Statistical aspects of Tricomi's function and modified Bessel functions of the second kind

Fitzgerald, D. L.

doi:10.1007/s004770000039

Cited by 3 publications

(9 citation statements)

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“…For HB and HIB, this happens because of the three fold use of the relationship represented by (2) that involves at least two moments each time. Moreover, the MM is valid under the condition m [ 1=2: Note that, the geometric mean (moment quasi-zero l 0 0 ; see Bobée et al 1993, p. 51), used in ML method, is not considered in the developed method and also by Fitzgerald (2000) for HA. The reason is that the geometric mean involves the derivative of the exponential factorial function ef m (a) with respect to its index m. Indeed, geometric mean of HB and HIB are given by (see Perreault et al 1999a):…”

Section: Discussionmentioning

confidence: 98%

“…However, in order to solve the MM system, four moment statistics are needed. Note that, also in MM for HA distribution, Fitzgerald (2000) used four moments rather than three in ML (see Table 1). For HB and HIB, this happens because of the three fold use of the relationship represented by (2) that involves at least two moments each time.…”

Section: Discussionmentioning

confidence: 98%

“…In the present paper, as a natural extension of the work by Fitzgerald (2000), we develop the method of moments to estimate the parameters of HB and HIB distributions. An advantage of this method is the explicit expression of the estimators, which makes their implementation very easy.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

“…A numerical method to solve the ML system for parameter estimation was presented by Perreault et al (1999b) and was implemented in HYFRAN software (CHS 2002). Fitzgerald (2000) developed a method of moments (MM) to estimate the HA distribution parameters using a recursive relation of Bessel function (which is the normalisation function of the HA density function). A three times use of this recursive relationship requires at least four moments.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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Method of moments of the Halphen distribution parameters

Chebana¹,

Adlouni²,

Bobée³

2007

Stoch Environ Res Risk Assess

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Halphen laws have been proposed as a complete system of distributions with sufficient statistics that lead to estimation with minimum variance. The Halphen system provides a flexibility to fit a large variety of data sets from natural events. In this paper we present the method of moments (MM) to estimate the Halphen type B and IB distribution parameters. Their computation is very fast when compared to those given by the maximum likelihood method (ML). Furthermore, this estimation method is very easy to implement since the formulae are explicit. Some simulations show the equivalence of both methods when estimating the quantiles for finite sample size.

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Section: Discussionmentioning

confidence: 98%

Section: Discussionmentioning

confidence: 98%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

See 2 more Smart Citations

Method of moments of the Halphen distribution parameters

Chebana¹,

Adlouni²,

Bobée³

2007

Stoch Environ Res Risk Assess

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“…1 Introduction Fitzgerald (2000) has shown that statistically interesting results involving Tricomi functions arise when the rate parameter of a Poisson process is randomised by an inverted beta or a Tricomi exponential variate. These results are now developed into a flexible method for treating series of events more clustered than the Poisson and the intervals between them.…”

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confidence: 99%

Tricomi and Kummer functions in occurrence, waiting time and exceedance statistics

Fitzgerald

2002

Stochastic Environmental Research and Risk Assessment (SERRA)

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Three and four parameter distributions of events and intervals based on the Kummer and Tricomi functions are shown to be statistically tractable. The Kummer distributions cover events less as well as more clustered than the Poisson while the Tricomi distributions have two and three parameter negative binomials as special cases. This flexibility and the moment relations between the parameters make them attractive in cases where the Poisson distribution proves inadequate. Fitzgerald (2000) has shown that statistically interesting results involving Tricomi functions arise when the rate parameter of a Poisson process is randomised by an inverted beta or a Tricomi exponential variate. These results are now developed into a flexible method for treating series of events more clustered than the Poisson and the intervals between them. The method offered there for computing Tricomi functions by statistical means is improved and a method of generating Tricomi exponential variates is also given. Then, using truncated beta distributions as randomisers, waiting-time distributions based on Kummer functions are derived; a statistical method of computing the Kummer function emerges and leads to another accurate method of computing the Tricomi function. IntroductionMoment relations between the parameters of the distributions are readily obtained as are the probabilities of a given number of occurrences and the statistical properties of the inter-occurrence times.A hyper-Poisson distribution based on Kummer functions and having four parameters is also treated as it covers series of events whether more or less clustered than the Poisson. In a very similar vein, but having a negative binomial as a special case, a four-parameter process based on the Tricomi function can be found. Mathematically and statistically both are generally as tractable as the three-parameter randomised versions but moment solutions in the four-parameter case present problems.With all these processes it is shown that, when the distribution for the exceedances possesses an inverse, the cumulative distribution functions of the return periods have the same analytic form as that of the occurrences and the values can be estimated by a method such as bisection.

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Some Poisson mixtures distributions with a hyperscale parameter

Laurent¹

2012

Braz. J. Probab. Stat.

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We mainly investigate certain mixtures of Poisson distributions with a scale parameter in the mixing distribution. They help us to derive the bivariate Poisson mixtures arising from the prior and posterior predictive distributions in the semi-conjugate family defined by Laurent and Legrand (ESAIM Probab. Stat. (2011) DOI:10.1051/ps/2010018) for the "two Poisson samples" model, which contains in particular the reference prior when the parameter of interest is the ratio of the two Poisson rates.

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Statistical aspects of Tricomi's function and modified Bessel functions of the second kind

Cited by 3 publications

References 0 publications

Method of moments of the Halphen distribution parameters

Method of moments of the Halphen distribution parameters

Tricomi and Kummer functions in occurrence, waiting time and exceedance statistics

Some Poisson mixtures distributions with a hyperscale parameter

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