Pitman estimator

An equivariant estimator for the shift parameter with respect to a group of real shifts, having minimal risk with respect to a quadratic loss function.

Let the components $ X _ {1} \dots X _ {n} $ of a random vector $ X = ( X _ {1} \dots X _ {n} ) $ be independent random variables having the same probability law, with probability density belonging to the family

$$ \{ f( x- \theta ) , | x | < \infty , \theta \in \Theta =(- \infty , + \infty ) \} , $$

and with

$$ {\mathsf E} _ \theta X _ {1} ^ {2} = \int\limits _ {- \infty } ^ { +\infty } x ^ {2} f( x- \theta ) dx < \infty $$

for any $ \theta \in \Theta $. Also, let $ G = \{ g \} $ be the group of real shifts operating in the realization space $ \mathbf R ^ {1} = (- \infty , + \infty ) $ of $ X _ {i} $ $ ( i = 1 \dots n) $:

$$ G = \{ {g } : {gX _ {i} = X _ {i} + g, | g | < \infty } \} . $$

In this case, the task of estimating $ \theta $ is invariant with respect to the quadratic loss function $ L( \theta , \widehat \theta ) = ( \theta - \widehat \theta ) ^ {2} $ if one uses an equivariant estimator $ \widehat \theta = \widehat \theta ( X) $ of $ \theta $, i.e. $ \widehat \theta ( gX) = g \widehat \theta ( X) $ for all $ g \in G $. E. Pitman [1] has shown that the equivariant estimator $ \widehat \theta ( X) $ for the shift parameter $ \theta $ with respect to the group $ G $ that has minimal risk with respect to the quadratic loss function takes the form

$$ \widehat \theta ( X) = X _ {(n1)} - \frac{\int\limits _ {- \infty } ^ { +\infty } xf( x) \prod_{i=2} ^ { n } f( x+ Y _ {i} ) dx }{\int\limits _ {- \infty } ^ { +\infty } f( x) \prod_{i=2}^ { n } f( x+ Y _ {i} ) dx } , $$

where $ Y _ {i} = X _ {(ni)} - X _ {(n1)} $, and $ X _ {(ni)} $ is the $ i $- th order statistic of the observation vector $ X $. The Pitman estimator is unbiased (cf. Unbiased estimator); it is a minimax estimator in the class of all estimators for $ \theta $ with respect to the quadratic loss function if all equivariant estimators for $ \theta $ have finite risk function [2].

Example 1. If

$$ f( x- \theta ) = e ^ {-( x- \theta ) } ,\ \ x \geq \theta , $$

i.e. $ X _ {i} $, $ i = 1 \dots n $, has exponential distribution with unknown shift parameter $ \theta $, then the Pitman estimator $ \widehat \theta ( X) $ for $ \theta $ is

$$ \widehat \theta ( X) = X _ {(n1)} - \frac{1}{n} , $$

and its variance is $ 1/n ^ {2} $.

Example 2. If

$$ f( x- \theta ) = \frac{1}{\sqrt {2 \pi } } e ^ {-( x- \theta ) ^ {2} /2 } ,\ \ | x | < \infty , $$

i.e. $ X _ {i} $, $ i = 1 \dots n $, has normal distribution $ N( \theta , 1) $ with unknown mathematical expectation $ \theta $, then the arithmetic mean

$$ \overline{X}\; = \frac{X _ {1} + \dots + X _ {n} }{n} $$

is the Pitman estimator.

References

[1]	E.J. Pitman, "The estimation of the location and scale parameters of a continuous population of any given form" Biometrika , 30 (1939) pp. 391–421
[2]	M.A. Girshick, L.J. Savage, "Bayes and minimax estimates for quadratic loss functions" J. Neyman (ed.) , Proc. 2-nd Berkeley Symp. Math. Statist. Prob. , Univ. California Press (1951) pp. 53–73
[3]	S. Zachs, "The theory of statistical inference" , Wiley (1971)