Beta Exponential Distributions Research Articles

Modelling Service Times Using Some Beta-Based Compound Distribution

The design of the queueing model involves modelling the arrival and service processes of the system. Conventionally, the arrival process is assumed to follow Poisson while service times are assumed to be exponentially distributed. Other distributions such as Weibull, uniform, lognormal have been used to model service times however, generalized distributions have not been used in this regard. In recent times, attention have been shifted to generalised families of distributions including Beta generalized family of distributions which led to the development of Beta-based distributions. Distributions generated from a mixture of beta random variables are quite numerous in literature with little or no application to service times data. In this study, six Beta-based compound distributions - Beta-Log-logistic distribution (BLlogD), Beta-Weibull distribution (BWeiD), Beta-Lomax distribution (BLomD), Beta-exponential distribution (BExD), Beta-Gompertz distribution (BGomD) and Beta-log-normal distribution (BLnormD) - were compared with the classical service times model on four service times data sets. Maximum likelihood estimator was employed in estimating parameters of the selected models while Akaike Information Criteria (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC) and Hannan Quin information criterion (HQIC) statistics were employed to select the best model. CDFs, PDFs and PP-plots were used to fit the data of the suggested models. Results from the study shows that Beta-Exponential distribution (BExpD) performed better for the datasets I (AIC=640.3, CAIC=640.5,BIC=648.1 and HQIC=643.4), Beta-weibull distribution (BWeiD) performed better for the data sets II and III (AIC=204.2, 2142.4,CAIC=204.2, 2142.7,BIC=212.8, 2154.9 and HQIC=207.6, 2147.5) while Beta-log-logistic distribution (BLlogD) performed better for the data sets IV (AIC=2275.3,CAIC=2275.5, BIC=2289.3 and HQIC=2280.9). Findings from this study revealed some useful Beta-based compound distributions which performed better than the conventional service time model. From the findings of this study, we recommend researchers to dive deep into queueing theory.

Normal-beta exponential stochastic frontier model: Maximum simulated likelihood approach

This paper considers the beta exponential distribution as a distribution function of inefficacy score in a stochastic frontier model. The beta exponential distribution is a three-parameter distribution, and it is more flexible than commonly used probability density functions in a stochastic frontier model (SFM). This new model, a “Normal-Beta Exponential SFM”, nests another five SFMs. This paper presents a simulated log-likelihood function and simulated inefficiency estimator of a normal-beta exponential SFM, a closed form log-likelihood function and closed form inefficiency estimator of a normal-weighted exponential SFM, and an empirical study using a normal-beta exponential SFM. In our empirical study, we have used a likelihood ratio test to compare the performance of SFMs and a normal-beta exponential SFM fits the data better than other nested special case SFMs. Furthermore, the empirical result shows that parameters of a normal-beta exponential SFM can be estimated with less standard error or high certainty than a normal-gamma SFM.

A shape space for generalized beta exponential distributions of the second type

The generalized beta exponential distribution of the second type contains subfamilies of logistic distributions, that are often used to control time arrays in complex systems. Currently, the choice of a model from possible implementations of a generalized family remains a rather time-consuming task. For this reason, it is often necessary to use complex mixtures of models. Displaying subfamilies of a generalized exponential distribution of the second type in the space of parametric and information features to choose the simplest form of the model from subfamilies and to evaluate the shape parameters is enabled. In the space of standardized moments of the third and fourth order the trajectories position of possible realizations limits the upper limit of distinguish-ability of the shape parameters of distribution models. A significant expansion of the distinguish-ability of trajectories it is possible to achieved by using the entropy coefficient of non-shifted distributions as an additional independent feature of the distribution shape. Mapping subfamilies of distributions in the space of entropy coefficient, antikurtosis and skewness one also to analyse the most important properties of distributions as well is allows. In article it is shows that for all models of the generalized distribution of the second type, the antikurtosis is in the range between the exponential and normal distributions.

A shape mapping for generalized beta exponential distribution of the first kind in the entropy-parametric space

Modern methods of tracking and controlling complex systems are associated with the analysis of the shape of statistical models of sample arrays. Often the values of the monitored parameters are maintained only on a semi-infinite range of values. In these cases, it is possible to use generalized beta exponential distribution models of the first kind to smooth the statistics, that it is including such models as exponential and generalized Gompertz-Verhulst distributions as special cases. The distributions with a large set of subfamilies rarely are use due to lack of methods of preliminary estimation of the distribution shape parameters. The paper illustrates that the displayed of distributions in the parametric space of skewness and kurtosis does not allow distinguishing the features of the position of the main subfamilies of distributions. Mapping the subfamilies of the Generalized Beta Exponential Distribution of the first kind in the Entropy-Parametric space makes it possible to distinguish between subfamilies and to comparative analysis of many their properties. It is convenient to use the space of the entropy coefficient and skewness when you are comparing the skew properties of distributions. The space of the entropy coefficient and antikurtosis is more suitable for comparing the weights of distribution tails and for analysing monotonicity. In particular, it is shown that the Generalized Beta Exponential Distribution of the first kind contains as monotonic and as non-monotonic leptokurtic distributions. The property of monotonicity is well discernible when compared with the antikurtosis of the exponential distribution.

Bayesian Inference for Negative Binomial—Beta Exponential Distribution and Its Regression Model

Bayesian Inference for Negative Binomial—Beta Exponential Distribution and Its Regression Model

A new statistical image watermark detector in RHFMs domain using beta-exponential distribution

The detection of watermarks can be achieved by statistical approaches. How to select robust modeling object, appropriate statistical model, and decision rules is one of the major issues in statistical image watermark detection. In this paper, we propose a new image watermark detector in robust fast radial harmonic Fourier moments (FRHFMs) magnitudes domain, wherein the Beta-exponential distribution model and locally most powerful (LMP) decision rule are used. We first investigate the statistical modeling of the robust FRHFMs magnitudes by the Beta-exponential distribution. It is shown that the Beta-exponential distribution model fits the empirical data more accurately than the formerly employed statistical distributions, such as the Cauchy, Weibull, BKF, and Exponential, do. Motivated by the statistical modeling results, we design a blind image watermark detector in FRHFMs magnitudes domain by using Beta-exponential distribution and LMP test. Also, we utilize the Beta-exponential model to derive the closed-form expressions for the watermark detector. We provide comparative experimental results to alternative approaches to demonstrate the advantages of the proposed image watermark detector.

Shape measures for the generalized beta exponential distribution

This paper contains parametric and informational shape measures for various families of the generalized beta exponential distribution since it is important to determination of the distribution shape for analysing an experimental data set. A logistic parameter is used to select independent types of beta exponential distributions, that it allows to combine the distributions of different subfamilies. In this paper the use of parametric shape measures to pre-define distribution shape is discusses. In particular, the initial and standard central moments for the main types of generalized beta exponential distribution are given. In the paper it is proposes to use the entropy coefficient of unshifted distribution as an independent information measure of the shape of unshifted generalized beta exponential distributions. In order to increase the reliability of the preliminary determination of the shape of the model, expressions for the entropy coefficient of shifted families both the generalized beta exponential distributions of the first and second types, and the generalized gamma exponential distribution were obtained. For practical applied the entropy coefficients of unshifted distributions for various subfamilies of generalized beta exponential distributions can be useful.

Measures of shape for the generalized beta exponential distribution

This paper contains parametric and informational measures of shape for various families of the generalized beta exponential distribution since it is important to determination of the distribution shape for analysing an experimental data set. A logistic parameter is used to select independent types of beta exponential distributions, that it allows to combine the distributions of different subfamilies. In this paper the use of parametric shape measures to predefine distribution shape is discusses. In particular, the initial and standard central moments for the main types of generalized beta exponential distribution are given. In the paper it is proposes to use the entropy coefficient of unshifted distribution as an independent information measure of the shape of unshifted generalized beta exponential distributions. In order to increase the reliability of the preliminary determination of the shape of the model, expressions for the entropy coefficient of shifted families both the generalized beta exponential distributions of the first and second types, and the generalized gamma exponential distribution were obtained. For practical applied the entropy coefficients of unshifted distributions for various subfamilies of generalized beta exponential distributions can be useful.

Statistical image watermarking using local RHFMs magnitudes and beta exponential distribution

The imperceptibility, robustness and data payload are widely considered as the three main properties vital for any image watermarking systems. They are complimentary to each other and hence challenging to attain the right balance between them. The statistical model-based multiplicative watermarking is an effective way to achieve the tradeoff among imperceptibility, robustness and data payload. Radial harmonic Fourier moments (RHFMs) is a strong tool in image processing, which has many advantages, such as lower noise sensitivity, powerful image description ability and geometric invariance feature. In this paper, we propose a new statistical image watermarking scheme using local RHFMs magnitudes and Beta exponential distribution model. Our image watermarking scheme consists of two parts, namely, embedding and extraction. In the embedding process, we divide the host image into no-overlapping blocks and compute the local RHFMs of image blocks, then insert the watermark signal into the robust local RHFMs magnitudes through multiplicative approach. In the extraction phase, robust local RHFMs magnitudes are firstly modeled by employing the Beta exponential distribution, where the statistical properties of local RHFMs magnitudes are captured accurately. Then the modified maximum likelihood parameter estimation (MMLE) approach is introduced to estimate the statistical parameters of Beta exponential distribution model. And finally an image watermark decoder for multiplicative watermarking is developed using Beta exponential distribution and maximum likelihood decision criterion. Experimental results on some test images and comparison with well-known existing methods demonstrate the efficacy and superiority of the proposed statistical image watermarking.

Zero-one inflated negative binomial - beta exponential distribution for count data with many zeros and ones

The characteristic of count data that have a high frequency of zeros and ones can be considered under a zero-one inflated distribution. In this article, we present a zero-one inflated negative binomial - beta exponential distribution to analyze for such data. This distribution shows that it is extended the mixture negative binomial with beta exponential distributions, was proposed by Pudprommarat, Bodhisuwan, and Zeephongsekul (2012). Some important properties of this distribution are discussed, which include probability mass function, moment generating function, moment about the origin, mean and variance. Additionally, some sub-models are presented. Its parameters are also derived based on the maximum likelihood estimation procedure. The applicability of the proposed distribution is demonstrated for fitting to three real data sets. We also evaluate the abilities of model selection relying on the negative log-likelihood, Akaike information criterion, mean absolute error, root mean squared error, discrete Kolmogorov–Smirnov test and Anderson-Darling tests. Results from this study indicate the zero-one inflated negative binomial - beta exponential distribution has shown the best fit for these data sets when it is compared with some sub-models and the zero-one inflated of traditional distributions.

On the Hurwitz zeta function with an application to the beta-exponential distribution

We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the beta-exponential distribution.

A New Class of Beta-Complementary Exponential Power Series Distributions

Abstract In this paper, we introduce a new class of beta-complementary exponential power series distributions, which is obtained by replacing the cumulative distribution function of the complementary exponential power series distributions in the logit of a beta random variable. This new class of distributions contains some new submodels, such as beta-complementary exponential geometric, beta-complementary exponential Poisson, beta-complementary exponential binomial, beta-complementary exponential logarithmic, and complementary exponential power series distributions. A general class of distributions is presented and some various properties are obtained in this paper. We characterize the beta-complementary exponential geometric distribution as one of the most applicable distributions in this class. Some mathematical properties of the beta-complementary exponential geometric distribution are reached in terms of the corresponding properties of the complementary exponential geometric and beta exponential distributions. We present expressions for the probability density function, cumulative distribution function, moment generating function, and moments. The estimation of parameters is approached by the maximum likelihood estimation procedure, and the expected information matrix is derived. The flexibility of the distribution is illustrated in application of a real data set.

The beta generalized Gompertz distribution

• A new continuous model called the beta generalized Gompertz distribution is introduced. • Some mathematical properties of the new model are studied. • The model parameters are estimated by maximum likelihood. • The usefulness of the new model is illustrated in an application to real data. A new five-parameter continuous model called the beta generalized Gompertz distribution is introduced and studied. This distribution contains the Gompertz, generalized Gompertz, beta Gompertz, generalized exponential, beta generalized exponential, exponential and beta exponential distributions as special sub-models. Some mathematical properties of the new model are derived. We show that the density function of the new distribution can be expressed as a linear combination of Gompertz densities. We obtain explicit expressions for the moments, moment generating function, quantile function, density function of the order statistics and their moments, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. The model parameters are estimated by using the maximum likelihood method of estimation and the observed information matrix is determined. Finally, an application to real data set is given to illustrate the usefulness of the proposed model.

Discrete Beta-Exponential Distribution

There are not many known distributions for modeling discrete data. In this paper, we shall introduce a discrete analogue of the beta-exponential distribution of Nadarajah and Kotz (2006), which is more plausible in modeling discrete data and exhibits both increasing and decreasing hazard rates. The discrete beta-exponential distribution can be viewed as a generalization of the discrete generalized exponential distribution introduced by Nekoukhou et al. (2012) and, thus, as an another generalization of the geometric distribution. We shall first study some basic distributional and moment properties of the new distribution. Then, certain structural properties of the distribution such as its unimodality, hazard rate behavior, and Rényi entropy are discussed. Using the maximum likelihood method, estimation of the model parameters is also investigated. Finally, the model is examined with a real data set and compared with its rival model, that is, the discrete generalized exponential distribution.

Libby and Novick's generalized beta exponential distribution

A new family of skewed distributions is presented. Some properties and estimation procedures for Libby and Novick's generalized beta exponential distribution, a particular member of the family, are derived. Real applications using two original data sets are described to show superior performance versus at least six known models.

The beta-Weibull geometric distribution

We propose a new distribution, the so-called beta-Weibull geometric distribution, whose failure rate function can be decreasing, increasing or an upside-down bathtub. This distribution contains special sub-models the exponential geometric [K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35–42], beta exponential [S. Nadarajah and S. Kotz, The exponentiated type distributions, Acta Appl. Math. 92 (2006), pp. 97–111; The beta exponential distribution, Reliab. Eng. Syst. Saf. 91 (2006), pp. 689–697], Weibull geometric [W. Barreto-Souza, A.L. de Morais, and G.M. Cordeiro, The Weibull-geometric distribution, J. Stat. Comput. Simul. 81 (2011), pp. 645–657], generalized exponential geometric [R.B. Silva, W. Barreto-Souza, and G.M. Cordeiro, A new distribution with decreasing, increasing and upside-down bathtub failure rate, Comput. Statist. Data Anal. 54 (2010), pp. 935–944; G.O. Silva, E.M.M. Ortega, and G.M. Cordeiro, The beta modified Weibull distribution, Lifetime Data Anal. 16 (2010), pp. 409–430] and beta Weibull [S. Nadarajah, G.M. Cordeiro, and E.M.M. Ortega, General results for the Kumaraswamy-G distribution, J. Stat. Comput. Simul. (2011). DOI: 10.1080/00949655.2011.562504] distributions, among others. The density function can be expressed as a mixture of Weibull density functions. We derive expansions for the moments, generating function, mean deviations and Rénvy entropy. The parameters of the proposed model are estimated by maximum likelihood. The model fitting using envelops was conducted. The proposed distribution gives a good fit to the ozone level data in New York.

General results for the beta Weibull distribution

In this paper, we study some mathematical properties of the beta Weibull (BW) distribution, which is a quite flexible model in analysing positive data. It contains the Weibull, exponentiated exponential, exponentiated Weibull and beta exponential distributions as special sub-models. We demonstrate that the BW density can be expressed as a mixture of Weibull densities. We provide their moments and two closed-form expressions for their moment-generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and two entropies. The density of the BW-order statistics is a mixture of Weibull densities and two closed-form expressions are derived for their moments. The estimation of the parameters is approached by two methods: moments and maximum likelihood. We compare the performances of the estimates obtained from both the methods by simulation. The expected information matrix is derived. For the first time, we introduce a log-BW regression model to analyse censored data. The usefulness of the BW distribution is illustrated in the analysis of three real data sets.

Reliability Characteristics of $k$-out-of-$n$ Warm Standby Systems

We study reliability characteristics of the k-out-of- n warm standby system with identical components subject to exponential lifetime distributions. We derive state probabilities of the warm standby system in a form that is similar to the state probabilities of the active redundancy system. Subsequently, the system reliability is expressed in several forms that can provide new insights into the system reliability characteristics. We also show that all properties and computational procedures that are applicable for active redundancy are also applicable for the warm standby redundancy. As a result, it is shown that the system reliability can be evaluated using robust algorithms within O(n-k+1) computational time. In addition, we provide closed-form expressions for the hazard rate, probability density function, and mean residual life function. We show that the time-to-failure distribution of the k-out-of-n warm standby system is equal to the beta exponential distribution. Subsequently, we derive closed-form expressions for the higher order moments of the system failure time. Further, we show that the reliability of the warm standby system can be calculated using well-established numerical procedures that are available for the beta distribution. We prove that the improvement in system reliability with an additional redundant component follows a negative binomial (Polya) distribution, and it is log-concave in n. Similarly, we prove that the system reliability function is log-concave in n. Because the k -out-of-n system with active redundancy can be considered as a special case of the k-out-of-n warm standby system, we indirectly provide some new results for the active redundancy case as well.

A New Mixed Negative Binomial Distribution

A negative binomial-beta exponential distribution is a new mixed negative binomial distribution obtained by mixing the negative binomial distribution with a beta exponential distribution. The generalized Waring and Waring and Yule distributions are presented as special cases of this negative binomial-beta exponential distribution. Various structural properties of the new distribution are derived, including expansions for its factorial moments, moments of the order statistics and so forth. We discuss maximum likelihood estimation method for estimating parameters of this distribution. The usefulness of the new distribution is illustrated through a real count data.

The McDonald extended distribution: properties and applications

We study a five-parameter lifetime distribution called the McDonald extended exponential model to generalize the exponential, generalized exponential, Kumaraswamy exponential and beta exponential distributions, among others. We obtain explicit expressions for the moments and incomplete moments, quantile and generating functions, mean deviations, Bonferroni and Lorenz curves and Gini concentration index. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. The applicability of the new model is illustrated by means of a real data set.