Invariant estimator: Difference between revisions

Revision as of 07:28, 20 January 2023

Revision as of 22:01, 30 January 2023

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If <math>x \sim C(\theta 1_n,I \sigma^2)\,\!</math> (independent components having a [[Cauchy distribution]] with scale parameter ''&sigma;'') then

<math>\delta_{pitman\text{Pitman}} \ne \delta_{ML}</math>,. However the result is

:<math>\delta_{pitman\text{Pitman}}=\sum_{k=1}^n{x_k\left[\frac{\text{Re}\{w_k\}}{\sum_{m=1}^{n}{\text{Re}\{w_k\}}}\right]}, \qquad n>1,</math>

with

:<math>w_k = \prod_{j\ne k}\left[\frac{1}{(x_k-x_j)^2+4\sigma^2}\right]\left[1-\frac{2\sigma}{(x_k-x_j)}i\right].</math>