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Generalized beta-generated distributions

Alexander, Carol; Cordeiro, Gauss M.; Ortega, Edwin Moises Marcos; Maria Sarabia, Jose
Fonte: ELSEVIER SCIENCE BV; AMSTERDAM Publicador: ELSEVIER SCIENCE BV; AMSTERDAM
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
45.91%
This article introduces generalized beta-generated (GBG) distributions. Sub-models include all classical beta-generated, Kumaraswamy-generated and exponentiated distributions. They are maximum entropy distributions under three intuitive conditions, which show that the classical beta generator skewness parameters only control tail entropy and an additional shape parameter is needed to add entropy to the centre of the parent distribution. This parameter controls skewness without necessarily differentiating tail weights. The GBG class also has tractable properties: we present various expansions for moments, generating function and quantiles. The model parameters are estimated by maximum likelihood and the usefulness of the new class is illustrated by means of some real data sets. (c) 2011 Elsevier B.V. All rights reserved.; Ministerio de Educacion of Spain [PR2009-0200, ECO2010-15455]; Ministerio de Educacion of Spain; CNPq-Brazil; CNPq (Brazil)

Modelos de regressão quando a função de taxa de falha não é monótona e o modelo probabilístico beta Weibull modificada; Regression models when the failure rate function is no monotone and the new beta modified Weibull model

Silva, Giovana Oliveira
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 05/02/2009 PT
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35.92%
Em aplicações na área de análise de sobrevivência, é freqüente a ocorrência de função de taxa de falha em forma de U ou unimodal, isto e, funções não-monótonas. Os modelos de regressão comumente usados para dados de sobrevivência são log-Weibull, função de taxa de falha monótona, e log-logística, função de taxa de falha decrescente ou unimodal. Um dos objetivos deste trabalho e propor os modelos de regressão, em forma de locação e escala, log-Weibull estendida que apresenta função de taxa de falha em forma de U e log- Burr XII que tem como caso particular o modelo de regressão log-logística. Considerando dados censurados, foram utilizados três métodos para estimação dos parâmetros, a saber, máxima verossimilhança, bayesiana e jackkinife. Para esses modelos foram calculadas algumas medidas de diagnósticos de influência local e global. Adicionalmente, desenvolveu-se uma análise de resíduos baseada no resíduo tipo martingale. Para diferentes parâmetros taxados, tamanhos de amostra e porcentagens de censuras, várias simulações foram feitas para avaliar a distribuição empírica do resíduo tipo martingale e compará-la com a distribuição normal padrão. Esses estudos sugerem que a distribuição empírica do resíduo tipo martingale para o modelo de regressão log-Weibull estendida com dados censurados aproxima-se de uma distribuição normal padrão quando comparados com outros resíduos considerados neste estudo. Para o modelo de regressão log-Burr XII...

Modelos de regressão estatísticos e dinâmicos para taxas ou proporções: uma abordagem bayesiana; Regression of static and dynamic models for proportions or rates: a Bayesian approach

Correia, Leandro Tavares
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 01/06/2015 PT
Relevância na Pesquisa
35.76%
Este trabalho apresenta um estudo de dados com resposta em intervalos limitados, mais especificamente no intervalo [0,1], como no caso de taxas e proporções. Em diversos casos práticos esta estrutura de dados apresenta uma quantidade não negligenciável de valores extremos (0 e 1) e que modelos usuais não são adequados para sua análise. Para esta situação propomos, por meio de um enfoque Bayesiano, modelos de regressão beta inflacionado de zeros e uns (BIZU) e modelos de regressão Tobit duplamente censurado adaptados nesse intervalo. Técnicas de diagnóstico e qualidade do ajuste também são discutidas. Apresentamos a análise desta estrutura de dados no contexto de série de tempo por meio da abordagem Bayesiana de modelos dinâmicos. Estudos de comportamento e previsão de séries de tempo foram explorados utilizando técnicas de Monte Carlo sequencial, conhecidas como filtro de partículas. Particularidades e competitividade entre as duas classes de modelos também foram discutidas.; This paper presents a study focused on observations in a limited interval , more specifically in [0,1] , such as rate and proportion data. In many practical cases this data structure has a considerable amount of extreme values (0 and 1) and usual classical models are not suitable for this type of data set. We propose two class of regression models to deal with this context: beta inflated of zeros and ones (BIZU) models and Tobit doubly censored models adapted in this interval. Fit quality and diagnostic techniques are also discussed. Time series of proportions are also developed through Bayesian dynamic models. Forecasting and behavioral analysis were explored using sequential Monte Carlo techniques...

A semi-parametric model for circular data based on mixtures of beta distributions

Carnicero, José Antonio; Wiper, Michael P.
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: Trabalho em Andamento Formato: application/pdf
Publicado em /03/2008 ENG
Relevância na Pesquisa
45.71%
This paper introduces a new, semi-parametric model for circular data, based on mixtures of shifted, scaled, beta (SSB) densities. This model is more general than the Bernstein polynomial density model which is well known to provide good approximations to any density with finite support and it is shown that, as for the Bernstein polynomial model, the trigonometric moments of the SSB mixture model can all be derived. Two methods of fitting the SSB mixture model are considered. Firstly, a classical, maximum likelihood approach for fitting mixtures of a given number of SSB components is introduced. The Bayesian information criterion is then used for model selection. Secondly, a Bayesian approach using Gibbs sampling is considered. In this case, the number of mixture components is selected via an appropriate deviance information criterion. Both approaches are illustrated with real data sets and the results are compared with those obtained using Bernstein polynomials and mixtures of von Mises distributions.

Rôle de l'estérification des acides gras dans la régulation de la sécrétion d'insuline et le stress métabolique induits par le glucose

Barbeau, Annie
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Thèse ou Mémoire numérique / Electronic Thesis or Dissertation
FR
Relevância na Pesquisa
25.93%
Le diabète est une maladie chronique de l’homéostasie du glucose caractérisée par une hyperglycémie non contrôlée qui est le résultat d’une défaillance de la sécrétion d’insuline en combinaison ou non avec une altération de l’action de l’insuline. La surnutrition et le manque d’activité physique chez des individus qui ont des prédispositions génétiques donnent lieu à la résistance à l’insuline. Pendant cette période dite de compensation où la concentration d’acides gras plasmatiques est élevée, l’hyperinsulinémie compense pleinement pour la résistance à l’insuline des tissus cibles et la glycémie est normale. Le métabolisme du glucose par la cellule pancréatique bêta entraîne la sécrétion d’insuline. Selon le modèle classique de la sécrétion d’insuline induite par le glucose, l’augmentation du ratio ATP/ADP résultant de la glycolyse et de l’oxydation du glucose, induit la fermeture des canaux KATP-dépendant modifiant ainsi le potentiel membranaire suivi d’un influx de Ca2+. Cet influx de Ca2+ permet l’exocytose des granules de sécrétion contenant l’insuline. Plusieurs nutriments comme les acides gras sont capables de potentialiser la sécrétion d’insuline. Cependant...

Distribution of beta/A4 protein and amyloid precursor protein in hereditary cerebral hemorrhage with amyloidosis-Dutch type and Alzheimer's disease.

Rozemuller, A. J.; Roos, R. A.; Bots, G. T.; Kamphorst, W.; Eikelenboom, P.; Van Nostrand, W. E.
Fonte: PubMed Publicador: PubMed
Tipo: Artigo de Revista Científica
Publicado em /05/1993 EN
Relevância na Pesquisa
25.85%
Brain amyloidosis with abundant beta/A4 protein deposition in plaques and cortical and meningeal vessels is found in Alzheimer's disease (AD) and hereditary cerebral hemorrhage with amyloidosis-Dutch type (HCHWA-D). In contrast to AD, no neuritic pathology or classical congophilic plaques are found in HCHWA-D. Unlike most AD cases, the congophilic angiopathy in HCHWA-D is very severe. It is still unknown whether beta/A4 deposits in plaques and vessels have the same origin. In this study, we have used frozen cortical tissue of HCHWA-D and AD patients to investigate the beta/A4 amyloid protein and the amyloid precursor protein (APP) in different types of plaques and congophilic angiopathy. Immunohistochemical staining was conducted using antibodies against synthetic beta/A4 proteins and antibodies against APP including MAbP2-1, a monoclonal antibody against purified protease nexin-2, which is the secreted form of APP. In contrast to immunohistochemical studies on formalin-fixed, paraffin-embedded tissue, frozen tissue of HCHWA-D patients revealed a very high number of beta/A4 plaques resembling AD. All plaques were of the diffuse type. Double-staining with MabP2-1 and beta/A4 antisera revealed: 1) the presence of APP immunoreactivity in classical plaques and transitional forms; 2) the absence of APP immunoreactivity in diffuse plaques in HCHWA-D and AD; and 3) pronounced APP immunoreactivity in congophilic vessels in HCHWA-D in contrast to weak APP staining in congophilic vessels in AD. Together these findings suggest that: a) the presence of APP in plaques is related to neuritic changes; b) different processes occur in amyloid formation in plaques and vessels; and c) differences exist between the process of amyloid formation in HCHWA-D and AD.

Asymptotic zero distribution of a class of hypergeometric polynomials

Driver, K. A.; Johnston, S. J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/07/2011
Relevância na Pesquisa
35.73%
We prove that the zeros of ${}_2F_1(-n,\frac{n+1}{2};\frac{n+3}{2};z)$ asymptotically approach the section of the lemniscate $\{z: |z(1-z)^2|=4/27; \textrm{Re}(z)>1/3\}$ as $n\rightarrow \infty$. In recent papers (cf. \cite{KMF}, \cite{orive}), Mart\'inez-Finkelshtein and Kuijlaars and their co-authors have used Riemann-Hilbert methods to derive the asymptotic zero distribution of Jacobi polynomials $P_n^{(\alpha_n,\beta_n)}$ when the limits $\ds A=\lim_{n\rightarrow \infty}\frac{\alpha_n}{n}$ and $\ds B=\lim_{n\rightarrow \infty}\frac{\beta_n}{n}$ exist and lie in the interior of certain specified regions in the $AB$-plane. Our result corresponds to one of the transitional or boundary cases for Jacobi polynomials in the Kuijlaars Mart\'inez-Finkelshtein classification.

Dynamics of Totally Constrained Systems I. Classical Theory

Kodama, Hideo
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/07/1995
Relevância na Pesquisa
25.91%
This is the first of a series of papers in which a new formulation of quantum theory is developed for totally constrained systems, that is, canonical systems in which the hamiltonian is written as a linear combination of constraints $h_\alpha$ with arbitrary coefficients. The main purpose of the present paper is to make clear that classical dynamics of a totally constrained system is nothing but the foliation of the constraint submanifold in phase space by the involutive system of infinitesimal canonical transformations $Y_\alpha$ generated by the constraint functions. From this point of view it is shown that statistical dynamics for an ensemble of a totally constrained system can be formulated in terms of a relative distribution function without gauge fixing or reduction. There the key role is played by the fact that the canonical measure in phase space and the vector fields $Y_\alpha$ induce natural conservative measures on acausal submanifolds, which are submanifolds transversal to the dynamical foliation. Further it is shown that the structure coefficients $c^\gamma_{\alpha\beta}$ defined by $\{h_\alpha,h_\beta\}=\sum_\gamma c^\gamma_{\alpha\beta}h_\gamma$ should weakly commute with $h_\alpha$, $\sum_\gamma\{h_\gamma,c^\gamma_{\alpha\beta}\}\approx0$...

Random Dirichlet series arising from records

Peled, Ron; Peres, Yuval; Pitman, Jim; Tanaka, Ryokichi
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
25.84%
We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by $$ S=\sum_{n=1}^{\infty}\frac{I_n}{n^s}, $$ where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value $1$ with probability $1/n^\beta$ and value $0$ otherwise. Random series of this type are motivated by the record indicator sequences which have been studied in extreme value theory in statistics. We show that when $s>0$ and $0< \beta \le 1$ with $s+\beta>1$ the distribution of $S$ has a density; otherwise it is purely atomic or not defined because of divergence. In particular, in the case when $s>0$ and $\beta=1$, we prove that for every $01$ it is unbounded. In the case when $s>0$ and $0<\beta<1$ with $s+\beta>1$, the density is smooth. To show the absolute continuity, we obtain estimates of the Fourier transforms, employing van der Corput's method to deal with number-theoretic problems. We also give further regularity results of the densities, and present an example of non atomic singular distribution which is induced by the series restricted to the primes.; Comment: 16 pages, 1 figure

Sharp bounds for cumulative distribution functions

Segura, Javier
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/06/2015
Relevância na Pesquisa
35.77%
Ratios of integrals can be bounded in terms of ratios of integrands under certain monotonicity conditions. This result, related with L'H\^{o}pital's monotone rule, can be used to obtain sharp bounds for cumulative distribution functions. We consider the case of noncentral cumulative gamma and beta distributions. Three different types of sharp bounds for the noncentral gamma distributions (also called Marcum functions) are obtained in terms of modified Bessel functions and one additional type of function: a second modified Bessel function, two error functions or one incomplete gamma function. For the noncentral beta case the bounds are expressed in terms of Kummer functions and one additional Kummer function or an incomplete beta function. These bounds improve previous results with respect to their range of application and/or its sharpness.

Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space

Ferger, Dietmar; Vogel, Daniel
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 13/06/2015
Relevância na Pesquisa
25.81%
We prove the asymptotic independence of the empirical process $\alpha_n = \sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $\beta_n = n (F_n(\tau+\frac{\cdot}{n})-F_n(\tau))$, where $F$ is an arbitrary cdf, differentiable at some point $\tau$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n \in N$, there is a deterministic correspondence between $\alpha_n$ and $\beta_n$. Precisely, we show that the pair $(\alpha_n,\beta_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge and a two-sided Poisson process. Since these processes have jumps, in particular if $F$ itself has jumps, the Skorokhod product space $D(R) \times D(R)$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D(R) \times D(R)$ by establishing the classical principle, devised by Yu. V. Prokhorov, that finite-dimensional convergence and tightness imply weak convergence. Several tightness criteria are given. Finally, the convergence of the pair $(\alpha_n,\beta_n)$ implies convergence of each of its components, thus, in passing, we provide a thorough proof of these known convergence results in a very general setting. In fact...

Compactly Supported One-cyclic Wavelets Derived from Beta Distributions

de Oliveira, H. M.; de Araujo, G. A. A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/02/2015
Relevância na Pesquisa
45.71%
New continuous wavelets of compact support are introduced, which are related to the beta distribution. They can be built from probability distributions using 'blur'derivatives. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters a and b. Close expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem applied for compactly supported signals.; Comment: 7 pages, 4 figures. Journal of Communication and Information Systems, former Journal of the Brazilian Telecommunications Society, ISSN 1980-6604 pp.105-111

Nearest-neigbor spacing distributions of the beta-Hermite ensemble of random matrices

Caer, G. Le; Male, C.; Delannay, R.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/04/2007
Relevância na Pesquisa
25.85%
The distributions of the spacing s between nearest-neighbor levels of unfolded spectra of random matrices from the beta-Hermite ensemble (beta-HE) is investigated by Monte Carlo simulations. The random matrices from the beta-HE are real-symmetric and tridiagonal where beta, which can take any positive value, is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues. Generalized gamma distributions are shown to be excellent approximations of the nearest-neighbor spacing (NNS) distributions for any beta while being still simple. They account both for the level repulsion when s tends to zero and for the whole shape of the NNS distributions in the range of s which is accessible to experiment or to most numerical simulations. The exact NNS distribution of the GOE (beta=1) is in particular significantly better described by a generalized gamma distribution than it is by the Wigner surmise while the best generalized gamma approximation coincides essentially with the Wigner surmise for beta larger than ~2. The distributions of the minimum NN spacing between eigenvalues of matrices from the beta-HE, obtained both from as-calculated eigenvalues and from unfolded eigenvalues are Brody distributions.; Comment: Physica A (in press...

Some Asymptotic Results for the Transient Distribution of the Halfin-Whitt Diffusion Process

Zhen, Qiang; Knessl, Charles
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/09/2013
Relevância na Pesquisa
35.58%
We consider the Halfin-Whitt diffusion process $X_d(t)$, which is used, for example, as an approximation to the $m$-server $M/M/m$ queue. We use recently obtained integral representations for the transient density $p(x,t)$ of this diffusion process, and obtain various asymptotic results for the density. The asymptotic limit assumes that a drift parameter $\beta$ in the model is large, and the state variable $x$ and the initial condition $x_0$ (with $X_d(0)=x_0>0$) are also large. We obtain some alternate representations for the density, which involve sums and/or contour integrals, and expand these using a combination of the saddle point method, Laplace method and singularity analysis. The results give some insight into how steady state is achieved, and how if $x_0>0$ the probability mass migrates from $X_d(t)>0$ to the range $X_d(t)<0$, which is where it concentrates as $t\to\infty$, in the limit we consider. We also discuss an alternate approach to the asymptotics, based on geometrical optics and singular perturbation techniques.; Comment: 43 pages and 8 figures

Products of Beta distributed random variables

Dunkl, Charles F.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/04/2013
Relevância na Pesquisa
45.67%
This is an expository note on useful expressions for the density function of a product of independent random variables where each variable has a Beta distribution.; Comment: 16 pages, 4 figures

A random matrix decimation procedure relating $\beta = 2/(r+1)$ to $\beta = 2(r+1)$

Forrester, Peter J.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/11/2007
Relevância na Pesquisa
25.95%
Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $\beta$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $\beta$-ensembles with $\beta = 2/(r+1)$ is equal to that of another $\beta$-ensemble with $\beta = 2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.; Comment: 19 pages, 1 figure

Normalized incomplete beta function: log-concavity in parameters and other properties

Karp, Dmitrii
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/09/2015
Relevância na Pesquisa
35.73%
The normalized incomplete beta function can be defined either as cumulative distribution function of beta density or as the Gauss hypergeometric function with one of the upper parameters equal to unity. Logarithmic concavity/convexity of this function in parameters was established by Finner and Roters in 1997. Their proof is indirect and rather difficult; it is based on generalized reproductive property of certain more general distributions. These authors remark that these results "seems to be very hard to obtain by usual analytic methods". In the first part of this paper we provide such proof based on standard tools of analysis. In the second part we go one step further and investigate the sign of generalized Tur\'{a}n determinants formed by shifts of the normalized incomplete beta function. Under some additional restrictions we demonstrate that these coefficients are of the same sign. We further conjecture that such restrictions can be removed without altering the results. Our method of proof also leads to various companion results which may be of independent interest. In particular, we establish linearization formulas and two-sided bounds for the above mentioned Tur\'{a}n determinants. Further, we find two combinatorial style identities for finite sums which we believe to be new.; Comment: 17 pages...

The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I

Wang, Dong
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
25.83%
We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric ($\beta$ = 1), Hermitian ($\beta$ = 2), and Hermitian self-dual ($\beta$ = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the one-band condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all $\beta$ > 0.; Comment: 45 pages, 4 figures, typos corrected

Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles

Popescu, Ionel
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
35.67%
We give a short proof of the free analogue of the Talagrand inequality for the transportation cost to the semicircular which was originally proved by Biane and Voiculescu. The proof is based on a convexity argument and is in the spirit of the original Talagrand's proof. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of beta ensembles, which gives also yet another proof of the convergence of the eigenvalue distribution to the semicircle law.; Comment: This is the final version. There were some typos and minor errors fixed

Bivariate beta-generated distributions with applications to well-being data

Sarabia Alegría, José María; Prieto Mendoza, Faustino; Jordá Gil, Vanesa
Fonte: Springer Publicador: Springer
Tipo: info:eu-repo/semantics/article; publishedVersion
ENG
Relevância na Pesquisa
65.92%
ABSTRACT: The class of beta-generated distributions (Commun. Stat. Theory Methods 31:497–512, 2002; TEST 13:1–43, 2004) has received a lot of attention in the last years. In this paper, three new classes of bivariate beta-generated distributions are proposed. These classes are constructed using three different definitions of bivariate distributions with classical beta marginals and different covariance structures. We work with the bivariate beta distributions proposed in (J. Educ. Stat. 7:271–294, 1982; Metrika 54:215–231, 2001; Stat. Probability Lett. 62:407–412, 2003) for the first proposal, in (Stat. Methods Appl. 18: 465–481, 2009) for the second proposal and (J. Multivariate Anal. 102:1194–1202, 2011) for the third one. In each of these three classes, the main properties are studied. Some specific bivariate beta-generated distributions are studied. Finally, some empirical applications with well-being data are presented.