prucha, I. R. (2007). Panel data models with spatially correlated error components. Journal of Econometrics 127(1):97–130.[Crossref], [Web of Science ®] , [Google Scholar]) for a spatial autoregressive panel data model. We derive the asymptotic distributions of these estimators and suggest a Hausman test a la Mutl and Pfaffermayr (2011 Mutl, J., Pfaffermayr, M. (2011). The Hausman test in a Cliff and Ord panel model. Econometrics Journal 14:48–76.[Crossref], [Web of Science ®] , [Google Scholar]) based on the difference between these estimators. Monte Carlo experiments are performed to investigate the performance of these estimators as well as the corresponding Hausman test.     

Minimum Divergence,Generalized Empirical Likelihoods,and Higher Order Expansions

This article studies the minimum divergence (MD) class of estimators for econometric models specified through moment restrictions. We show that MD estimators can be obtained as solutions to a tractable lower dimensional optimization problem. This problem is similar to the one solved by the generalized empirical likelihood estimators of Newey and Smith (2004 Newey , W. K. , Smith , R. J. ( 2004 ). Higher order properties of GMM and Generalized Empirical Likelihood estimators . Econometrica 72 : 219 – 255 .[Crossref], [Web of Science ®] , [Google Scholar]), but it is equivalent to it only for a subclass of divergences. The MD framework provides a coherent testing theory: tests for overidentification and parametric restrictions in this framework can be interpreted as semiparametric versions of Pearson-type goodness of fit tests. The higher order properties of MD estimators are also studied and it is shown that MD estimators that have the same higher order bias as the empirical likelihood (EL) estimator also share the same higher order mean square error and are all higher order efficient. We identify members of the MD class that are not only higher order efficient, but also, unlike the EL estimator, well behaved when the moment restrictions are misspecified.     

MULTI-LEVEL FACTORIAL DESIGNS WITH MINIMUM NUMBERS OF LEVEL CHANGES

The order of experimental runs in a fractional factorial experiment is essential when the cost of level changes in factors is considered. The generalized foldover scheme given by [1] Coster, D. C. and Cheng, C. S. 1988. Minimum cost trend free run orders of fractional factorial designs. The Annals of Statistics, 16: 1188–1205. [Crossref], [Web of Science ®] , [Google Scholar]gives an optimal order to experimental runs in an experiment with specified defining contrasts. An experiment can be specified by a design requirement such as resolution or estimation of some interactions. To meet such a requirement, we can find several sets of defining contrasts. Applying the generalized foldover scheme to these sets of defining contrasts, we obtain designs with different numbers of level changes and then the design with minimum number of level changes. The difficulty is to find all the sets of defining contrasts. An alternative approach is investigated by [2] Cheng, C. S., Martin, R. J. and Tang, B. 1998. Two-level factorial designs with extreme numbers of level changes. The Annals of Statistics, 26: 1522–1539. [Crossref], [Web of Science ®] , [Google Scholar]for two-level fractional factorial experiments. In this paper, we investigate experiments with all factors in slevels.     

A Generalized Correlation Coefficient for the General Two-Treatment Crossover Design

This article extends the correlation methodology developed by Chinchilli et al. (2005 Chinchilli , V. M. , Phillips , B. R. , Mauger , D. T. , Szefler , S. J. ( 2005 ). A general class of correlation coefficients for the 2 × 2 crossover design . Biometr. J. 47 : 1 – 10 . [Google Scholar]) for the 2 × 2 crossover design to more complex crossover designs for clinical trials. We describe how the methodology can be adapted to a general type of two-treatment crossover design which includes either at least two sequences or at least two treatment periods or both. We then derive the asymptotic theory for the corresponding correlation statistics, investigate the statistical accuracy of the estimators via bootstrap analyses, and demonstrate their use with two real data examples.     

Second-Order Covariance Matrix Formula for Heteroskedastic Generalized Linear Models

For the first time, we provide a matrix formula for second-order covariances of maximum likelihood estimates in heteroskedastic generalized linear models, thus generalizing the results of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro et al. (2006 Cordeiro , G. M. , Barroso , L. P. , Botter , D. A. (2006). Covariance matrix formula for generalized linear models with unknown dispersion. Commun. Statist. Theor. Meth. 35:113–120.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) related to the generalized linear models with known and unknown dispersion parameter, respectively. The covariance matrix formula does not involve cumulants of log-likelihood derivatives and can be easily obtained using simple matrix operations. We apply our main result to a simple model. Some simulations show that the second-order covariances can be quite pronounced in small to moderate samples. The usual covariances of the maximum likelihood estimates can be corrected by these second-order covariances.     

Relations for Moments of Progressively Type-II Censored Order Statistics from Log-Logistic Distribution with Applications to Inference

In this article, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a log-logistic distribution. The use of these relations in a systematic recursive manner would enable the computation of all the means, variances and covariances of progressively Type-II right censored order statistics from the log-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R ₁,…, R _m ). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan and Malik (1987 Balakrishnan , N. , Malik , H. J. ( 1987 ). Moments of order statistics from truncated log-logistic distribution . J. Statist. Plann. Infer. 17 : 251 – 267 .[Crossref], [Web of Science ®] , [Google Scholar]) and Balakrishnan et al. (1987 Balakrishnan , N. , Malik , H. J. , Puthenpura , S. ( 1987 ). Best linear unbiased estimation of location and scale parameters of the log-logistic distribution . Commun. Statist. Theor. Meth. 16 : 3477 – 3495 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The moments so determined are then utilized to derive best linear unbiased estimators for the scale- and location-scale log-logistic distributions. A comparison of these estimates with the maximum likelihood estimates is made through Monte Carlo simulation. The best linear unbiased predictors of progressively censored failure times is then discussed briefly. Finally, a numerical example is presented to illustrate all the methods of inference developed here.     

Generalized linear failure rate power series distribution

Abstract

In this article, we introduce a new class of lifetime distributions. This new class includes several previously known distributions such as those of Chahkandi and Ganjali (2009 Chahkandi, M., Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computat. Statist. Data Anal. 53:4433–4440.[Crossref], [Web of Science ®] , [Google Scholar]), Mahmoudi and Jafari (2012 Mahmoudi, E., Jafari, A.A. (2012). Generalized exponential power series distributions. Comput. Statist. Data Anal. 56(12):4047–4066.[Crossref], [Web of Science ®] , [Google Scholar]), and Nadarajah et al. (2012 Nadarajah, S., Shahsanaei, F., Rezaei, S. (2012). A new four-parameter lifetime distribution. J. Statist. Computat. Simul.. ifirst, 1–16. [Google Scholar]). This new class of four-parameter distributions allows for flexible failure rate behavior. Indeed, the failure rate function here can be increasing, decreasing, bathtub-shaped or upside-down bathtub-shaped. Several distributional properties of the new class including moments, quantiles and order statistics are studied. An EM algorithm for computing the estimates of the parameters involved is proposed and some maximum entropy characterizations are discussed. Finally, to show the flexibility and potential of the new class of distributions, applications to two real data sets are provided.     

On an Extended Version of Skew Generalized Normal Distribution and Some of its Properties

Abstract

A generalization of Chauvenet's test (see Bol'shev, L. N. 1969 Bol'shev, L. N. 1969. On tests for rejecting outlying observations. Trudy In-ta prikladnoi Mat. Tblissi Gosudart. univ., 2: 159–177. (In Russian) [Google Scholar]. On tests for rejecting outlying observations. Trudy In-ta prikladnoi Mat. Tblissi Gosudart. univ. 2:159–177. (In Russian); Voinov, V. G., Nikulin, M. N. 1996 Voinov, V. G. and Nikulin, M. N. 1996. Unbaised Estimators and Their Applications Vol. 2, Kluwer Academic Publishers.  [Google Scholar]. Unbaised Estimators and Their Applications. Vol. 2. Kluwer Academic Publishers.) suitable to applied the problem of detecting r outliers in an univariate data set is proposed. In the exponential case, the Chauvenet's test can be used. Various modifications of this test were considered by Bol'shev, Ibrakimov and Khalfina (Ibrakimov, I. A., Khalfina 1978 Ibrakimov, I. A. and Khalfina. 1978. Some asymptotic results concerning the Chauvenet test. Ter. Veroyatnost. i Primenen., 23(3): 593–597.  [Google Scholar]. Some asymptotic results concerning the Chauvenet test. Ter. Veroyatnost. i Primenen. 23(3):593–597.), Greenwood and Nikulin (Greenwood, Nikulin, P. E. 1996 Greenwood and Nikulin, P. E. 1996. A Guide to Chi-Squared Testing New York: John Wiley and Sons, Inc..  [Google Scholar]. A Guide to Chi-Squared Testing. New York: John Wiley and Sons, Inc.) depending on the choice of the estimation method used: MLE or MVUE. As procedures for testing one outlier in exponential model have been investigated by a number of authors including Chikkagoudar and Kunchur (Chikkagoudar, M. S., Kunchur, S. H. 1983 Chikkagoudar, M. S. and Kunchur, S. H. 1983. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth., 12: 2127–2142. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth. 12:2127–2142.), Lewis and Fieller (Lewis, T., Fiellerm N. R. J. 1979 Lewis, T. and Fiellerm, N. R. J. 1979. A recursive algorithm for null distribution for outliers: I. Gamma samples. Technometrics, 21: 371–376. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. A recursive algorithm for null distribution for outliers : I. Gamma samples. Technometrics 21:371–376.), Likes (Likes, J. 1966 Likes, J. 1966. Distribution of Dixon's statistics in the case of an exponential population. Metrika, 11: 46–54. (91, 96, 136, 198–200, 204, 209, 210)[Crossref] , [Google Scholar]. Distribution of Dixon's statistics in the case of an exponential population. Metrika 11:46–54. (91, 96, 136, 198–200, 204, 209, 210).) and Kabe (Kabe, D. G. 1970 Kabe, D. G. 1970. Testing outliers from an exponential population. Metrika, 15: 15–18. [Crossref], [Web of Science ®] , [Google Scholar]. Testing outliers from an exponential population. Metrika 15:15–18.); only two types of statistics for testing multiple outliers exist. First is Dixon's while the second is based on the ratio of the sum of the observations suspected to be outliers to the sum of all observations of the sample. In fact, most of these authors have considered a general case of gamma model and the results for exponential model are given a special case. The object of the present communication is to focus on alternative models, namely slippage alternatives (see Barnett, Vic., Toby Lewis 1978 Barnett, Vic. and Toby, Lewis. 1978. Outlier in Statistical Data New York: John Wiley and Sons, Inc..  [Google Scholar]. Outlier in Statistical Data. New York: John Wiley and Sons, Inc.) in exponential samples. We propose a statistic different from the well known Dixon's statistic D_r to test for multiple outliers. Distribution of the test based on this new statistic under slippage alternatives is obtained and hence the tables of critical values are given, for various n (size of the sample) and r (the number of outliers). The power of the new test is also calculated, it is compared to the power of the Dixon's statistic (Chikkagoudar, M. S., Kunchur, S. H. 1983 Chikkagoudar, M. S. and Kunchur, S. H. 1983. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth., 12: 2127–2142. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]. Distribution of test statistics for multiple outliers in exponential samples. Comm. Stat. Theory. and Meth. 12:2127–2142.). Notice that the new statistic based test power is greater the Dixon's statistic based test one.     

Natural Exponential Families and Generalized Hypergeometric Measures

Let ν be a positive Borel measure on ?ⁿ and _pF_q(a₁,…, a_p; b₁,…, b_q; s) be a generalized hypergeometric series. We define a generalized hypergeometric measure, μ_p,q := _pF_q(a₁,…, a_p; b₁,…, b_q;ν), as a series of convolution powers of the measure ν, and we investigate classes of probability distributions which are expressible as such a measure. We show that the Kemp (1968 Kemp , A. W. ( 1968 ). A wide class of discrete distributions and the associated differential equations . Sankhyā, Ser. A 30 : 401 – 410 . [Google Scholar]) family of distributions is an example of μ_p,q in which ν is a Dirac measure on ?. For the case in which ν is a Dirac measure on ?ⁿ, we relate μ_p,q to the diagonal natural exponential families classified by Bar-Lev et al. (1994 Bar-Lev , S. K. , Bshouty , D. , Enis , P. , Letac , G. , Lu , I. , Richards , D. ( 1994 ). The diagonal natural exponential families on ? ⁿ and their classification . J. Theoret. Probab. 7 : 883 – 929 .[Crossref] , [Google Scholar]). For p < q, we show that certain measures μ_p,q can be expressed as the convolution of a sequence of independent multi-dimensional Bernoulli trials. For p = q, q + 1, we show that the measures μ_p,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures.     

Marginal correlation from an extended random-effects model for repeated and overdispersed counts

Vangeneugden et al. [15 Vangeneugden, T., Molenberghs, G., Laenen, A., Geys, H., Beunckens, C. and Sotto, C. 2007. Marginal correlation in longitudinal binary data based on generalized linear mixed models, Tech. Rep., Hasselt University. submitted for publication [Google Scholar]] derived approximate correlation functions for longitudinal sequences of general data type, Gaussian and non-Gaussian, based on generalized linear mixed-effects models (GLMM). Their focus was on binary sequences, as well as on a combination of binary and Gaussian sequences. Here, we focus on the specific case of repeated count data, important in two respects. First, we employ the model proposed by Molenberghs et al. [13 Molenberghs, G., Verbeke, G. and Demétrio, C. G.B. 2007. An extended random-effects approach to modeling repeated, overdispersed count data. Lifetime Data Anal., 13: 513–531. [Crossref], [PubMed], [Web of Science ®] [Google Scholar]], which generalizes at the same time the Poisson-normal GLMM and the conventional overdispersion models, in particular the negative-binomial model. The model flexibly accommodates data hierarchies, intra-sequence correlation, and overdispersion. Second, means, variances, and joint probabilities can be expressed in closed form, allowing for exact intra-sequence correlation expressions. Next to the general situation, some important special cases such as exchangeable clustered outcomes are considered, producing insightful expressions. The closed-form expressions are contrasted with the generic approximate expressions of Vangeneugden et al. [15 Vangeneugden, T., Molenberghs, G., Laenen, A., Geys, H., Beunckens, C. and Sotto, C. 2007. Marginal correlation in longitudinal binary data based on generalized linear mixed models, Tech. Rep., Hasselt University. submitted for publication [Google Scholar]]. Data from an epileptic-seizures trial are analyzed and correlation functions derived. It is shown that the proposed extension strongly outperforms the classical GLMM.