# Is entropy relative

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**Contents**

Consistency and Relative Entropy

**Notice**-
- We look again at the hypotheses or.for any two but predefined parameter values With .
- For the sake of simplicity, we assume that the distributions are absolutely continuous and that the densities of are positive everywhere.
- It follows in particular that
(47)

- We look again at the hypotheses

The following limit theorem is used in the literature *Lemma of Stein* named (in honor of Charles Stein, Professor Emeritus in Statistics, Stanford University, USA)

**Theorem 4.3**Be . For each sample size be a Neyman-Pearson test with . Then applies

(48) |

**proof**-
- We use similar considerations as in the proof of Theorem 2.10.
- Be and
(49)

in which is the likelihood quotient defined in (28). - Then applies , and
- according to Theorem 4.2 there are constants and , so that for the Neyman-Pearson tests With
(50) - We first show that
(51) - From (50) it follows that and so too if .
- This results in the validity of the inequalities or.
- Thus, in order to prove (51) it suffices to show that it is for every a natural number there so that for each .
- But this follows from the fact that for each and for each , where the last convergence results from (49) and from the strong law of large numbers, see the proof of Theorem 2.10.
- To end the proof, let us show that
(52) - Here we can o.B.d.A. assume that .
- For each then applies

for each sufficiently large , where the last inequality results from the strong law of large numbers (cf. Theorem WR-5.15), because under . - It follows that
- This proves (52).

**Notice****example***Normally distributed sample variables*

**Next:**Sufficiency and the monotonic likelihood quotient

**Up:**Consistently best tests

**Previous:**Lemma of Neyman-Pearson & nbsp

**Contents**Roland Maier 2003-03-06

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