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)