Ordered Multi-Category Logits Model Under Dirichlet Distribution

2021 ◽  
Vol 23 (2) ◽  
pp. 513-524
Author(s):  
Kwang Mo Jeog
Keyword(s):  
Dirichlet Distribution

A COMPREHENSIVC GENERALIZED MEAN VALUE BASED ON THE DIRICHLET DISTRIBUTION

1985 ◽  
Vol 8 (4) ◽  
pp. 343-360
Author(s):  
James M. Dickey
Keyword(s):  
Dirichlet Distribution ◽  
Mean Value ◽  
Generalized Mean

scholarly journals Gibbs partitions, Riemann–Liouville fractional operators, Mittag–Leffler functions, and fragmentations derived from stable subordinators

2021 ◽  
Vol 58 (2) ◽  
pp. 314-334
Author(s):  
Man-Wai Ho ◽  
Lancelot F. James ◽  
John W. Lau
Keyword(s):  
Special Functions ◽  
Dirichlet Distribution ◽  
Random Trees ◽  
Fractional Integrals ◽  
Biased Sampling ◽  
Scaling Limits ◽  
Fractional Operators ◽  
Random Partitions ◽  
Two Parameter ◽  
Potential Use

AbstractPitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson–Dirichlet distribution, $\textrm{PD}(\alpha,\theta)$, whose corresponding $\alpha$-diversity/local time have generalized Mittag–Leffler distributions, denoted by $\textrm{ML}(\alpha,\theta)$. Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann–Liouville fractional integrals and size-biased sampling, and in decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide characterizations of general laws associated with nested families of $\textrm{PD}(\alpha,\theta)$ mass partitions that are constructed from fragmentation operations described in Dong et al. (2014). These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits. A centerpiece of our work is results related to Mittag–Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\textrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation within the context of $\textrm{PD}(\alpha,\theta)$ laws conditioned on Poisson point process counts over intervals of scaled lengths of the $\alpha$-diversity.

scholarly journals Rate of convergence for traditional Pólya urns

2020 ◽  
Vol 57 (4) ◽  
pp. 1029-1044
Author(s):  
Svante Janson
Keyword(s):  
Rate Of Convergence ◽  
Dirichlet Distribution ◽  
Fixed Number ◽  
Kolmogorov Distance ◽  
Initial Composition ◽  
Exact Distributions ◽  
Polya Urn ◽  
The One ◽  
Lévy Distance ◽  
Direct Calculations

AbstractConsider a Pólya urn with balls of several colours, where balls are drawn sequentially and each drawn ball is immediately replaced together with a fixed number of balls of the same colour. It is well known that the proportions of balls of the different colours converge in distribution to a Dirichlet distribution. We show that the rate of convergence is $\Theta(1/n)$ in the minimal $L_p$ metric for any $p\in[1,\infty]$, extending a result by Goldstein and Reinert; we further show the same rate for the Lévy distance, while the rate for the Kolmogorov distance depends on the parameters, i.e. on the initial composition of the urn. The method used here differs from the one used by Goldstein and Reinert, and uses direct calculations based on the known exact distributions.

Conditional Genotypic Probabilities for Microsatellite Loci

Genetics ◽  
2000 ◽  
Vol 155 (4) ◽  
pp. 1973-1980
Author(s):  
Jinko Graham ◽  
James Curran ◽  
B S Weir
Keyword(s):  
Short Tandem Repeats ◽  
Tandem Repeats ◽  
Dirichlet Distribution ◽  
Forensic Dna ◽  
Match Probability ◽  
Microsatellite Genotype ◽  
Allelic State ◽  
Forensic Assessments ◽  
Dna Profiles ◽  
Short Tandem

Abstract Modern forensic DNA profiles are constructed using microsatellites, short tandem repeats of 2–5 bases. In the absence of genetic data on a crime-specific subpopulation, one tool for evaluating profile evidence is the match probability. The match probability is the conditional probability that a random person would have the profile of interest given that the suspect has it and that these people are different members of the same subpopulation. One issue in evaluating the match probability is population differentiation, which can induce coancestry among subpopulation members. Forensic assessments that ignore coancestry typically overstate the strength of evidence against the suspect. Theory has been developed to account for coancestry; assumptions include a steady-state population and a mutation model in which the allelic state after a mutation event is independent of the prior state. Under these assumptions, the joint allelic probabilities within a subpopulation may be approximated by the moments of a Dirichlet distribution. We investigate the adequacy of this approximation for profiled loci that mutate according to a generalized stepwise model. Simulations suggest that the Dirichlet theory can still overstate the evidence against a suspect with a common microsatellite genotype. However, Dirichlet-based estimators were less biased than the product-rule estimator, which ignores coancestry.

scholarly journals A fractional generalization of the dirichlet distribution and related distributions

2021 ◽  
Vol 24 (1) ◽  
pp. 112-136
Author(s):  
Elvira Di Nardo ◽  
Federico Polito ◽  
Enrico Scalas
Keyword(s):  
Monte Carlo ◽  
Probability Density Function ◽  
Monte Carlo Simulations ◽  
Probability Density ◽  
Dirichlet Distribution ◽  
One Dimensional ◽  
Reasonable Approximation ◽  
Generalized Dirichlet Distribution ◽  
The One ◽  
Generalized Dirichlet

Abstract This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the n partitions of the interval [0, Wn ] are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented.

scholarly journals The spherical-Dirichlet distribution

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Jose H. Guardiola
Keyword(s):  
Dirichlet Distribution

A novel strategy to assimilate category variables in land-use models based on Dirichlet distribution

2022 ◽  
pp. 105324
Author(s):  
Xiaoli Hu ◽  
Feng Liu ◽  
Xin Li ◽  
Yuan Qi ◽  
Jinlong Zhang
Keyword(s):  
Land Use ◽  
Dirichlet Distribution ◽  
Land Use Models ◽  
Novel Strategy

scholarly journals The Dirichlet Distribution as a Model of Brand Choice: Further Testing

1981 ◽  
Author(s):  
Albert C. Bemmaor
Keyword(s):  
Dirichlet Distribution ◽  
Brand Choice

Information measures of Dirichlet distribution with applications

2011 ◽  
Vol 27 (2) ◽  
pp. 131-150 ◽  
Author(s):  
Nader Ebrahimi ◽  
Ehsan S. Soofi ◽  
Shaoqiong Annie Zhao
Keyword(s):  
Dirichlet Distribution ◽  
Information Measures
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