arXiv subjects

Man-Wai Ho

Publications and source records attributed to Man-Wai Ho.

6 recordsLinked to original sources

arXiv2018

Gibbs Partitions, Riemann-Liouville Fractional Operators, Mittag-Leffler Functions, and Fragmentations Derived From Stable Subordinators

Pitman(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 $α\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson-Dirichlet distribution, $\mathrm{PD}(α,θ)$, which are induced by mixing over variables with generalized Mittag-Leffler distributions, denoted by $\mathrm{ML}(α,θ).$ 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, decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide novel characterizations of general laws associated with two nested families of $\mathrm{PD}(α,θ)$ mass partitions that are constructed from notable fragmentation operations described in Dong, Goldschmidt and Martin(2006) and Pitman(1999), respectively. These operations are known to be related in distribution to various constructions of discrete random trees/graphs in $[n],$ and their scaling limits, such as stable trees. A centerpiece of our work are results related to Mittag-Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\mathrm{ML}(α,θ)$ variables. Notably, this leads to an interpretation of $\mathrm{PD}(α,θ)$ laws within a mixed Poisson waiting time framework based on $\mathrm{ML}(α,θ)$ variables, which suggests connections to recent construction of Pólya urn models with random immigration by Peköz, Röllin and Ross(2018). Simplifications in the Brownian case are highlighted.

math.PR↗

arXiv2007

Gibbs Partitions (EPPF's) Derived From a Stable Subordinator are Fox H and Meijer G Transforms

This paper derives explicit results for the infinite Gibbs partitions generated by the jumps of an $α-$stable subordinator, derived in Pitman \cite{Pit02, Pit06}. We first show that for general $α$ the conditional EPPF can be represented as ratios of Fox-$H$ functions, and in the case of rational $α,$ Meijer-G functions. Furthermore the results show that the resulting unconditional EPPF's, can be expressed in terms of H and G transforms indexed by a function h. Hence when h is itself a H or G function the EPPF is also an H or G function. An implication, in the case of rational $α,$ is that one can compute explicitly thousands of EPPF's derived from possibly exotic special functions. This would also apply to all $α$ except that computations for general Fox functions are not yet available. However, moving away from special functions, we demonstrate how results from probability theory may be used to obtain calculations. We show that a forward recursion can be applied that only requires calculation of the simplest components. Additionally we identify general classes of EPPF's where explicit calculations can be carried out using distribution theory.

math.PR↗

arXiv2007

A Bayes method for a Bathtub Failure Rate via two $\mathbf{S}$-paths

A class of semi-parametric hazard/failure rates with a bathtub shape is of interest. It does not only provide a great deal of flexibility over existing parametric methods in the modeling aspect but also results in a closed and tractable Bayes estimator for the bathtub-shaped failure rate (BFR). Such an estimator is derived to be a finite sum over two $\mathbf{S}$-paths due to an explicit posterior analysis in terms of two (conditionally independent) $\mathbf{S}$-paths. These, newly discovered, explicit results can be proved to be a Rao-Blackwellization of counterpart results in terms of partitions that are readily available by a specialization of James (2005)'s work. We develop both iterative and non-iterative computational procedures based on existing efficient Monte Carlo methods for sampling one single $\mathbf{S}$-path. Nmerical simulations are given to demonstrate the practicality and the effectiveness of our methodology. Last but not least, two applications of the proposed method are discussed, of which one is about a Bayesian test for failure rates and the other is related to modeling with covariates.

stat.ME↗

arXiv2006

Bayesian Nonparametric Estimation of a Unimodal Density via two $\mathbf{S}$-paths

A Bayesian nonparametric method for unimodal densities on the real line is provided by considering a class of species sampling mixture models containing random densities that are unimodal and not necessarily symmetric. This class of densities generalize the model considered by Brunner (1992, Statist. Probab. Lett.), in which the Dirichlet process is replaced by a more general class of species sampling models. A novel and explicit characterization of the posterior distribution via a finite mixture of two dependent $\mathbf{S}$-paths is derived. This results in a closed-form and tractable Bayes estimator for any unimodal density in terms of a finite sum over two $\mathbf{S}$-paths. To approximate this class of estimates, we propose a sequential importance sampling algorithm that exploits the idea of the accelerated path sampler, an efficient path-sampling Markov chain Monte Carlo method. Numerical simulations are given to demonstrate the practicality and the effectiveness of our methodology.

math.ST↗

arXiv2006

A Bayes method for a monotone hazard rate via S-paths

A class of random hazard rates, which is defined as a mixture of an indicator kernel convolved with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of S-paths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over S-paths. The path characterization or the estimator is proved to be a Rao--Blackwellization of an existing partition characterization or partition-sum estimator. This accentuates the importance of S-paths in Bayesian modeling of monotone hazard rates. An efficient Markov chain Monte Carlo (MCMC) method is proposed to approximate this class of estimates. It is shown that S-path characterization also exists in modeling with covariates by a proportional hazard model, and the proposed algorithm again applies. Numerical results of the method are given to demonstrate its practicality and effectiveness.

math.ST↗

arXiv2006

Coagulation Fragmentation Laws Induced By General Coagulations of Two-Parameter Poisson-Dirichlet Processes

Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize $(α,θ)$} and $(β,θ/α)$, wherein $PD(α, θ)$ is coagulated by $PD(β,θ/α)$ for $0<α<1$, $0 \leqβ<1$ and $-β<θ/α$. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of $PD(α,θ)$ coagulated with some law $Q$. The analysis exploits distributional relationships between compositions of species sampling random probability measures and coagulation operators and recent work on Cauchy-Stieltjes transforms of random probability measures by Vershik, Yor and Tsilevich (2004) and James (2002). We use this to obtain explicit descriptions in the case where {\footnotesize $Q$} corresponds to a large class of power tempered Poisson Kingman models analyzed in James~(2002). That is, explicit results are obtained for models outside of the $PD(β,θ/α)$ family.

math.PR↗