Abstract
Consider a random vector $(V_{1}, \dots , V_{n})$ where $\{V_{k}\}_{k=1, \dots , n}$ are the first $n$ components of a two-parameter Poisson-Dirichlet distribution $PD(\alpha , \theta )$. In this paper, we derive a decomposition for the components of the random vector, and propose an exact simulation algorithm to sample from the random vector. Moreover, a special case arises when $\theta /\alpha $ is a positive integer, for which we present a very fast modified simulation algorithm using a compound geometric representation of the decomposition. Numerical examples are provided to illustrate the accuracy and effectiveness of our algorithms.
Highlights
The two-parameter Poisson-Dirichlet distribution is a probability distribution on the set of decreasing positive sequences with sum 1
In this paper we develop two exact simulation algorithms for the first n components, (V1, V2, . . . , Vn), of the P D(α, θ) distribution
We provide two decompositions for 1/Vk, k = 1, . . . , n under the probability measure Pα,θ. These decompositions will lead to the exact simulation algorithms
Summary
The two-parameter Poisson-Dirichlet distribution is a probability distribution on the set of decreasing positive sequences with sum 1 It can be defined in terms of independent Beta random variables as the following. These decompositions will lead to the exact simulation algorithms. We present the decomposition for the first n components of the P D(α, θ) distribution, this result will permit us to use the subordinator algorithm developed in [5]. The theorem gives another decomposition for the components of P D(α, θ), which will permit us to use a faster simulation algorithm when θ/α is a positive integer. Dzdr1 . . . drn−1, and the theorem is a direct consequence of this result
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