Asymptotic consistency of the James-stein shrinkage estimator
Date
2019
Authors
Mungo, Alex Samuel
Journal Title
Journal ISSN
Volume Title
Publisher
University of Zambia
Abstract
This study establishes the asymptotic consistency of the James-Stein shrinkage
estimator (JSSE) ˆ!⇤
n of some parameter ✓, which is obtained by shrinking a maximum
likelihood estimator (MLE) ˆ✓n. Our shrinking strategy involves creating a
subspace to shrink to by partitioning the parameter space ⌦ into two components.
From this partition, our interest is in the whole parameter space ⌦ and one of the
partitioned components which we refer to as the sub-parameter space ⌦o. Due to
this partition, we have another maximum likelihood estimator for the parameter
in the sub-parameter space ⌦o and we call it the restricted maximum likelihood
estimator (RMLE) ˜✓
o
n which is the shrinkage target. Therefore in this framework we
consider three estimators, the JSSE ˆ!⇤
n , RMLE ˜✓
o
n and MLE ˆ✓n. We use Hansen’s
approach to derive the asymptotic distribution of the James-Stein shrinkage estimator
(JSSE). With regularity conditions for the MLE considered, we obtain the
asymptotic distribution as a multivariate normal distribution with some shrinkage
e↵ect values. We use the Taylor’s theorem and limit theorems on this distribution
to show that the James-Stein shrinkage estimator is asymptotically consistent as
long as the initial (MLE) estimator is consistent. The asymptotic distributional bias
(ADB) is evaluated for each of the three estimators. Results show that the JSSE
ˆ!⇤
n and RMLE ˜✓
o
n are asymptotically biased while the unrestricted MLE ˆ✓n is asymptotically
unbiased. Furthermore we show that the JSSE ˆ!⇤
n is also asymptotically
efficient. Lastly, simulation plots are done in R for the mean squared error (MSE)
for sample size values of 30, 2000, 8000, 50000 and 100000 using the R multivariate
model to compare the unbiased estimator (MLE) and the James-Stein shrinkage
estimator in order to show lower MSE of the latter. Results also show that the
James-Stein shrinkage estimator converges faster compared to the MLE. We conclude
from the study that the James-Stein shrinkage estimator (JSSE) obtained by
shrinking a maximum likelihood estimator (MLE) is asymptotically consistent and
efficient.
Keywords: Convergence, Efficiency, Maximum likelihood estimator, Mean squared
error, Shrinkage.
Description
Keywords
Llikelihood estimator , James-Stein shrinkage estimator , Restricted Maximum Likelihood Estimator