Estimation for the negative binomial distribution based on the conditional likelihood

Anraku, Kazuo; Yanagimoto, Takemi

doi:10.1080/03610919008812887

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…Anraku & Yanagimoto (1990, eq. 1.1) and Hilbe (2007, eqs 5.9 or 5.29) propose the following NBD density form, which they obtain as a mixture of the Poisson distributions with the gamma distributed rate parameter λ (see, for example, Hilbe 2007, eqs 5.1–5.9 or 7.21–7.33) which converges to the geometric distribution if α= 1 , but to the Poisson distribution when α→ 0 , Comparison with shows that α= 1/τ and λ=τ (1 −θ)/θ .…”

Section: Theoretical Considerationsmentioning

confidence: 99%

Statistical distributions of earthquake numbers: consequence of branching process

Kagan¹

2010

Geophysical Journal International

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S U M M A R YWe discuss various statistical distributions of earthquake numbers. Previously, we derived several discrete distributions to describe earthquake numbers for the branching model of earthquake occurrence: these distributions are the Poisson, geometric, logarithmic and the negative binomial (NBD). The theoretical model is the 'birth and immigration' population process. The first three distributions above can be considered special cases of the NBD. In particular, a point branching process along the magnitude (or log seismic moment) axis with independent events (immigrants) explains the magnitude/moment-frequency relation and the NBD of earthquake counts in large time/space windows, as well as the dependence of the NBD parameters on the magnitude threshold (magnitude of an earthquake catalogue completeness). We discuss applying these distributions, especially the NBD, to approximate event numbers in earthquake catalogues. There are many different representations of the NBD. Most can be traced either to the Pascal distribution or to the mixture of the Poisson distribution with the gamma law. We discuss advantages and drawbacks of both representations for statistical analysis of earthquake catalogues. We also consider applying the NBD to earthquake forecasts and describe the limits of the application for the given equations. In contrast to the one-parameter Poisson distribution so widely used to describe earthquake occurrence, the NBD has two parameters. The second parameter can be used to characterize clustering or overdispersion of a process. We determine the parameter values and their uncertainties for several local and global catalogues, and their subdivisions in various time intervals, magnitude thresholds, spatial windows, and tectonic categories. The theoretical model of how the clustering parameter depends on the corner (maximum) magnitude can be used to predict future earthquake number distribution in regions where very large earthquakes have not yet occurred.

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Section: Theoretical Considerationsmentioning

confidence: 99%

Statistical distributions of earthquake numbers: consequence of branching process

Kagan¹

2010

Geophysical Journal International

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“…We can expect from the results in the previous section the superiority of the CMLE in such a case. Anraku and Yanagimoto (1988) conducted simulation studies using bias and the MSE for 0, 1/0 and 0/(1 + 0), which are summarized as follows. The CMLE has the smaller bias for 1/0 and 0/(1 + 0), for any 00 and /.to they employed.…”

Section: Case Study--the Discrete Modelsmentioning

confidence: 99%

Possible superiority of the conditional MLE over the unconditional MLE

Yanagimoto

Anraku

1989

Ann Inst Stat Math

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“…The usual estimator of $\mu$ is the sample mean ${\bar{X}}$ , which is the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (Anraku and Yanagimoto, 1990). Estimating $k$ can be problematic.…”

Section: Introductionmentioning

confidence: 99%

“…See for example Piegorsch (1990), Willson et al . (1984), and Anraku and Yanagimoto (1990). Alternative estimators have been proposed, including those by Willson et al .…”

Section: Introductionmentioning

confidence: 99%

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Mapping spatial aggregation from counts data: a penalized likelihood approach

Grevstad

2010

Environmetrics

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SUMMARYA model for counts data on a spatial domain is presented. The counts are assumed to follow the negative binomial distribution, with both the mean and dispersion parameter allowed to vary spatially. The ratio of the mean to the dispersion parameter is estimated via non-parametric penalized likelihood regression, and is equivalent to David and Moore's ecological index of aggregation. The dispersion parameter is assumed to follow a parametric model whose parameters are estimated via minimization of a variant of the well-known generalized approximate cross-validation (GACV) score, which is simultaneously used to estimate the smoothing parameter associated with estimation of the mean to dispersion parameter ratio. The increased flexibility of the model permitted by allowing the dispersion parameter to vary spatially is illustrated through spatial maps of David and Moore's and other indices of aggregation, such as Lloyd's indices of crowding and patchiness, using real and simulated data.

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Estimation for the negative binomial distribution based on the conditional likelihood

Cited by 10 publications

References 11 publications

Statistical distributions of earthquake numbers: consequence of branching process

Statistical distributions of earthquake numbers: consequence of branching process

Possible superiority of the conditional MLE over the unconditional MLE

Mapping spatial aggregation from counts data: a penalized likelihood approach

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