Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications

Lindsay, Bruce G.; Pilla, Ramani S.; Basak, Prasanta

doi:10.1023/a:1004105603806

Cited by 55 publications

(35 citation statements)

References 20 publications

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“…The null asymptotic distribution is χ 2 ν , with ν = ( j w j ) 2 / j w 2 j . Matching of moments up to higher order have been also be considered (Wood, 1989;Lindsay et al, 2000). These corrections to pw(θ) might be inaccurate because they correct only moments of the distribution.…”

Section: Background On Pairwise Likelihoodmentioning

confidence: 99%

“…However, one drawback with pairwise likelihood methods is that the null distribution of the pairwise likelihood ratio statistic does not converge to the standard chi-square distribution, but to a linear combination of independent chi-square variates with coefficients given by the eigenvalues of a matrix related to the Godambe information (Kent, 1982). Several adjustments of the pairwise likelihood ratio to approximate the usual chi-square distribution have been proposed, which typically require the computation of the elements of the Godambe information (see Satterthwaite, 1946;Wood, 1989;Geys et al, 1999;Lindsay et al, 2000;Chandler and Bate, 2007;Pace et al, 2011).…”

Section: Introductionmentioning

confidence: 99%

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A note on empirical likelihoods derived from pairwise score functions

Lunardon

Pauli

Ventura

2013

Journal of Statistical Computation and Simulation

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Pairwise likelihood functions are convenient surrogates for the ordinary likelihood, useful when the latter is too difficult or even impractical to compute. One drawback of pairwise likelihood inference is that, for a multidimensional parameter of interest, the pairwise likelihood analogue of the likelihood ratio statistic does not have the standard chi-square asymptotic distribution. Invoking the theory of unbiased estimating functions, this paper proposes and discusses a computationally and theoretically attractive approach based on the derivation of empirical likelihood functions from the pairwise scores. This approach produces alternatives to the pairwise likelihood ratio statistic, which allow reference to the usual asymptotic chi-square distribution useful when the elements of the Godambe information are troublesome to evaluate or in the presence of large datasets with relative small sample sizes. Monte Carlo studies are performed in order to assess the finite-sample performance of the proposed empirical pairwise likelihoods.

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Section: Background On Pairwise Likelihoodmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A note on empirical likelihoods derived from pairwise score functions

Lunardon

Pauli

Ventura

2013

Journal of Statistical Computation and Simulation

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“…In the statistical literature, mixtures of probability densities are widely used in theory and applications (see, for instance, [7,Chapters 17 and 19], [9], and the references therein). If P is concentrated on the strictly increasing sequence (…”

Section: Mixtures Of Probability Densitiesmentioning

confidence: 99%

On a function studied by Ramanujan and connected with discrete mixtures of gamma densities

Adell

Alzer²

2009

Ramanujan J

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Abstract. We consider discrete mixtures of gamma densities with appropriate parameters, and characterize the complete monotonicity on any interval (c, ∞) (c ≥ 0) in terms of its defining parameters. This extends a previous result of Boas, after observing that the functionis, up to a constant, a special case of such mixtures. Moreover, we study convexity and subadditivity properties of φ λ (λ ∈ R) and we present several functional inequalities involving φ.

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“…The HallBuckley-Eagleson (Hall 1983;Buckley and Eagleson 1988) and Wood F (Wood 1989) methods match the first three moments of Q N to other distributions in a similar fashion. The Lindsay-Pilla-Basak method (Lindsay et al 2000) matches the first 2n moments of Q N to a mixture distribution. These four moment-matching methods are described in Sect.…”

Section: Introductionmentioning

confidence: 99%

A comparison of efficient approximations for a weighted sum of chi-squared random variables

Bodenham

Adams

2015

Stat Comput

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In many applications, the cumulative distribution function (cdf) F Q N of a positively weighted sum of N i.i.d. chi-squared random variables Q N is required. Although there is no known closed-form solution for F Q N , there are many good approximations. When computational efficiency is not an issue, Imhof's method provides a good solution. However, when both the accuracy of the approximation and the speed of its computation are a concern, there is no clear preferred choice. Previous comparisons between approximate methods could be considered insufficient. Furthermore, in streaming data applications where the computation needs to be both sequential and efficient, only a few of the available methods may be suitable. Streaming data problems are becoming ubiquitous and provide the motivation for this paper. We develop a framework to enable a much more extensive comparison between approximate methods for computing the cdf of weighted sums of an arbitrary random variable. Utilising this framework, a new and comprehensive analysis of four efficient approximate methods for computing F Q N is performed. This analysis procedure is much more thorough and statistically valid than previous approaches described in the literature. A surprising result of this analysis is that the accuracy of these approximate methods increases with N . Electronic supplementary materialThe online version of this article (

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Moment-Based Approximations of Distributions Using Mixtures: Theory and Applications

Cited by 55 publications

References 20 publications

A note on empirical likelihoods derived from pairwise score functions

A note on empirical likelihoods derived from pairwise score functions

On a function studied by Ramanujan and connected with discrete mixtures of gamma densities

A comparison of efficient approximations for a weighted sum of chi-squared random variables

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