0

Suppose $T$ is minimal sufficient statistics for a family of distribution. If $T = H(U)$ where $U$ has the same dimension as that of $T$, then does it imply that $U$ is also minimal sufficient?

I think the answer to this question should be NO.

I will give one example for this...

Consider for a single observation $X$ for the family $\{f_0(x), f_0(x) \;| \;x\in \mathbb{R}\}$ where $f_0(x) = \frac{1}{\sqrt{2\pi}}\exp(-\frac{x^2}2)$ and $f_1(x) = \frac{1}{\pi(1+x^2)}$

Then, we know that $\frac{f_1(x)}{f_0(x)} = \frac{\pi(1+x^2)\exp(-x^2/2)}{\sqrt{2\pi}} = T(x)$ is minimal sufficient for this family. Let $U = x^2$, then clearly $T$ is a function of $U$, however, $U$ is not a function of $T$. Hence, $U$ has the same dimension as $T$, but not minimal sufficent

Abhay
  • 159
  • 5
  • Your question has some difficulties in definitions. The subscripts $f_0$ and $f_1$ are not very clear, where does $f_1$ come from and what does it mean? The derivation of $T(x)$ is also not clear. – Sextus Empiricus Feb 03 '21 at 14:25
  • No I said that it means $U$ is not minimal sufficient. ofcouse, $U$ is sufficient at $T$ is a function of $U$ – Abhay Feb 03 '21 at 14:27

0 Answers0