Near-exact distributions for the sphericity likelihood ratio test statistic

Marques, Filipe J.; Coelho, Carlos A.

doi:10.1016/j.jspi.2007.01.002

Cited by 18 publications

(21 citation statements)

References 12 publications

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“…Well-fitting near-exact distributions have already been developed for this statistic by Marques and Coelho in [6]. In this paper we will show that even better near-exact distributions may be obtained for this statistic by taking as a basis the nearexact distributions developed for the statistic Λ 1 in (3) and the decomposition performed on its characteristic function.…”

Section: Introductionmentioning

confidence: 86%

“…we obtain as near-exact distributions for W 1 , a GNIG distribution, for h = 2, or, for h = 4 or 6, a mixture of 2 or 3 GNIG distributions, with cdf's given by (using the notation in (19) of [6])…”

Section: Near-exact Distributions For the Likelihood Ratio Test Statimentioning

confidence: 99%

“…We may write for H 0 in (4), H 01 in (2) and H 02 in (8), This way to look at this test will enable us to obtain even better near-exact distributions than the already available ones in [6].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 86%

Section: Near-exact Distributions For the Likelihood Ratio Test Statimentioning

confidence: 99%

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

Coelho

Marques

2010

Journal of Multivariate Analysis

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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%

“…Our near-exact distributions have links with phase-type approximations (Aldous and Shepp 1987;O'Cinneide 1990) and, as we discuss below, their accuracy can be controlled effectively through a precision parameter. Our first near-exact distribution is based on the Generalized Integer Gamma (GIG) and Generalized NearInteger Gamma (GNIG) distributions, which have a wealth of applications in multivariate analysis (Marques and Coelho 2008;Coelho andMarques 2010, 2012;Marques et al 2011;Coelho et al 2013). The GIG distribution corresponds to the distribution of the sum of independent Gamma random variables with integer shape parameters (Amari and Misra 1997;Coelho 1998), while the GNIG distribution corresponds to the distribution of the sum of a GIG random variable with an independent Gamma random variable with a non-integer shape parameter (Coelho 2004); further details on the GIG and GNIG distributions are given in Appendix A.…”

Section: Introductionmentioning

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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The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Marques

Coelho

2011

Math Methods in App Sciences

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Communicated by W. SprößigUsing a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near-exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent-induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near-exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method.

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Near-exact distributions for the sphericity likelihood ratio test statistic

Cited by 18 publications

References 12 publications

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

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

On the distribution of linear combinations of independent Gumbel random variables

The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

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