A new lifetime model: the Kumaraswamy generalized Rayleigh distribution

The generalized Rayleigh (GR) distribution [V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, I, Appl. Math. 21 (1976), pp. 395–412; V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, II, Appl. Math. 21 (1976), pp. 413–419] has been applied in several areas such as health, agriculture, biology and other sciences. For the first time, we propose the Kumaraswamy GR (KwGR) distribution for analysing lifetime data. The new density function can be expressed as a mixture of GR density functions. Explicit formulae are derived for some of its statistical quantities. The density function of the order statistics can be expressed as a mixture of GR density functions. We also propose a linear log-KwGR regression model for analysing data with real support to extend some known regression models. The estimation of parameters is approached by maximum likelihood. The importance of the new models is illustrated in two real data sets.     

Mean residual weighted versus the length-biased Rayleigh distribution

In some statistical applications, data may not be considered as a random sample of the whole population and some subjects have less probability of belonging to the sample. Consequently, statistical inferences for such data sets, usually yields biased estimation. In such situations, the length-biased version of the original random variable as a special weighted distribution often produces better inferences. An alternative weighted distribution based on the mean residual life is suggested to treat the biasedness. The Rayleigh distribution is applied in many real applications, hence the proposed method of weighting is performed to produce a new lifetime distribution based on the Rayleigh model. In addition, statistical properties of the proposed distribution is investigated. A simulation study and a real data set are prepared to illustrate that the mean residual weighted Rayleigh distribution gives a better fit than the original and also the length-biased Rayleigh distribution.     

Modified slashed-Rayleigh distribution

A new family of slash distributions, the modified slashed-Rayleigh distribution, is proposed and studied. This family is an extension of the ordinary Rayleigh distribution, being more flexible in terms of distributional kurtosis. It arises as a quotient of two independent random variables, one being a Rayleigh distribution in the numerator and the other a power of the exponential distribution in denominator. We present properties of the proposed family. In addition, we carry out estimation of the model parameters by moment and maximum likelihood methods. Finally, we conduct a small-scale simulation study to evaluate the performance of the maximum likelihood estimators and apply the results to a real data set, revealing its good performance.     

The Rayleigh–Lindley model: properties and applications

In this paper, the Rayleigh–Lindley (RL) distribution is introduced, obtained by compounding the Rayleigh and Lindley discrete distributions, where the compounding procedure follows an approach similar to the one previously studied by Adamidis and Loukas in some other contexts. The resulting distribution is a two-parameter model, which is competitive with other parsimonious models such as the gamma and Weibull distributions. We study some properties of this new model such as the moments and the mean residual life. The estimation was approached via EM algorithm. The behavior of these estimators was studied in finite samples through a simulation study. Finally, we report two real data illustrations in order to show the performance of the proposed model versus other common two-parameter models in the literature. The main conclusion is that the model proposed can be a valid alternative to other competing models well established in the literature.     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

Efficient estimation of the PDF and the CDF of the inverse Rayleigh distribution

In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.     

Bayesian Prediction for Progressively Type-II Censored Data from the Rayleigh Model

ABSTRACT

The Rayleigh distribution is proposed to be the underlying model from which observables are to be predicted by using Bayesian approach. Progressively Type-II censored data from the Rayleigh distribution is considered and the two-sample prediction technique is used. Numerical computations and a simulation are given to illustrate the performance of the procedures.     

The inverted Nadarajah–Haghighi distribution: estimation methods and applications

The inverted (or inverse) distributions are sometimes very useful to explore additional properties of the phenomenons which non-inverted distributions cannot. We introduce a new inverted model called the inverted Nadarajah–Haghighi distribution which exhibits decreasing and unimodal (right-skewed) density while the hazard rate shapes are decreasing and upside-down bathtub. Our main focus is the estimation (from both frequentist and Bayesian points of view) of the unknown parameters along with some mathematical properties of the new model. The Bayes estimators and the associated credible intervals are obtained using Markov Chain Monte Carlo techniques under squared error loss function. The gamma priors are adopted for both scale and shape parameters. The potentiality of the distribution is analysed by means of two real data sets. In fact, it is found to be superior in its ability to sufficiently model the data as compared to the inverted Weibull, inverted Rayleigh, inverted exponential, inverted gamma, inverted Lindley and inverted power Lindley models.     

On the comparison of the Fisher information of the log-normal and generalized Rayleigh distributions

Surles and Padgett recently considered two-parameter Burr Type X distribution by introducing a scale parameter and called it the generalized Rayleigh distribution. It is observed that the generalized Rayleigh and log-normal distributions have many common properties and both distributions can be used quite effectively to analyze skewed data set. In this paper, we mainly compare the Fisher information matrices of the two distributions for complete and censored observations. Although, both distributions may provide similar data fit and are quite similar in nature in many aspects, the corresponding Fisher information matrices can be quite different. We compute the total information measures of the two distributions for different parameter ranges and also compare the loss of information due to censoring. Real data analysis has been performed for illustrative purposes.     

Linear regression model with new symmetric distributed errors

Regression models play a dominant role in analyzing several data sets arising from areas like agricultural experiment, space experiment, biological experiment, financial modeling, etc. One of the major strings in developing the regression models is the assumption of the distribution of the error terms. It is customary to consider that the error terms follow the Gaussian distribution. However, there are some drawbacks of Gaussian errors such as the distribution being mesokurtic having kurtosis three. In many practical situations the variables under study may not be having mesokurtic but they are platykurtic. Hence, to analyze these sorts of platykurtic variables, a two-variable regression model with new symmetric distributed errors is developed and analyzed. The maximum likelihood (ML) estimators of the model parameters are derived. The properties of the ML estimators with respect to the new symmetrically distributed errors are also discussed. A simulation study is carried out to compare the proposed model with that of Gaussian errors and found that the proposed model performs better when the variables are platykurtic. Some applications of the developed model are also pointed out.     

The beta generalized Rayleigh distribution with applications to lifetime data

For the first time, we propose a new distribution so-called the beta generalized Rayleigh distribution that contains as special sub-models some well-known distributions. Expansions for the cumulative distribution and density functions are derived. We obtain explicit expressions for the moments, moment generating function, mean deviations, Bonferroni and Lorenz curves and densities of the order statistics and their moments. We estimate the parameters by maximum likelihood and provide the observed information matrix. The usefulness of the new distribution is illustrated through two real data sets that show that it is quite flexible in analyzing positive data instead of the generalized Rayleigh and Rayleigh distributions.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.     

Heavy or semi-heavy tail,that is the question

While there has been considerable research on the analysis of extreme values and outliers by using heavy-tailed distributions, little is known about the semi-heavy-tailed behaviors of data when there are a few suspicious outliers. To address the situation where data are skewed possessing semi-heavy tails, we introduce two new skewed distribution families of the hyperbolic secant with exciting properties. We extend the semi-heavy-tailedness property of data to a linear regression model. In particular, we investigate the asymptotic properties of the ML estimators of the regression parameters when the error term has a semi-heavy-tailed distribution. We conduct simulation studies comparing the ML estimators of the regression parameters under various assumptions for the distribution of the error term. We also provide three real examples to show the priority of the semi-heavy-tailedness of the error term comparing to heavy-tailedness. Online supplementary materials for this article are available. All the new proposed models in this work are implemented by the shs R package, which can be found on the GitHub webpage.     

Discriminating between the generalized Rayleigh and Weibull distributions: Some comparative studies

The generalized Rayleigh distribution was introduced and studied quite effectively in the literature. The closeness and separation between the distributions are extremely important for analyzing any lifetime data. In this spirit, both the generalized Rayleigh and Weibull distributions can be used for analyzing skewed datasets. In this article, we compare these two distributions based on the Fisher information measures and use it for discrimination purposes. It is evident that the Fisher information measures play an important role in separating between the distributions. The total information measures and the variances of the different percentile estimators are computed and presented. A real life dataset is analyzed for illustration purposes and a numerical comparison study is performed to assess our procedures in separating between these two distributions.     

The Kumaraswamy generalized gamma distribution with application in survival analysis

We introduce and study the so-called Kumaraswamy generalized gamma distribution that is capable of modeling bathtub-shaped hazard rate functions. The beauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a large number of well-known lifetime special sub-models such as the exponentiated generalized gamma, exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma, generalized Rayleigh, among others. Some structural properties of the new distribution are studied. We obtain two infinite sum representations for the moments and an expansion for the generating function. We calculate the density function of the order statistics and an expansion for their moments. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. The usefulness of the new distribution is illustrated in two real data sets.     

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

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

The exponentiated-log-logistic geometric distribution: Dual activation

ABSTRACT

The log-logistic distribution is commonly used to model lifetime data. We propose a wider distribution, named the exponentiated log-logistic geometric distribution, based on a double activation approach. We obtain the quantile function, ordinary moments, and generating function. The method of maximum likelihood is used to estimate the model parameters. We propose a new extended regression model based on the logarithm of the exponentiated log-logistic geometric distribution. This regression model can be very useful in the analysis of real data and could provide better fits than other special regression models. The potentiality of the new models is illustrated by means of two applications to real lifetime data sets.     

Modified slashed generalized exponential distribution

Abstract

In this article, we introduce a new distribution for modeling positive data sets with high kurtosis, the modified slashed generalized exponential distribution. The new model can be seen as a modified version of the slashed generalized exponential distribution. It arises as a quotient of two independent random variables, one being a generalized exponential distribution in the numerator and a power of the exponential distribution in the denominator. We studied various structural properties (such as the stochastic representation, density function, hazard rate function and moments) and discuss moment and maximum likelihood estimating approaches. Two real data sets are considered in which the utility of the new model in the analysis with high kurtosis is illustrated.     

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.