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The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation for some properties and an analytical study of its bimodality. The hazard rate function and the limiting behavior are examined. We derive explicit expressions for moments, generating function, mean deviations using a power series expansion for the quantile function, and Shannon entropy., 18 pages, 9 figures

A new extension of the generalized gamma distribution with six- parameter called the Kummer beta generalized gamma distribution is introduced and studied. It contains at least 28 special models such as the beta generalized gamma, beta Weibull, beta exponential, generalized gamma, Weibull and gamma distributions and thus could be a better model for analyzing positive skewed data. The new density function can be expressed as a linear combination of generalized gamma densities. Various mathematical properties of the new distribution including explicit expressions for the ordinary and incomplete moments, generating function, mean deviations, entropy, density function of the order statistics and their moments are derived. The elements of the observed information matrix are provided. We discuss the method of maximum likelihood and a Bayesian approach to fit the model parameters. The superiority of the new model is illustrated by means of three real data sets.

Research background: In applied welfare economics, the constant relative inequality aversion function is routinely used as the model of a social decisionmaker’s or a society’s preferences over income distributions. This function is entirely determined by the parameter, ε, of inequality aversion. However, there is no authoritative answer to the question of what the range of ε an analyst should select for empirical work. Purpose of the article: The aim of this paper is elaborating the method of deriving ε from a parametric distribution of disposable incomes. Methods: We assume that households’ disposable incomes obey the generalised beta distribution of the second kind GB2(a,b,p,q). We have proved that, under this assumption, the social welfare function exists if and only if ε belongs to (0,ap+1) interval. The midpoint εmid of this interval specifies the inequality aversion of the median social-decisionmaker. Findings & Value added: The maximum likelihood estimator of εmid has been developed. Inequality aversion for Poland 1998–2015 has been estimated. If inequality is calculated on the basis of disposable incomes, the standard inequality–development relationship might be complemented by inequality aversion. The “augmented” inequality–development relationship reveals new phenomena; for instance, the stage of economic development might matter when assessing the impact of inequality aversion on income inequality.

The probability distributions whether discrete or continuous can encounter barriers when we talk about data that contains "zeros in excess" LORD et al. (2005). In order for these distributions to measure such data, mixed distributions called inflated distributions are created. The way in which this inflation will occur will depend of the distribution support set . In this research, we propose the construction of the zero inflated beta prime distribution (BPIZ) from the reparametrization presented in BOURGUIGNON et al. (2018). We also determined maximum likelihood estimators and confidence intervals for the model BPIZ. We evaluate numerically estimators and confidence intervals where we check its asymptotic eficiency. Finally, we performed an application to check also its applicability in relation to real data. Nenhuma As distribuições de probabilidade, sejam elas discretas ou contínuas, podem encontrar barreiras quando falamos de dados que contenham "zeros em excessos" LORD et al. (2005). Para que essas distribuições possam mensurar tais dados, criam-se distribuições mistas chamadas de distribuições inflacionadas. A forma como esse in acionamento vai ocorrer dependerá do conjunto suporte da distribui- ção. Nesta pesquisa, propomos a construção da distribuição beta prime in acionada em zero (BPIZ) a partir da reparametrização apresentada em BOURGUIGNON et al. (2018). Determinamos também estimadores de máxima verossimilhança e intervalos de confiança para o modelo BPIZ. Avaliamos numericamente os estimadores e os intervalos de confiança, onde verificamos sua eficiência assintótica. Por fim, realizamos uma aplicação para verificar, também, sua aplicabilidade em relação à dados reais.

We introduce the free analogue of the classical beta prime distribution by the multiplicative free convolution of the free Poisson and the reciprocal of free Poisson distributions, and related free analogues of the classical $F$, $T$, and beta distributions. We show the rationales of our free analogues via the score functions and the potentials. We calculate the moments of the free beta prime distribution explicitly in combinatorial by using non-crossing linked partitions, and demonstrate that the free beta prime distribution belongs to the class of the free negative binomials in the free Meixner family., 36 pages

This paper proposes a Bayesian predictive density estimator of time to goal in a hockey game, using ancillary information such as performance in the past, points, and specialists’ opinions about teams. To be more specific, we model time to r-th goal as a gamma distribution. The proposed Bayesian predictive density estimator using the ancillary information belongs to an interesting new version of a weighted beta prime distribution and it outperforms the other estimators in the literature such as the one that does not incorporate this information as well as the plug-in estimator. The efficiency of our estimator is evaluated using frequentist risk along with measuring the prediction error from the old dataset, 2016–2017, to the season 2018–2019 of the National Hockey League.

In this paper, we propose a regression model where the response variable is beta prime distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The proposed regression model is useful for situations where the variable of interest is continuous and restricted to the positive real line and is related to other variables through the mean and precision parameters. The variance function of the proposed model has a quadratic form. In addition, the beta prime model has properties that its competitor distributions of the exponential family do not have. Estimation is performed by maximum likelihood. Furthermore, we discuss residuals and influence diagnostic tools. Finally, we also carry out an application to real data that demonstrates the usefulness of the proposed model.

This paper proposes a general but tractable approach to price volatility derivatives which includes most existing stochastic volatility models as special cases. Specifically, we use the generalized beta prime distribution to model the dynamics of the volatility underlying, which is flexible enough to capture the empirical marginal density function and yield desired implied volatility slope consistent with empirical data. Based on that, we use a Jacobi process to add a diffusion component, and further extend our model through stochastic time changes to capture jumps as well as to accommodate the term structure of volatility index futures. Our approach is supported by calibrations.

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 Renyi 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.

In this paper, we compare the performance of dynamic conditional score (DCS) and standard financial time-series models for Central American energy prices. We extend the Student’s t and the exponential generalised beta distribution of the second kind stochastic location and stochastic seasonal DCS models. We consider the generalised t distribution as an alternative for the error term and also consider dynamic specifications of volatility. We use a unique dataset of spot electricity prices for El Salvador, Guatemala and Panama. We consider two data windows for each country, which are defined with respect to the liberalisation and development process of the energy market in Central America. We study the identification of a wide range of DCS specifications, likelihood-based model performance, time-series components of energy prices, maximum likelihood parameter estimates, the discounting property of conditional score, and out-of-sample forecast performance. Our main results are the following. (i) We determine the most robust models of energy prices, with respect to parameter identification, from a wide range of DCS specifications. (ii) For most of the cases, the in-sample statistical performance of DCS is superior to that of the standard model. (iii) For El Salvador and Panama, the standard model provides better point forecasts than DCS, and for Guatemala the point forecast precision of standard and DCS models does not differ significantly. (iv) For El Salvador, the standard model provides better density forecasts than DCS, and for Guatemala and Panama, the density forecast precision of standard and DCS models does not differ significantly.

The wide range of values observed in a measured concentration time series after the release of a dispersing airborne pollutant from a point source in the atmospheric boundary layer, and the hazard level associated with the peak values, demonstrate the necessity of predicting the concentration probability distribution. For this, statistical models describing the probability of occurrence are preferably employed. In this paper a concentration database pertaining to a field experiment of dispersion in an urban-like area (MUST experiment) from a continuously emitting source is used for the selection of the best performing statistical model between the Gamma and the Beta distributions. The skewness, the kurtosis as well as the inverses of the cumulative distribution function were compared between the two statistical models and the experiment. The evaluation is performed in the form of validation metrics such as the Fractional Bias (FB), the Normalized Mean Square Error and the factor-of-2 percentage. The Beta probability distribution agreed with the experimental results better than the Gamma probability distribution except for the 25th percentile. Also according to the significant tests using the BOOT software the Beta model presented FB and NMSE values that are statistical different than the ones of the Gamma model except the 75th percentiles and the FB of the 99th percentiles. The effect of the stability conditions and source heights on the performance of the statistical models is also examined. For both cases the performance of the Beta distribution was slightly better than that of the Gamma.

We provide an analytically treatable model that describes in a unified manner income distribution for all income categories. The approach is based on a master equation with growth and reset terms. The model assumptions on the growth and reset rates are tested on an exhaustive database with incomes on individual level spanning a nine year period in the Cluj county (Romania). In agreement with our theoretical predictions we find that income distributions computed for several years collapse on a master-curve when a properly normalised income is considered. The Beta Prime distribution is appropriate to fit the collapsed data and it is shown that distributions derived for other countries are following similar trends with different fit parameters. The non-universal feature of the fit parameters suggests that for a more realistic modelling the model parameters have to be linked with specific socio-economic regulations., 10 pages, 6 Figures

We present a destructive testing procedure based on absorption cross section (ACS) measurements defined by the recorded power in a reverberation chamber (RC). Destructive testing of light-emitting diodes (LED) are performed at frequencies with maximum absorption cross section ratio (ACSR) and at frequencies with lower ACSR. The failure outcome scales with the expected maximum value distribution as a function of the number of uncorrelated samples in the RC. The required power to break the diodes at lower ACSR is consistent with the theory. The statistical distribution of the composite ACSR is modelled with a generalized beta prime distribution.

Let $X_1,\ldots,X_N$, $N>n$, be independent random points in $\mathbb{R}^n$, distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension $n$ tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated., 26 pages

We study distributions of realized variance (squared realized volatility) and squared implied volatility, as represented by VIX and VXO indices. We find that Generalized Beta distribution provide the best fits. These fits are much more accurate for realized variance than for squared VIX and VXO -- possibly another indicator that the latter have deficiencies in predicting the former. We also show that there are noticeable differences between the distributions of the 1970-2017 realized variance and its 1990-2017 portion, for which VIX and VXO became available. This may be indicative of a feedback effect that implied volatility has on realized volatility. We also discuss the distribution of the difference between squared implied volatility and realized variance and show that, at the basic level, it is consistent with Pearson's correlations obtained from linear regression., Comment: 16 pages, 11 figures, 10 tables

The Weibull distribution is commonly used to model wind speed data, mainly due to its good fit to asymmetric positive variables. Several proposals have extended this approach to accommodate realistic features of wind data such as nonstationary behavior due to changes in atmospheric regimes. The present work considers wind speed modeling over time through the dynamic Weibull and Gamma state space models. Properties of both models are presented and filtering, smoothing and prediction equations are analytically obtained. Efficient simulation of scenarios is obtained through the beta prime distribution, which allows fast online forecasts of wind speed. The models are compared regarding fit and predictive performance for the analysis of two wind speed datasets in different regions of Brazil. Results indicate that the dynamic Gamma model is competitive with the Weibull model for wind prediction.

In this paper, a new five-parameter lifetime distribution called beta generalized linear exponential distribution (BGLED) is introduced. It includes at least 17 popular sub-models as special cases such as the beta linear exponential, the beta generalized exponential, and the exponentiated generalized linear distributions. Mathematical and statistical properties of the proposed distribution are discussed in details. In particular, explicit expression for the density function, moments, asymptotics distributions for sample extreme statistics, and other statistical measures are obtained. The estimation of the parameters by the method of maximum-likelihood is discussed and the finite sample properties of the maximum-likelihood estimators (MLEs) are investigated numerically. A real data set is used to demonstrate its superior performance fit over several existing popular lifetime models.

We study high-dimensional Bayesian linear regression with a general beta prime distribution for the scale parameter. Under the assumption of sparsity, we show that appropriate selection of the hyperparameters in the beta prime prior leads to the (near) minimax posterior contraction rate when $p \gg n$. For finite samples, we propose a data-adaptive method for estimating the hyperparameters based on marginal maximum likelihood (MML). This enables our prior to adapt to both sparse and dense settings, and under our proposed empirical Bayes procedure, the MML estimates are never at risk of collapsing to zero. We derive efficient Monte Carlo EM and variational EM algorithms for implementing our model, which are available in the R package NormalBetaPrime. Simulations and analysis of a gene expression data set illustrate our model's self-adaptivity to varying levels of sparsity and signal strengths., 37 pages, 4 figures, 3 tables. We have added a section on posterior computation and corrected the theoretical results. Sections on normal means estimation were removed in this updated technical report

In studies on the application of machine learning such as Information Retrieval (IR), the focus is typically on the estimation of the $F_1$ measure of algorithm performance. Approximate symmetrical confidence intervals constructed by the $F_1$ value based on cross-validated $t$ distribution are commonly used in the literature. However, theoretical analysis on the distribution of $F_1$ values shows that such distribution is actually non-symmetrical. Thus, simply using symmetrical distribution to approximate non-symmetrical distribution may be inappropriate and may result in a low degree of confidence and long interval length for the confidence interval. In the present study, a non-symmetrical confidence interval of the $F_1$ measure based on Beta prime distribution is constructed by using the $F_1$ value computed based on the average confusion matrix of a blocked $3\times2$ cross-validation. Experimental results show that in most cases, our method has high degrees of confidence. With an acceptable degree of confidence, our method has a shorter interval length than the approximate symmetrical confidence intervals based on the blocked $3\times 2$ and $5 \times 2$ cross-validated $t$ distributions. The approximate symmetrical confidence interval based on the $10$ -fold cross-validated $t$ distribution has the shortest interval length of the four confidence intervals but with low degrees of confidence in all cases. Taking these two factors into consideration, our method is recommended.

The beta distribution is a basic distribution serving several purposes. It is used to model data, and also, as a more flexible version of the uniform distribution, it serves as a prior distribution for a binomial probability. The bivariate beta distribution plays a similar role for two probabilities that have a bivariate binomial distribution. We provide a new multivariate distribution with beta marginal distributions, positive probability over the unit square, and correlations over the full range. We discuss its extension to three or more dimensions., Comment: 10 pages, 1 table, 1 figure

In this study, the difference between the mean of the Rayleigh Probability Density Function and the mean obtained from the graph of Rayleigh Probability Density Function is minimized by changing the coefficient in the equation yielding the mean. By using various numbers of data and K values, Rayleigh Probability Density Function is plotted with the means mentioned above.

The gamma and beta functions have been generalized in several ways. The multivariate beta and multivariate gamma functions due to Ingham and Siegel have been defined as integrals having the integrand as a scalar function of the real symmetric matrix. In this article, we define extended matrix variate gamma and extended matrix variate beta functions thereby generalizing multivariate gamma and multivariate beta functions defined by Ingham and Siegel. We study a number of properties of these newly defined functions. We also give some applications of these functions to statistical distribution theory.

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.

The inverse Weibull distribution is one of the widely applied distribution for problems in reliability theory. In this article, we introduce a generalization—referred to as the Beta Inverse-Weibull distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the Beta Inverse-Weibull distribution. The shapes of the corresponding probability density function and the hazard rate function have been obtained and graphical illustrations have been given. The distribution is found to be unimodal. Results for the non central moments are obtained. The relationship between the parameters and the mean, variance, skewness, and kurtosis are provided. The method of maximum likelihood is proposed for estimating the model parameters. We hope that this generalization will attract wider applicability to the problems in reliability theory and mechanical engineering.

Let X and Y be independent random variables, X having a gamma distribution with shape parameter a and Y having a non-central gamma distribution with shape and non-centrality parameters b and δ, respectively. Define Z = X/(X + 2Y ). Then, the random variable Z has a non-central beta type 3 distribution, Z ∼ NCB3(a, b; δ). In this article we derive density functions of sum, difference, product and quotient of two independent random variables each having noncentral beta type 3 distribution. These density functions are expressed in series involving first hypergeometric function of Appell.

In this paper, we develop the Size-biased form of the weighted Generalized Beta distribution of first kind (WGBD1); a particular case of the weighted Generalized Beta distribution of first kind, taking the weights as the variate values has been defined. The structural property of Size-biased Generalized Beta distribution of first kind (SBGBD1) including moments, variance, mode and harmonic mean, coefficient of variation, skewness and kurtosis has been derived. The estimates of the parameters of Size- biased Generalized Beta distribution of first kind (SBGBD1) is obtained by employing a new method of moments. Also, a test for detecting the size- baisedness is conducted. Keywords: Generalized Beta distribution of first kind, Beta function, Size-biased generalized beta distribution of first kind, Structural properties and moment estimator.

In the literature, numerous works have modeled user activity on smartphones and the effects on battery life. Power-saving modes prolong battery life by saving energy, but application performance is limited as a result. We investigate performance-energy trade-offs of smartphone applications by investigating three strategies: first, we propose an M/M/1 discriminatory processor sharing queue to act as a smartphone server and measure delays of Android applications; secondly, we form a performance-energy trade-off that takes into account cellular radio transfers using an objective cost function incorporating mean delay and power consumption; and thirdly, we build an online HMM to act as a power consumption model that predicts battery life given recent data transfers. For all three strategies, we obtain logged smartphone activity of over 750 users via an open-source smartphone data-collection application. Hence, we obtain three hypotheses from our strategies: first, delay of applications is approximated well using the beta prime distribution; secondly, power consumption increases as mean delay decreases with battery life prolonged if adjustments are made to cellular radio usage; and thirdly, burstiness is captured by HMMs in both data transfers and rates of power consumption.

Prior distributions for high-dimensional linear regression require specifying a joint distribution for the unobserved regression coefficients, which is inherently difficult. We instead propose a new class of shrinkage priors for linear regression via specifying a prior first on the model fit, in particular, the coefficient of determination, and then distributing through to the coefficients in a novel way. The proposed method compares favorably to previous approaches in terms of both concentration around the origin and tail behavior, which leads to improved performance both in posterior contraction and in empirical performance. The limiting behavior of the proposed prior is1/x, both around the origin and in the tails. This behavior is optimal in the sense that it simultaneously lies on the boundary of being an improper prior both in the tails and around the origin. None of the existing shrinkage priors obtain this behavior in both regions simultaneously. We also demonstrate that our proposed prior leads to the same near-minimax posterior contraction rate as the spike-and-slab prior. Supplementary materials for this article are available online.

In this letter, the small-scale variations of cross-polar discrimination (XPD) are shown to have a generalized Beta prime distribution in Nakagami-m fading channels. Closed-form expressions for the expectation and standard deviation of the XPD are given and the impact of Nakagami-m parameters on the statistical behavior of the XPD is analyzed. The accuracy of the XPD model has been investigated by Monte Carlo simulations. Indoor measurements have been conducted at 2.4 GHz to validate the derived XPD model. The theoretical distribution shows a better fit with the experiment results compared with the conventional XPD model derived in Ricean fading channels.

We introduce a five-parameter continuous model, called the McDonald inverted beta distribution, to extend the two-parameter inverted beta distribution and provide new four- and three-parameter sub-models. We give a mathematical treatment of the new distribution including expansions for the density function, moments, generating and quantile functions, mean deviations, entropy and reliability. The model parameters are estimated by maximum likelihood and the observed information matrix is derived. An application of the new model to real data shows that it can give consistently a better fit than other important lifetime models.

The study of proportions is a common topic in many fields of study. The standard beta distribution or the inflated beta distribution may be a reasonable choice to fit a proportion in most situations. However, they do not fit well variables that do not assume values in the open interval (0, c), 0

The lower and the upper estimates with explicit coefficients for the beta distribution with a > 1,b > 1 are given. Furthermore, using these results, the lower and the upper estimates of the beta distribution of the second kind and F-distribution, and also partial estimates of Student’s t-distribution are obtained. Mathematics subject classification (2010): 33B15, 26D15, 60E15.

The beta-κ distribution is a distinct case of the generalized beta distribution of the second kind. In previous studies, beta-p and beta-κ distributions have played important roles in representing extreme events, and thus, the present paper uses the beta-κ distribution. Further, this paper uses the method of moments and the method of L-moments to estimate the parameters from the beta-κ distribution, and to demonstrate the performance of the proposed model, the paper presents a simulation study using three estimation methods (including the maximum likelihood estimation method) and beta-κ and non beta-κ samples. In addition, this paper evaluates the performance of the beta-κ distribution by employing two widely used extreme value distributions (i.e., the GEV and Gumbel distributions) and two sets of actual data on extreme events.

Abtsrcat This paper deals with the approximation of distribution func- tions of complicated statistics using Beta distribution. The problem is motivated and the performance of our method is evaluated by several examples. Two methods are suggested to estimate the parameters of Beta distribution. We approximate distributions of cusum statistic and Hsu's G test statistic by this method. Error analysis is given. Fitting the mixture of Beta distributions are examined. Finally, a discussion Section is given.

In this paper we present some Turán type inequalities for the probability density function (pdf) of the non-central chi-squared distribution, non-central chi distribution and Student distribution, respectively. Moreover, we improve a result of Laforgia and Natalini concerning a Turán type inequality for the modified Bessel functions of the second kind.