The inverse power Lindley distribution

Several probability distributions have been proposed in the literature, especially with the aim of obtaining models that are more flexible relative to the behaviors of the density and hazard rate functions. Recently, two generalizations of the Lindley distribution were proposed in the literature: the power Lindley distribution and the inverse Lindley distribution. In this article, a distribution is obtained from these two generalizations and named as inverse power Lindley distribution. Some properties of this distribution and study of the behavior of maximum likelihood estimators are presented and discussed. It is also applied considering two real datasets and compared with the fits obtained for already-known distributions. When applied, the inverse power Lindley distribution was found to be a good alternative for modeling survival data.     

A comparison of several adaptive tests for paired data

In the last few years, two adaptive tests for paired data have been proposed. One test proposed by Freidlin et al. [On the use of the Shapiro–Wilk test in two-stage adaptive inference for paired data from moderate to very heavy tailed distributions, Biom. J. 45 (2003), pp. 887–900] is a two-stage procedure that uses a selection statistic to determine which of three rank scores to use in the computation of the test statistic. Another statistic, proposed by O'Gorman [Applied Adaptive Statistical Methods: Tests of Significance and Confidence Intervals, Society for Industrial and Applied Mathematics, Philadelphia, 2004], uses a weighted t-test with the weights determined by the data. These two methods, and an earlier rank-based adaptive test proposed by Randles and Hogg [Adaptive Distribution-free Tests, Commun. Stat. 2 (1973), pp. 337–356], are compared with the t-test and to Wilcoxon's signed-rank test. For sample sizes between 15 and 50, the results show that the adaptive test proposed by Freidlin et al. and the adaptive test proposed by O'Gorman have higher power than the other tests over a range of moderate to long-tailed symmetric distributions. The results also show that the test proposed by O'Gorman has greater power than the other tests for short-tailed distributions. For sample sizes greater than 50 and for small sample sizes the adaptive test proposed by O'Gorman has the highest power for most distributions.     

Power-normal distribution

Recently, Gupta and Gupta [Analyzing skewed data by power-normal model, Test 17 (2008), pp. 197–210] proposed the power-normal distribution for which normal distribution is a special case. The power-normal distribution is a skewed distribution, whose support is the whole real line. Our main aim of this paper is to consider bivariate power-normal distribution, whose marginals are power-normal distributions. We obtain the proposed bivariate power-normal distribution from Clayton copula, and by making a suitable transformation in both the marginals. Lindley–Singpurwalla distribution also can be used to obtain the same distribution. Different properties of this new distribution have been investigated in detail. Two different estimators are proposed. One data analysis has been performed for illustrative purposes. Finally, we propose some generalizations to multivariate case also along the same line and discuss some of its properties.     

On confidence regions of semiparametric nonlinear reproductive dispersion models

This article investigates the confidence regions for semiparametric nonlinear reproductive dispersion models (SNRDMs), which is an extension of nonlinear regression models. Based on local linear estimate of nonparametric component and generalized profile likelihood estimate of parameter in SNRDMs, a modified geometric framework of Bates and Wattes is proposed. Within this geometric framework, we present three kinds of improved approximate confidence regions for the parameters and parameter subsets in terms of curvatures. The work extends the previous results of Hamilton et al. [in Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions, Ann. Statist. 10, pp. 386–393, 1982], Hamilton [in Confidence regions for parameter subset in nonlinear regression, Biometrika, 73, pp. 57–64, 1986], Wei [in On confidence regions of embedded models in regular parameter families (a geometric approch), Austral. J. Statist. 36, pp. 327–338, 1994], Tang et al. [in Confidence regions in quasi-likelihood nonlinear models: a geometric approach, J. Biomath. 15, pp. 55–64, 2000b] and Zhu et al. [in On confidence regions of semiparametric nonlinear regression models, Acta. Math. Scient. 20, pp. 68–75, 2000].     

The exponentiated exponential–geometric distribution: a distribution with decreasing,increasing and unimodal failure rate

In this paper, we proposed a new family of distributions namely exponentiated exponential–geometric (E2G) distribution. The E2G distribution is a straightforwardly generalization of the exponential–geometric (EG) distribution proposed by Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], which accommodates increasing, decreasing and unimodal hazard functions. It arises on a latent competing risk scenarios, where the lifetime associated with a particular risk is not observable but only the minimum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime and modal value. Maximum-likelihood inference is implemented straightforwardly. From a mis-specification simulation study performed in order to assess the extent of the mis-specification errors when testing the EG distribution against the E2G, and we observed that it is usually possible to discriminate between both distributions even for moderate samples with presence of censoring. The practical importance of the new distribution was demonstrated in three applications where we compare the E2G distribution with several lifetime distributions.     

Discrete generalized exponential distribution of a second type

In this paper, we shall attempt to introduce another discrete analogue of the generalized exponential distribution of Gupta and Kundu [Generalized exponential distributions, Aust. N. Z. J. Stat. 41(2) (1999), pp. 173–188], different to that of Nekoukhou et al. [A discrete analogue of the generalized exponential distribution, Comm. Stat. Theory Methods, to appear (2011)]. This new discrete distribution, which we shall call a discrete generalized exponential distribution of the second type (DGE₂(α, p)), can be viewed as another generalization of the geometric distribution. We shall first study some basic distributional and moment properties, as well as order statistics distributions of this family of new distributions. Certain compounded DGE₂(α, p) distributions are also discussed as the results of which some previous lifetime distributions such as that of Adamidis and Loukas [A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42] follow as corollaries. Then, we will investigate estimation of the parameters involved. Finally, we will examine the model with a real data set.     

A new estimator of entropy and its application in testing normality

In this paper, we introduce a new estimator of entropy of a continuous random variable. We compare the proposed estimator with the existing estimators, namely, Vasicek [A test for normality based on sample entropy, J. Roy. Statist. Soc. Ser. B 38 (1976), pp. 54–59], van Es [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Correa [A new estimator of entropy, Commun. Statist. Theory and Methods 24 (1995), pp. 2439–2449] and Wieczorkowski-Grzegorewski [Entropy estimators improvements and comparisons, Commun. Statist. Simulation and Computation 28 (1999), pp. 541–567]. We next introduce a new test for normality. By simulation, the powers of the proposed test under various alternatives are compared with normality tests proposed by Vasicek (1976) and Esteban et al. [Monte Carlo comparison of four normality tests using different entropy estimates, Commun. Statist.–Simulation and Computation 30(4) (2001), pp. 761–785].     

General results for the Kumaraswamy-G distribution

For any continuous baseline G distribution [G.M. Cordeiro and M. de Castro, A new family of generalized distributions, J. Statist. Comput. Simul. 81 (2011), pp. 883–898], proposed a new generalized distribution (denoted here with the prefix ‘Kw-G’ (Kumaraswamy-G)) with two extra positive parameters. They studied some of its mathematical properties and presented special sub-models. We derive a simple representation for the Kw-G density function as a linear combination of exponentiated-G distributions. Some new distributions are proposed as sub-models of this family, for example, the Kw-Chen [Z.A. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statist. Probab. Lett. 49 (2000), pp. 155–161], Kw-XTG [M. Xie, Y. Tang, and T.N. Goh, A modified Weibull extension with bathtub failure rate function, Reliab. Eng. System Safety 76 (2002), pp. 279–285] and Kw-Flexible Weibull [M. Bebbington, C.D. Lai, and R. Zitikis, A flexible Weibull extension, Reliab. Eng. System Safety 92 (2007), pp. 719–726]. New properties of the Kw-G distribution are derived which include asymptotes, shapes, moments, moment generating function, mean deviations, Bonferroni and Lorenz curves, reliability, Rényi entropy and Shannon entropy. New properties of the order statistics are investigated. We discuss the estimation of the parameters by maximum likelihood. We provide two applications to real data sets and discuss a bivariate extension of the Kw-G distribution.     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

Comment on “Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring”

Lin et al. [Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring, J. Stat. Comput. Simul. 81 (2011), pp. 873–882] claimed to have derived exact Bayesian sampling plans for exponential distributions with progressive hybrid censoring. We comment on the accuracy of the design parameters of their proposed sampling plans and the associated Bayes risks given in tables of Lin et al. [Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring, J. Stat. Comput. Simul. 81 (2011), pp. 873–882]. Counter-examples to their claim are provided.     

On the entropy estimators

The paper introduces an estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Ebrahimi et al. [Two measures of sample entropy, Statist. Probab. Lett. 20 (1994), pp. 225–234]. The consistency of the estimator is proved and comparisons are made with Vasicek's estimator [A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B 38 (1976), pp. 54–59], van Es estimator [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Ebrahimi et al. estimator and Correa estimator [A new estimator of entropy, Comm. Statist. Theory Methods 24 (1995), pp. 2439–2449]. The results indicate that the proposed estimator has smaller mean-squared error than above estimators. A real example is presented and analysed.     

Pfeifer records process

In this article, we study the limit distribution of sums of Pfeifer records. Motivated by the results obtained by Arnold and Villaseñor [Generalized order statistics process and Pfeifer records, Statistics 46(3) (2012), pp. 373–385], we show that the partial sum process of Pfeifer records converge to a function of the Brownian motion. The normalization is either a sequence of appropriate constants or a sequence of functions, depending on the tail behaviour of the underlying variables. These results, in particular, prove stronger version of results obtained in Villaseñor and Arnold [On limit laws for sums of Pfeifer records, Extremes 10 (2007), pp. 235–248] and Bose and Gangopadhyay [Convergence of linear functions of Pfeifer records, Extremes 13 (2010), pp. 89–106] and extends results of Bose et al. [Partial sum process for records, Extremes 8 (2005), pp. 43–56] from classical records to Pfeifer records.     

Semiparametric location mixtures with distinct components

We consider a two-component location mixture model with symmetric components, one of which is assumed to be known, the other is unknown. We show identifiability under assumptions on the tails of the characteristic function for the true underlying mixture, and also construct asymptotically normal estimates. The model is an extension of the contamination model in Bordes et al. [Semiparametric estimation of a two-component mixture model when a component is known, Scand. J. Statist. 33 (2006), pp. 733–752], and also related to a location mixture of one symmetric density as in Bordes et al. [Semiparametric estimation of a two component mixture model, Ann. Statist. 34 (2006), pp. 1204–1232]. We show by simulation that estimating the additional location parameter leads to a slight loss of efficiency as compared with the contamination model.     

Construction of fixed width confidence intervals for a Bernoulli success probability using sequential sampling: a simulation study

This article considers the construction of level 1?α fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229–1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than ‘exact’ for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119–126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209–212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101–133] have close to 1?α confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.     

The beta Weibull-geometric distribution

A new five-parameter distribution called the beta Weibull-geometric (BWG) distribution is proposed. The new distribution is generated from the logit of a beta random variable and includes the Weibull-geometric distribution of Barreto-Souza et al. [The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], beta Weibull (BW), beta exponential, exponentiated Weibull, and some other lifetime distributions as special cases. A comprehensive mathematical treatment of this distribution is provided. The density function can be expressed as an infinite mixture of BW densities and then we derive some mathematical properties of the new distribution from the corresponding properties of the BW distribution. The density function of the order statistics and also estimation of the stress–strength parameter are obtained using two general expressions. To estimate the model parameters, we use the maximum likelihood method and the asymptotic distribution of the estimators is also discussed. The capacity of the new distribution are examined by various tools, using two real data sets.     

Bayes estimation of Gompertz distribution parameters and acceleration factor under partially accelerated life tests with type-I censoring

In this paper, the Bayesian approach is applied to the estimation problem in the case of step stress partially accelerated life tests with two stress levels and type-I censoring. Gompertz distribution is considered as a lifetime model. The posterior means and posterior variances are derived using the squared-error loss function. The Bayes estimates cannot be obtained in explicit forms. Approximate Bayes estimates are computed using the method of Lindley [D.V. Lindley, Approximate Bayesian methods, Trabajos Estadistica 31 (1980), pp. 223–237]. The advantage of this proposed method is shown. The approximate Bayes estimates obtained under the assumption of non-informative priors are compared with their maximum likelihood counterparts using Monte Carlo simulation.     

A new four-parameter lifetime distribution

Generalizing lifetime distributions is always precious for applied statisticians. In this paper, we introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley geometric (EPLG) distribution, obtained by compounding EPL and geometric distributions. The new distribution arises in a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the maximum lifetime value among all risks. The distribution exhibits decreasing, increasing, unimodal and bathtub-shaped hazard rate functions, depending on its parameters. It contains several lifetime distributions as particular cases: EPL, new generalized Lindley, generalized Lindley, power Lindley and Lindley geometric distributions. We derive several properties of the new distribution such as closed-form expressions for the density, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Simulation studies are also provided. Finally, two real data applications are given for showing the flexibility and potentiality of the new distribution.     

Fisher information in progressive hybrid censoring schemes

The hybrid censoring scheme, which is a mixture of Type-I and Type-II censoring schemes, has been extended to the case of progressive censoring schemes by Kundu and Joarder [Analysis of Type-II progressively hybrid censored data, Comput. Stat. Data Anal. 50 (2006), pp. 2509–2528] and Childs et al. [Exact likelihood inference for an exponential parameter under progressive hybrid censoring schemes, in Statistical Models and Methods for Biomedical and Technical Systems, F. Vonta, M. Nikulin, N. Limnios, and C. Huber-Carol, eds., Birkhäuser, Boston, MA, 2007, pp. 323–334]. In this paper, we derive a simple expression for the Fisher information contained in Type-I and Type-II progressively hybrid censored data. An illustrative example is provided applicable to a scaled-exponential distribution to demonstrate our methodologies.     

Tests of fit for the Laplace distribution based on correcting moments of entropy estimators

ABSTRACT

In this paper, we first consider the entropy estimators introduced by Vasicek [A test for normality based on sample entropy. J R Statist Soc, Ser B. 1976;38:54–59], Ebrahimi et al. [Two measures of sample entropy. Stat Probab Lett. 1994;20:225–234], Yousefzadeh and Arghami [Testing exponentiality based on type II censored data and a new cdf estimator. Commun Stat – Simul Comput. 2008;37:1479–1499], Alizadeh Noughabi and Arghami [A new estimator of entropy. J Iran Statist Soc. 2010;9:53–64], and Zamanzade and Arghami [Goodness-of-fit test based on correcting moments of modified entropy estimator. J Statist Comput Simul. 2011;81:2077–2093], and the nonparametric distribution functions corresponding to them. We next introduce goodness-of-fit test statistics for the Laplace distribution based on the moments of nonparametric distribution functions of the aforementioned estimators. We obtain power estimates of the proposed test statistics with Monte Carlo simulation and compare them with the competing test statistics against various alternatives. Performance of the proposed new test statistics is illustrated in real cases.     

Statistical diagnostics for nonlinear regression models based on scale mixtures of skew-normal distributions

The purpose of this paper is to develop diagnostics analysis for nonlinear regression models (NLMs) under scale mixtures of skew-normal (SMSN) distributions introduced by Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124]. This novel class of models provides a useful generalization of the symmetrical NLM [Vanegas LH, Cysneiros FJA. Assessment of diagnostic procedures in symmetrical nonlinear regression models. Comput. Statist. Data Anal. 2010;54:1002–1016] since the random terms distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as the skew-t, skew-slash, skew-contaminated normal distributions, among others. Motivated by the results given in Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124], we presented a score test for testing the homogeneity of the scale parameter and its properties are investigated through Monte Carlo simulations studies. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. The newly developed procedures are illustrated considering a real data set.