A new lifetime model with decreasing failure rate

In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Rényi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.     

A Modified Kolmogorov–Smirnov Test for Normality

In this article we propose an improvement of the Kolmogorov-Smirnov test for normality. In the current implementation of the Kolmogorov-Smirnov test, given data are compared with a normal distribution that uses the sample mean and the sample variance. We propose to select the mean and variance of the normal distribution that provide the closest fit to the data. This is like shifting and stretching the reference normal distribution so that it fits the data in the best possible way. A study of the power of the proposed test indicates that the test is able to discriminate between the normal distribution and distributions such as uniform, bimodal, beta, exponential, and log-normal that are different in shape but has a relatively lower power against the student's, t-distribution that is similar in shape to the normal distribution. We also compare the performance (both in power and sensitivity to outlying observations) of the proposed test with existing normality tests such as Anderson–Darling and Shapiro–Francia.     

A new useful three-parameter extension of the exponential distribution

A three-parameter extension of the exponential distribution is introduced and studied in this paper. The new distribution is quite flexible and can be used effectively in modelling survival data, reliability problems, fatigue life studies and hydrological data. It can have constant, decreasing, increasing, upside-down bathtub (unimodal), bathtub-shaped and decreasing–increasing–decreasing hazard rate functions. We provide a comprehensive account of the mathematical properties of the new distribution and various structural quantities are derived. We discuss maximum likelihood estimation of the model parameters for complete sample and for censored sample. An empirical application of the new model to real data is presented for illustrative purposes. We hope that the new distribution will serve as an alternative model to other models available in the literature for modelling real data in many areas.     

A flexible parametric survival model which allows a bathtub-shaped hazard rate function

A new parametric (three-parameter) survival distribution, the lognormal–power function distribution, with flexible behaviour is introduced. Its hazard rate function can be either unimodal, monotonically decreasing or can exhibit a bathtub shape. Special cases include the lognormal distribution and the power function distribution, with finite support. Regions of parameter space where the various forms of the hazard-rate function prevail are established analytically. The distribution lends itself readily to accelerated life regression modelling. Applications to five data sets taken from the literature are given. Also it is shown how the distribution can behave like a Weibull distribution (with negative aging) for certain parameter values.     

Multivariate normal mean–variance mixture distribution based on Birnbaum–Saunders distribution

A multivariate normal mean–variance mixture based on a Birnbaum–Saunders (NMVMBS) distribution is introduced and several properties of this new distribution are discussed. A new robust non-Gaussian ARCH-type model is proposed in which there exists a relation between the variance of the observations, and the marginal distributions are NMVMBS. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of this normal mean–variance mixture distribution is given. A simulation study and some real data are used to demonstrate the modelling strength of this new model.     

Multivariate Birnbaum–Saunders distribution: properties and associated inference

The univariate fatigue life distribution proposed by Birnbaum and Saunders [A new family of life distributions. J Appl Probab. 1969;6:319–327] has been used quite effectively to model times to failure for materials subject to fatigue and for modelling lifetime data and reliability problems. In this article, we introduce a Birnbaum–Saunders (BS) distribution in the multivariate setting. The new multivariate model arises in the context of conditionally specified distributions. The proposed multivariate model is an absolutely continuous distribution whose marginals are univariate BS distributions. General properties of the multivariate BS distribution are derived and the estimation of the unknown parameters by maximum likelihood is discussed. Further, the Fisher's information matrix is determined. Applications to real data of the proposed multivariate distribution are provided for illustrative purposes.     

A new continuous distribution on the unit interval applied to modelling the points ratio of football teams

We introduce a new flexible distribution to deal with variables on the unit interval based on a transformation of the sinh–arcsinh distribution, which accommodates different degrees of skewness and kurtosis and becomes an interesting alternative to model this type of data. We also include this new distribution into the generalised additive models for location, scale and shape (GAMLSS) framework in order to develop and fit its regression model. For different parameter settings, some simulations are performed to investigate the behaviour of the estimators. The potentiality of the new regression model is illustrated by means of a real dataset related to the points rate of football teams at the end of a championship from the four most important leagues in the world: Barclays Premier League (England), Bundesliga (Germany), Serie A (Italy) and BBVA league (Spain) during three seasons (2011–2012, 2012–2013 and 2013–2014).     

Exponentiated power Lindley–Poisson distribution: Properties and applications

A new four-parameter distribution called the exponentiated power Lindley–Poisson distribution which is an extension of the power Lindley and Lindley–Poisson distributions is introduced. Statistical properties of the distribution including the shapes of the density and hazard functions, moments, entropy measures, and distribution of order statistics are given. Maximum likelihood estimation technique is used to estimate the parameters. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators, and width of the confidence interval for each parameter. Finally, applications to real data sets are presented to illustrate the usefulness of the proposed distribution.     

Goodness-of-Fit Tests for the Skew-Normal Distribution When the Parameters Are Estimated from the Data

In this article, tests are developed which can be used to investigate the goodness-of-fit of the skew-normal distribution in the context most relevant to the data analyst, namely that in which the parameter values are unknown and are estimated from the data. We consider five test statistics chosen from the broad Cramér–von Mises and Kolmogorov–Smirnov families, based on measures of disparity between the distribution function of a fitted skew-normal population and the empirical distribution function. The sampling distributions of the proposed test statistics are approximated using Monte Carlo techniques and summarized in easy to use tabular form. We also present results obtained from simulation studies designed to explore the true size of the tests and their power against various asymmetric alternative distributions.     

A robust multivariate Birnbaum–Saunders distribution: EM estimation

We propose here a robust multivariate extension of the bivariate Birnbaum–Saunders (BS) distribution derived by Kundu et al. [Bivariate Birnbaum–Saunders distribution and associated inference. J Multivariate Anal. 2010;101:113–125], based on scale mixtures of normal (SMN) distributions that are used for modelling symmetric data. This resulting multivariate BS-type distribution is an absolutely continuous distribution whose marginal and conditional distributions are of BS-type distribution of Balakrishnan et al. [Estimation in the Birnbaum–Saunders distribution based on scalemixture of normals and the EM algorithm. Stat Oper Res Trans. 2009;33:171–192]. Due to the complexity of the likelihood function, parameter estimation by direct maximization is very difficult to achieve. For this reason, we exploit the nice hierarchical representation of the proposed distribution to propose a fast and accurate EM algorithm for computing the maximum likelihood (ML) estimates of the model parameters. We then evaluate the finite-sample performance of the developed EM algorithm and the asymptotic properties of the ML estimates through empirical experiments. Finally, we illustrate the obtained results with a real data and display the robustness feature of the estimation procedure developed here.     

Uncertainty estimation in heterogeneous capture–recapture count data

The Conway–Maxwell–Poisson estimator is considered in this paper as the population size estimator. The benefit of using the Conway–Maxwell–Poisson distribution is that it includes the Bernoulli, the Geometric and the Poisson distributions as special cases and, furthermore, allows for heterogeneity. Little emphasis is often placed on the variability associated with the population size estimate. This paper provides a deep and extensive comparison of bootstrap methods in the capture–recapture setting. It deals with the classical bootstrap approach using the true population size, the true bootstrap, and the classical bootstrap using the observed sample size, the reduced bootstrap. Furthermore, the imputed bootstrap, as well as approximating forms in terms of standard errors and confidence intervals for the population size, under the Conway–Maxwell–Poisson distribution, have been investigated and discussed. These methods are illustrated in a simulation study and in benchmark real data examples.     

On entropy-based goodness-of-fit test for asymmetric Student-t and exponential power distributions

This paper examines the goodness-of-fit (GOF) test for a generalized asymmetric Student-t distribution (ASTD) and asymmetric exponential power distribution (AEPD). These distributions are known to include a broad class of distribution families and are quite suitable to modelling the innovations of financial time series. Despite their popularity, to our knowledge, no studies in the literature have so far investigated their affinity and differences in implementation. To fill this gap, we examine the empirical power behaviour of entropy-based GOF tests for hypotheses wherein the ASTD and AEPD play the role of null and alternative distributions. Our findings through a simulation study and real data analysis indicate that the two distributions are generally hard to distinguish and that the ASTD family accommodates AEPDs to a greater degree than the other way around for larger samples.     

Modelling claim number using a new mixture model: negative binomial gamma distribution

ABSTRACT

In actuarial applications, mixed Poisson distributions are widely used for modelling claim counts as observed data on the number of claims often exhibit a variance noticeably exceeding the mean. In this study, a new claim number distribution is obtained by mixing negative binomial parameter p which is reparameterized as p?=?exp( ?λ) with Gamma distribution. Basic properties of this new distribution are given. Maximum likelihood estimators of the parameters are calculated using the Newton–Raphson and genetic algorithm (GA). We compared the performance of these methods in terms of efficiency by simulation. A numerical example is provided.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

Modelling interval data with Normal and Skew-Normal distributions

A parametric modelling for interval data is proposed, assuming a multivariate Normal or Skew-Normal distribution for the midpoints and log-ranges of the interval variables. The intrinsic nature of the interval variables leads to special structures of the variance–covariance matrix, which is represented by five different possible configurations. Maximum likelihood estimation for both models under all considered configurations is studied. The proposed modelling is then considered in the context of analysis of variance and multivariate analysis of variance testing. To access the behaviour of the proposed methodology, a simulation study is performed. The results show that, for medium or large sample sizes, tests have good power and their true significance level approaches nominal levels when the constraints assumed for the model are respected; however, for small samples, sizes close to nominal levels cannot be guaranteed. Applications to Chinese meteorological data in three different regions and to credit card usage variables for different card designations, illustrate the proposed methodology.     

An extended fatigue life distribution

A five-parameter extended fatigue life model called the McDonald–Birnbaum–Saunders (McBS) distribution is proposed. It extends the Birnbaum–Saunders and beta Birnbaum–Saunders [G.M. Cordeiro and A.J. Lemonte, The β-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling. Comput. Statist. Data Anal. 55 (2011), pp. 1445–1461] distributions and also the new Kumaraswamy–Birnbaum–Saunders distribution. We obtain the ordinary moments, generating function, mean deviations and quantile function. The method of maximum likelihood is used to estimate the model parameters and its potentiality is illustrated with an application to a real fatigue data set. Further, we propose a new extended regression model based on the logarithm of the McBS distribution. This model can be very useful to the analysis of real data and could give more realistic fits than other special regression models.     

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.     

A density-based empirical likelihood ratio goodness-of-fit test for the Rayleigh distribution and power comparison

The Rayleigh distribution has been used to model right skewed data. Rayleigh [On the resultant of a large number of vibrations of the some pitch and of arbitrary phase. Philos Mag. 1880;10:73–78] derived it from the amplitude of sound resulting from many important sources. In this paper, a new goodness-of-fit test for the Rayleigh distribution is proposed. This test is based on the empirical likelihood ratio methodology proposed by Vexler and Gurevich [Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy. Comput Stat Data Anal. 2010;54:531–545]. Consistency of the proposed test is derived. It is shown that the distribution of the proposed test does not depend on scale parameter. Critical values of the test statistic are computed, through a simulation study. A Monte Carlo study for the power of the proposed test is carried out under various alternatives. The performance of the test is compared with some well-known competing tests. Finally, an illustrative example is presented and analysed.     

A log-Birnbaum–Saunders regression model with asymmetric errors

This paper introduces a skewed log-Birnbaum–Saunders regression model based on the skewed sinh-normal distribution proposed by Leiva et al. [A skewed sinh-normal distribution and its properties and application to air pollution, Comm. Statist. Theory Methods 39 (2010), pp. 426–443]. Some influence methods, such as the local influence and generalized leverage, are presented. Additionally, we derived the normal curvatures of local influence under some perturbation schemes. An empirical application to a real data set is presented in order to illustrate the usefulness of the proposed model.     

On the effect of inducted negative correlation rate for beta acceptance–rejection algorithms

One of the variance reduction methods in simulation experiments is negative correlation induction, and in particular the use of the antithetic variates. The simultaneous use of antithetic variates and an acceptance–rejection method has been studied in some papers, where the inducted negative correlation has been calculated. In this study, the factors affecting the inducted negative correlation rate are addressed. To do this, the beta distribution is first selected to generate negatively correlated random variates using the acceptance–rejection method. The effects of both the efficiency of the acceptance–rejection method and the initial negative correlation rate on the inducted negative correlation are explored. Results show that both factors have significant effects; therefore, a combination of both can lead to algorithms better able to generate negative correlations.