A family of skew-symmetric-Laplace distributions

Skew-symmetric distributions of various types have been the center of attraction by many researchers in the literature. In this article, we shall introduce another more general class of skew distributions, specially related to the Laplace distribution. This new class contains some previously known skew distributions. We shall investigate different characteristics of members of this class such as its moments, thus generalizing a result of Umbach (Stat Probab Lett 76:507?C512, 2006), limiting behavior, moment generating function, unimodality and reveal its natural occurrence as the distribution of some order statistics. In addition, we will generalize a result of Aryal and Rao (Nonlinear Anal 63:639?C646, 2005) in connection with truncated skew-Laplace distribution and study its certain stochastic orderings. Some illustrative examples are also provided.     

One- and two-sample Bayesian prediction intervals based on progressively Type-II censored data

In this article, one- and two-sample Bayesian prediction intervals based on progressively Type-II censored data are derived. For the illustration of the developed results, the exponential, Pareto, Weibull and Burr Type-XII models are used as examples. Some of the previous results in the literature such as Dunsmore (Technometrics 16:455–460, 1974), Nigm and Hamdy (Commun Stat Theory Methods 16:1761–1772, 1987), Nigm (Commun Stat Theory Methods 18:897–911, 1989), Al-Hussaini and Jaheen (Commun Stat Theory Methods 24:1829–1842, 1995), Al-Hussaini (J Stat Plan Inference 79:79–91, 1999), Ali Mousa (J Stat Comput Simul 71: 163–181, 2001) and Ali Mousa and Jaheen (Stat Pap 43:587–593, 2002) can be achieved as special cases of our results. Finally, some numerical computations are presented for illustrating all the proposed inferential procedures.     

On the exact joint distribution of a linear combination of order statistics and their concomitants in an exchangeable multivariate normal distribution

In this work we prove that for an exchangeable multivariate normal distribution the joint distribution of a linear combination of order statistics and a linear combination of their concomitants together with an auxiliary variable is skew normal. We also investigate some special cases, thus extending the results of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995), Loperfido (Test 17:370–380, 2008a) and Sheikhi and Jamalizadeh (Paper 52:885–892, 2011).     

An extension of the generalized half-normal distribution

In this paper we propose an extension of the generalized half-normal distribution studied in Cooray and Ananda (Commun Stat 37:1323–1337, 2008). This new distribution is defined by considering the quotient of two random variables, the one in the numerator being a generalized half normal distribution and the one in the denominator being a power of the uniform distribution on \((0,1)\) , respectively. The resulting distribution has greater kurtosis than the generalized half normal distribution. The density function of this more general distribution is derived jointly with some of its properties and moments. We discuss stochastic representation, maximum likelihood and moments estimation. Applications to real data sets are reported revealing that the proposed distribution can fit real data better than the slashed half-normal, generalized half-normal and Birnbaum–Saunders distributions.     

On tests of radial symmetry for bivariate copulas

The unique copula of a continuous random pair \((X,Y)\) is said to be radially symmetric if and only if it is also the copula of the pair \((-X,-Y)\) . This paper revisits the recently considered issue of testing for radial symmetry. Three rank-based statistics are proposed to this end which are asymptotically equivalent but simpler to compute than those of Bouzebda and Cherfi (J Stat Plan Inference 142:1262–1271, 2012). Their limiting null distribution and its approximation using the multiplier bootstrap are discussed. The finite-sample properties of the resulting tests are assessed via simulations. The asymptotic distribution of one of the test statistics is also computed under an arbitrary alternative, thereby correcting an error in the recent work of Dehgani et al. (Stat Pap 54:271–286, 2013).     

The log-beta Weibull regression model with application to predict recurrence of prostate cancer

We study the properties of the called log-beta Weibull distribution defined by the logarithm of the beta Weibull random variable (Famoye et al. in J Stat Theory Appl 4:121–136, 2005; Lee et al. in J Mod Appl Stat Methods 6:173–186, 2007). An advantage of the new distribution is that it includes as special sub-models classical distributions reported in the lifetime literature. We obtain formal expressions for the moments, moment generating function, quantile function and mean deviations. We construct a regression model based on the new distribution to predict recurrence of prostate cancer for patients with clinically localized prostate cancer treated by open radical prostatectomy. It can be applied to censored data since it represents a parametric family of models that includes as special sub-models several widely-known regression models. The regression model was fitted to a data set of 1,324 eligible prostate cancer patients. We can predict recurrence free probability after the radical prostatectomy in terms of highly significant clinical and pathological explanatory variables associated with the recurrence of the disease. The predicted probabilities of remaining free of cancer progression are calculated under two nested models.     

Influence diagnostics for Grubbs’s model with asymmetric heavy-tailed distributions

Grubbs’s model (Grubbs, Encycl Stat Sci 3:42–549, 1983) is used for comparing several measuring devices, and it is common to assume that the random terms have a normal (or symmetric) distribution. In this paper, we discuss the extension of this model to the class of scale mixtures of skew-normal distributions. Our results provide a useful generalization of the symmetric Grubbs’s model (Osorio et al., Comput Stat Data Anal, 53:1249–1263, 2009) and the asymmetric skew-normal model (Montenegro et al., Stat Pap 51:701–715, 2010). We discuss the EM algorithm for parameter estimation and the local influence method (Cook, J Royal Stat Soc Ser B, 48:133–169, 1986) for assessing the robustness of these parameter estimates under some usual perturbation schemes. The results and methods developed in this paper are illustrated with a numerical example.     

Robust parameter estimation for the Ornstein–Uhlenbeck process

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein–Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et?al. (Stat Methods Appl 19:333–354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator.     

Efficient tests for one sample correlated binary data with applications

Four testing procedures are considered for testing the response rate of one sample correlated binary data with a cluster size of one or two, which often occurs in otolaryngologic and ophthalmologic studies. Although an asymptotic approach is often used for statistical inference, it is criticized for unsatisfactory type I error control in small sample settings. An alternative to the asymptotic approach is an unconditional approach. The first unconditional approach is the one based on estimation, also known as parametric bootstrap (Lee and Young in Stat Probab Lett 71(2):143–153, 2005). The other two unconditional approaches considered in this article are an approach based on maximization (Basu in J Am Stat Assoc 72(358):355–366, 1977), and an approach based on estimation and maximization (Lloyd in Biometrics 64(3):716–723, 2008a). These two unconditional approaches guarantee the test size and are generally more reliable than the asymptotic approach. We compare these four approaches in conjunction with a test proposed by Lee and Dubin (Stat Med 13(12):1241–1252, 1994) and a likelihood ratio test derived in this article, in regards to type I error rate and power for sample sizes from small to medium. An example from an otolaryngologic study is provided to illustrate the various testing procedures. The unconditional approach based on estimation and maximization using the test in Lee and Dubin (Stat Med 13(12):1241–1252, 1994) is preferable due to the power advantageous.     

Distribution of product and quotient of bivariate generalized exponential distribution

In this work we derive closed form expressions for the probability density functions and moments of the quotient and product of the components of the bivariate generalized exponential distribution introduced by Kundu and Gupta (J Multivariate Anal, 100:581–593, 2009) and compute the percentage points. The derivations will be useful for practitioners of this bivariate model. We then give a real data application of the product.     

An empirical Bayes procedure for the selection of Gaussian graphical models

A new methodology for model determination in decomposable graphical Gaussian models (Dawid and Lauritzen in Ann. Stat. 21(3), 1272?C1317, 1993) is developed. The Bayesian paradigm is used and, for each given graph, a hyper-inverse Wishart prior distribution on the covariance matrix is considered. This prior distribution depends on hyper-parameters. It is well-known that the models??s posterior distribution is sensitive to the specification of these hyper-parameters and no completely satisfactory method is registered. In order to avoid this problem, we suggest adopting an empirical Bayes strategy, that is a strategy for which the values of the hyper-parameters are determined using the data. Typically, the hyper-parameters are fixed to their maximum likelihood estimations. In order to calculate these maximum likelihood estimations, we suggest a Markov chain Monte Carlo version of the Stochastic Approximation EM algorithm. Moreover, we introduce a new sampling scheme in the space of graphs that improves the add and delete proposal of Armstrong et al. (Stat. Comput. 19(3), 303?C316, 2009). We illustrate the efficiency of this new scheme on simulated and real datasets.     

A discrete version of the half-normal distribution and its generalization with applications

A new discrete distribution depending on two parameters $\alpha >-1$ and $\sigma >0$ is obtained by discretizing the generalized normal distribution proposed in García et al. (Comput Stat and Data Anal 54:2021–2034, 2010), which was derived from the normal distribution by using the Marshall and Olkin (Biometrika 84(3):641–652, 1997) scheme. The particular case $\alpha =1$ leads us to the discrete half-normal distribution which is different from the discrete half-normal distribution proposed previously in the statistical literature. This distribution is unimodal, overdispersed (the responses show a mean sample greater than the variance) and with an increasing failure rate. We revise its properties and the question of parameter estimation. Expected frequencies were calculated for two overdispersed and underdispersed (the responses show a variance greater than the mean) examples, and the distribution was found to provide a very satisfactory fit.     

On diagnostics in multivariate measurement error models under asymmetric heavy-tailed distributions

In this paper, we discuss the extension of some diagnostic procedures to multivariate measurement error models with scale mixtures of skew-normal distributions (Lachos et?al., Statistics 44:541?C556, 2010c). This class provides a useful generalization of normal (and skew-normal) measurement error models since the random term distributions cover symmetric, asymmetric and heavy-tailed distributions, such as skew-t, skew-slash and skew-contaminated normal, among others. Inspired by the EM algorithm proposed by Lachos et?al. (Statistics 44:541?C556, 2010c), we develop a local influence analysis for measurement error models, following Zhu and Lee??s (J R Stat Soc B 63:111?C126, 2001) approach. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and Cook??s well-known approach can be very difficult to apply to achieve local influence measures. Some useful perturbation schemes are also discussed. In addition, a score test for assessing the homogeneity of the skewness parameter vector is presented. Finally, the methodology is exemplified through a real data set, illustrating the usefulness of the proposed methodology.     

Classical and Bayesian estimation of P(X<Y) using upper record values from Kumaraswamy’s distribution

In this paper, maximum likelihood and Bayesian approaches have been used to obtain the estimation of \(P(X<Y)\) based on a set of upper record values from Kumaraswamy distribution. The existence and uniqueness of the maximum likelihood estimates of the Kumaraswamy distribution parameters are obtained. Confidence intervals, exact and approximate, as well as Bayesian credible intervals are constructed. Bayes estimators have been developed under symmetric (squared error) and asymmetric (LINEX) loss functions using the conjugate and non informative prior distributions. The approximation forms of Lindley (Trabajos de Estadistica 3:281–288, 1980) and Tierney and Kadane (J Am Stat Assoc 81:82–86, 1986) are used for the Bayesian cases. Monte Carlo simulations are performed to compare the different proposed methods.     

Approximate MLEs for the location and scale parameters of the skew logistic distribution

Azzalini (Scand J Stat 12:171–178, 1985) provided a methodology to introduce skewness in a normal distribution. Using the same method of Azzalini (1985), the skew logistic distribution can be easily obtained by introducing skewness to the logistic distribution. For the skew logistic distribution, the likelihood equations do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The coverage probabilities of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. To improve the coverage probabilities and for constructing confidence intervals, we suggest the use of simulated percentage points. Finally, we present a numerical example to illustrate the methods of inference developed here.     

Online signal extraction by robust regression in moving windows with data-adaptive width selection

Online (also ‘real-time’ or ‘sequential’) signal extraction from noisy and outlier-interfered data streams is a basic but challenging goal. Fitting a robust Repeated Median (Siegel in Biometrika 69:242–244, 1982) regression line in a moving time window has turned out to be a promising approach (Davies et al. in J. Stat. Plan. Inference 122:65–78, 2004; Gather et al. in Comput. Stat. 21:33–51, 2006; Schettlinger et al. in Biomed. Eng. 51:49–56, 2006). The level of the regression line at the rightmost window position, which equates to the current time point in an online application, is then used for signal extraction. However, the choice of the window width has a large impact on the signal extraction, and it is impossible to predetermine an optimal fixed window width for data streams which exhibit signal changes like level shifts and sudden trend changes. We therefore propose a robust test procedure for the online detection of such signal changes. An algorithm including the test allows for online window width adaption, meaning that the window width is chosen w.r.t. the current data situation at each time point. Comparison studies show that our new procedure outperforms an existing Repeated Median filter with automatic window width selection (Schettlinger et al. in Int. J. Adapt. Control Signal Process. 24:346–362, 2010).     

Likelihood-free parallel tempering

Approximate Bayesian Computational (ABC) methods, or likelihood-free methods, have appeared in the past fifteen years as useful methods to perform Bayesian analysis when the likelihood is analytically or computationally intractable. Several ABC methods have been proposed: MCMC methods have been developed by Marjoram et al. (2003) and by Bortot et al. (2007) for instance, and sequential methods have been proposed among others by Sisson et al. (2007), Beaumont et al. (2009) and Del Moral et al. (2012). Recently, sequential ABC methods have appeared as an alternative to ABC-PMC methods (see for instance McKinley et al., 2009; Sisson et al., 2007). In this paper a new algorithm combining population-based MCMC methods with ABC requirements is proposed, using an analogy with the parallel tempering algorithm (Geyer 1991). Performance is compared with existing ABC algorithms on simulations and on a real example.     

A new procedure for variance estimation in simple random sampling using auxiliary information

In this article we have envisaged an efficient generalized class of estimators for finite population variance of the study variable in simple random sampling using information on an auxiliary variable. Asymptotic expressions of the bias and mean square error of the proposed class of estimators have been obtained. Asymptotic optimum estimator in the proposed class of estimators has been identified with its mean square error formula. We have shown that the proposed class of estimators is more efficient than the usual unbiased, difference, Das and Tripathi (Sankhya C 40:139–148, 1978), Isaki (J. Am. Stat. Assoc. 78:117–123, 1983), Singh et al. (Curr. Sci. 57:1331–1334, 1988), Upadhyaya and Singh (Vikram Math. J. 19:14–17, 1999b), Kadilar and Cingi (Appl. Math. Comput. 173:2, 1047–1059, 2006a) and other estimators/classes of estimators. In the support of the theoretically results we have given an empirical study.     

A three-parameter generalized von Mises distribution

In this paper, we propose a three-parameter generalized von Mises distribution, called the asymmetric generalized von Mises (AGvM) distribution, which is an extension of the von Mises (vM) distribution, and a subclass of the generalized von Mises (GvM) distribution introduced by Gatto and Jammalamadaka (Stat Methodol 4:341–353, 2007). The three parameter model belongs to an exponential family of distributions and can be used to model both asymmetric and bimodal data. Some properties are studied and interpretation of the parameters is discussed in detail. It is shown that the parameters of the AGvM distribution are particularly easy to interpret and contain a skewness measure as one of its three parameters. A real environmental data set example is provided to illustrate the goodness of fit for AGvM distribution.     

Consistency of the likelihood depth estimator for the correlation coefficient

Denecke and Müller (CSDA 55:2724–2738, 2011) presented an estimator for the correlation coefficient based on likelihood depth for Gaussian copula and Denecke and Müller (J Stat Planning Inference 142: 2501–2517, 2012) proved a theorem about the consistency of general estimators based on data depth using uniform convergence of the depth measure. In this article, the uniform convergence of the depth measure for correlation is shown so that consistency of the correlation estimator based on depth can be concluded. The uniform convergence is shown with the help of the extension of the Glivenko-Cantelli Lemma by Vapnik- C? ervonenkis classes.