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MCMC Estimators; Bias and Fundamental Matrix

In this section we will investigate the characteristics of Monte-Carlo estimators for expectations.

Statistical Model
 

Remarks
 

Theorem 3.17   $ \;$ For all $ n\ge 1$,

$\displaystyle {\mathbb{E}\,}\,\widehat\theta_n= \frac{1}{n}\; {\boldsymbol{\alpha}}^\top\sum\limits_{k=0}^{n-1}{\mathbf{P}}^k{\boldsymbol{\varphi}}\,.$ (71)

Proof
 

Remarks
 

Apart from this, the asymptotic behavior of $ n\bigl({\mathbb{E}\,}\,\widehat\theta_n-\theta\bigr)$ for $ n\to\infty$ can be determined. For this purpose we need the following two lemmata.

Lemma 3.2   $ \;$ Let $ {\boldsymbol{\Pi}}$ be the $ \ell\times\ell$ matrix consisting of the $ \ell$ identical row vectors $ {\boldsymbol{\pi}}^\top$. Then

$\displaystyle ({\mathbf{P}}-{\boldsymbol{\Pi}})^n={\mathbf{P}}^n-{\boldsymbol{\Pi}}$ (72)

for all $ n\ge 1$ and in particular

$\displaystyle \lim\limits_{n\to\infty}({\mathbf{P}}-{\boldsymbol{\Pi}})^n={\,{\bf0}}\,.$ (73)

Proof
 

Remarks
 


Lemma 3.3   $ \;$ The fundamental matrix $ {\mathbf{Z}}=({\mathbf{I}}-({\mathbf{P}}-{\boldsymbol{\Pi}}))^{-1}$ of the irreducible and aperiodic transition matrix $ {\mathbf{P}}$ has the representation formulae

$\displaystyle {\mathbf{Z}}={\mathbf{I}}+\sum\limits_{k=1}^\infty ({\mathbf{P}}^k-{\boldsymbol{\Pi}})$ (75)

and

$\displaystyle {\mathbf{Z}}={\mathbf{I}}+ \lim\limits_{n\to\infty}\sum_{k=1}^{n-1}\;\frac{n-k}{n}\;({\mathbf{P}}^k-{\boldsymbol{\Pi}})\,.$ (76)


Proof
 

Theorem 3.17 and Lemma 3.3 enable us to give a more detailed description of the asymptotic behavior of the bias $ {\mathbb{E}\,}\,\widehat\theta_n-\theta$.

Theorem 3.18    

Proof
 


next up previous contents
Next: Asymptotic Variance of Estimation; Up: Error Analysis for MCMC Previous: Estimate for the Rate   Contents
Ursa Pantle 2006-07-20