Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Grilo, Luís M.; Coelho, Carlos A.

doi:10.1016/j.jspi.2006.09.005

Cited by 15 publications

(5 citation statements)

References 11 publications

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“…Parenthetically, we further note that to be ensured that we accurately approximate the tail of the exact distribution, we need to keep increasing the precision parameter γ as we move towards higher quantiles; further details on the measure ∆ can be found in Grilo and Coelho (2007), Marques and Coelho (2008), and Coelho andMarques (2010, 2012). …”

Section: Measuring Accuracymentioning

confidence: 99%

On the distribution of linear combinations of independent Gumbel random variables

2014

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The distribution of linear combinations of independent Gumbel random variables is of great interest for modeling risk and extremes in the most different areas of application. In this paper we develop near-exact approximations for the distribution of linear combination of independent Gumbel random variables based on a shifted Generalized NearInteger Gamma distribution and on the distribution of the difference of two independent Generalized Integer Gamma distributions. These near-exact distributions are computationally appealing and numerical studies confirm their accuracy, as assessed by a proximity measure used in related studies. We illustrate the proposed approximations on applied problems in networks engineering, computational biology, and flood risk management.

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Section: Measuring Accuracymentioning

confidence: 99%

On the distribution of linear combinations of independent Gumbel random variables

2014

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“…One of these distributions is a GNIG distribution that matches the first three exact moments and the other is a M2GNIG distribution which matches the first four exact moments. These distributions emerge as the direct application of the procedure used by Grilo (2005) and Grilo and Coelho (2007) to obtain two near-exact distributions for the product of particular independent Beta r.v.'s.…”

Section: Two Near-exact Distributions For the Generalized Wilks λ Stamentioning

confidence: 99%

“…To assess the performance of the asymptotic and near-exact distributions proposed we use two proximity measures, based on the difference between the exact and asymptotic or near-exact c.f.'s. These measures were used by Grilo and Coelho (2007) and they are directly derived from the inversion formulas respectively for the p.d.f. and the c.d.f.. Their expressions are…”

Section: Comparative Numerical Studiesmentioning

confidence: 99%

“…The concept of near-exact distribution has already been introduced in a number of papers (Coelho, 2003(Coelho, , 2004Grilo andCoelho, 2007, 2010) and also a similar derivation procedure has already been applied to obtain nearexact distributions for the product of an odd number of particular independent Beta r.v. 's (Grilo and Coelho, 2007). Now, based on a factorization of the exact c.f.…”

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Near-exact distributions for the generalized wilks lambda statistic

Grilo

Coelho

2010

Discuss. Math. Probability and Statistics

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Two near-exact distributions for the generalized Wilks Lambda statistic, used to test the independence of several sets of variables with a multivariate normal distribution, are developed for the case where two or more of these sets have an odd number of variables. Using the concept of near-exact distribution and based on a factorization of the exact characteristic function we obtain two approximations, which are very close to the exact distribution but far more manageable. These near-exact distributions equate, by construction, some of the first exact moments and correspond to cumulative distribution functions which are practical to use, allowing for an easy computation of quantiles. We also develop three asymptotic distributions which also equate some of the first exact moments. We assess the proximity of the asymptotic and near-exact distributions obtained to the exact distribution using two measures based on the Berry-Esseen bounds. In our comparative numerical study we consider different numbers of sets of variables, different numbers of variables per set and different sample sizes.

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“…9]). Near-exact distributions for this statistic are thus readily available from the results in [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

Coelho

Marques

2010

Journal of Multivariate Analysis

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a b s t r a c tIn this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain nearexact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.

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Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Cited by 15 publications

References 11 publications

On the distribution of linear combinations of independent Gumbel random variables

On the distribution of linear combinations of independent Gumbel random variables

Near-exact distributions for the generalized wilks lambda statistic

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

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