Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Large parameter asymptotic analysis for homogeneous normalized random measures with independent increments

Junxi Zhang, Shui Feng, Yaozhong Hu

Abstract

Homogeneous normalized random measures with independent increments (hNRMIs) represent a broad class of Bayesian nonparametric priors and thus are widely used. In this paper, we obtain the strong law of large numbers, the central limit theorem and the functional central limit theorem of hNRMIs when the concentration parameter $a$ approaches infinity. To quantify the convergence rate of the obtained central limit theorem, we further study the Berry-Esseen bound, which turns out to be of the form $O \left( \frac{1}{\sqrt{a}}\right)$. As an application of the central limit theorem, we present the functional delta method, which can be employed to obtain the limit of the quantile process of hNRMIs. As an illustration of the central limit theorems, we demonstrate the convergence numerically for the Dirichlet processes and the normalized inverse Gaussian processes with various choices of the concentration parameters.

Large parameter asymptotic analysis for homogeneous normalized random measures with independent increments

Abstract

Homogeneous normalized random measures with independent increments (hNRMIs) represent a broad class of Bayesian nonparametric priors and thus are widely used. In this paper, we obtain the strong law of large numbers, the central limit theorem and the functional central limit theorem of hNRMIs when the concentration parameter approaches infinity. To quantify the convergence rate of the obtained central limit theorem, we further study the Berry-Esseen bound, which turns out to be of the form . As an application of the central limit theorem, we present the functional delta method, which can be employed to obtain the limit of the quantile process of hNRMIs. As an illustration of the central limit theorems, we demonstrate the convergence numerically for the Dirichlet processes and the normalized inverse Gaussian processes with various choices of the concentration parameters.
Paper Structure (17 sections, 9 theorems, 76 equations, 1 figure)

This paper contains 17 sections, 9 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Normalized random measures with independent increments
  3. Construction of NRMIs
  4. Representation of CRMs
  5. Main results
  6. Strong law of large numbers
  7. CLT and functional CLT for hNRMIs
  8. Berry-Esseen bound
  9. Numerical illustration
  10. Discussion
  11. Supplementary materials
  12. Introduction of $D(\mathbb{R} ^d)$
  13. Proof of Proposition \ref{['prop.CRM']}
  14. Proof of Proposition \ref{['prop: order of var']}
  15. Proof of Theorem \ref{['thm: slln']}
  16. ...and 2 more sections

Key Result

Proposition 2

Let $A \in \mathcal{X}$ and let $(A_1, \cdots, A_n)$ be disjoint measurable subsets belonging to $\mathcal{X}$ such that $H(A_i)\in (0,1)$ for all $i \in \{1,\cdots,n\}$. Denote the random vector ${\bf{Z}}:=(Z_1,\cdots,Z_n)\sim N(0,\Sigma)$ with $\Sigma=(\sigma_{ij})_{1\leq i,j \leq n}$ being given

Figures (1)

  • Figure 1: The convergence of $\mathbf{D}_a$ for the Dirichlet process and the normalized inverse Gaussian process when $a=2,5,10,20,30$.

Theorems & Definitions (23)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Example 5
  • Theorem 6
  • Example 7
  • Remark 8
  • Corollary 9
  • Example 10
  • ...and 13 more