Asymptotic consistency of the James-stein shrinkage estimator

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.