Beta-normal distribution and its applications
14 May 2002-Communications in Statistics-theory and Methods (Taylor & Francis Group)-Vol. 31, Iss: 4, pp 497-512
TL;DR: In this article, the authors introduced a general class of distributions generated from the logit of the beta random variable, a special case of this family is the beta-normal distribution, which provides great flexibility in modeling not only symmetric heavy-tailed distributions, but also skewed and bimodal distributions.
Abstract: This paper introduces a general class of distributions generated from the logit of the beta random variable. A special case of this family is the beta-normal distribution. The shape properties of the beta-normal distribution are discussed. Estimation of parameters of the beta-normal distribution by the maximum likelihood method is also discussed. The beta-normal distribution provides great flexibility in modeling not only symmetric heavy-tailed distributions, but also skewed and bimodal distributions. The flexibility of this distribution is illustrated by applying it to two empirical data sets and comparing the results to previously used methods.
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01 Jan 2011
TL;DR: In this paper, a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions is presented.
Abstract: This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].
1,296 citations
TL;DR: In this paper, a new family of generalized distributions for double-bounded random processes with hydrological applications is described, including Kw-normal, Kw-Weibull and Kw-Gamma distributions.
Abstract: Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with a...
742 citations
Cites background or methods from "Beta-normal distribution and its ap..."
...[1] proposed a general class of distributions for a random variable defined from the logit of the beta random variable by employing two parameters whose role is to introduce skewness and to vary tail weight....
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...Following the idea of the class of beta generalized distributions [1] and the distribution by Kumaraswamy [5], we define a new family of Kw generalized (Kw-G) distributions to extend several widely known distributions such as the normal, Weibull, gamma and Gumbel distributions....
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...[1], since it does not involve any special function....
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...[1], who defined the beta normal distribution, Nadarajah and Kotz [2] introduced the beta Gumbel distribution, Nadarajah and Gupta [3] proposed the beta Fréchet distribution and Nadarajah and Kotz [4] worked with the beta exponential distribution....
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...[1] and Jones [11] (see also [12]) to construct a new class of Kw generalized (Kw-G) distributions....
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10 Apr 2013
TL;DR: In this article, a new method is proposed for generating families of continuous distributions, where a random variable is used to transform another random variable and the resulting family, the $$T$$¯¯ -=-=-=-=-=-=-=-=-=-=-=-=- family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function.
Abstract: In this paper, a new method is proposed for generating families of continuous distributions. A random variable $$X$$
, “the transformer”, is used to transform another random variable $$T$$
, “the transformed”. The resulting family, the $$T$$
-
$$X$$
family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable $$X$$
. Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.
694 citations
TL;DR: In this article, the authors define a family of univariate distributions generated by Stacy's generalized gamma variables and propose an expected ratio of quantile densities for the discrimination of members of these two broad families of distributions.
445 citations
Cites background from "Beta-normal distribution and its ap..."
...[3] defined the family of beta-normal distributions and discussed its properties....
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...[3], wherein the beta-normal distributionwas introduced and its propertieswere studied....
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TL;DR: In this article, a simple generalisation of the use of the collection of order statistic distributions associated with symmetric distributions is presented, and an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution.
Abstract: Consider starting from a symmetric distributionF on ℜ and generating a family of distributions from it by employing two parameters whose role is to introduce skewness and to vary tail weight. The proposal in this paper is a simple generalisation of the use of the collection of order statistic distributions associated withF for this purpose; an alternative derivation of this family of distributions is as the result of applying the inverse probability integral transformation to the beta distribution. General properties of the proposed family of distributions are explored. It is argued that two particular special cases are especially attractive because they appear to provide the most tractable instances of families with power and exponential tails; these are the skewt distribution and the logF distribution, respectively. Limited experience with fitting the distributions to data in their four-parameter form, with location and scale parameters added, is described, and hopes for their incorporation into complex modelling situations expressed. Extensions to the multivariate case and to ℜ+ are discussed, and links are forged between the distributions underlying the skewt and logF distributions and Tadikamalla and Johnson'sLU family.
440 citations
References
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TL;DR: In this article, a method for making statistical inferences about the upper tail of a distribution function is presented for estimating the probabilities of future extremely large observations, where the underlying distribution function satisfies a condition which holds for all common continuous distribution functions.
Abstract: A method is presented for making statistical inferences about the upper tail of a distribution function. It is useful for estimating the probabilities of future extremely large observations. The method is applicable if the underlying distribution function satisfies a condition which holds for all common continuous distribution functions.
3,504 citations
3,324 citations
"Beta-normal distribution and its ap..." refers background in this paper
...Since then numerous other authors have developed various classes of generalized distributions including (2) and (3) who generalized the inverse Gaussian distribution....
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Book•
01 Jan 1982TL;DR: This book summarizes the author's research into basic concepts and Distributions for Product Life and presents a meta-analyses of Inspection Data (Qualtal--Response and Interval Data) and Maximum Likelihood Comparisons (Multiply Censored and Other Data).
Abstract: Preface to the Paperback Edition. Preface. About the Author. 1. Overview and Background. 2. Basic Concepts and Distributions for Product Life. 3. Probability Plotting of Complete and Singly Censored Data. 4. Graphical Analysis of Multiply Censored Data. 5. Series Systems and Competing Risks. 6. Analysis of Complete Data. 7. Linear Methods for Singly Censored Data. 8. Maximum Likelihood Analysis of Multiply Censored Data. 9. Analyses of Inspection Data (Qualtal--Response and Interval Data). 10. Comparisons (Hypothesis Tests) For Complete Data. 11. Comparisons with Linear Estimators (Singly Censored and Complete Data). 12. Maximum Likelihood Comparisons (Multiply Censored and Other Data). 13. Survey of Other Topics. Appendix A. Tables. References. Index.
1,684 citations
Additional excerpts
...One way we can achieve this, as suggested by (10) is to use a general model that is likely to include a simpler model as a limiting case....
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IBM1
TL;DR: In this paper, the authors show that unless the sample size is 500 or more, estimators derived by either the method of moments or probability-weighted moments are more reliable.
Abstract: The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.
1,233 citations
TL;DR: In this article, two generalized beta distributions have been used as models for the distribution of income and a unified method of comparing many models previously considered is presented, which includes many of these models as special or limiting cases.
Abstract: Many distributions have been used as descriptive models for the size distribution of income. This paper considers two generalized beta distributions which include many of these models as special or limiting cases. These generalized distributions have not been used as models for the distribution of income and provide a unified method of comparing many models previously considered.
883 citations
"Beta-normal distribution and its ap..." refers background in this paper
...The generalized beta of the first and second kind was introduced by (7) to study the distribution of income and later (8) used the generalized beta of the second kind for describing stock price returns....
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...For example, consider the generalized gamma and generalized beta of the first and second type of (7)....
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...According to (7), these three distributions together have the beta of the first kind considered by (9), the beta of the second kind, the Sing-Maddala, the lognormal, the gamma, the Weibull, the Fisk and the exponential families as limiting or special cases....
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