Sequential Statistics
Designed for students with familiarity in advanced calculus, probability and statistical inference, this text provides only elementary proofs, and maintains mathematics and statistics at a basic level.
TABLE OF CONTENTS Sequential Statistics
4.3: Cram r-Rao Lower Bound
4.3 Cram r-Rao Lower Bound
Suppose the statistician is interested in solving problem (i) posed in Section 4.1, that the terminal decision is an estimate for ?, and that r( ?(x); ?)= [ ?(x) ? ?] 2. In this case, if we restrict ourselves to unbiased estimates of ? we would then be interested in lower bounds for the variance of such estimates. The Cram r-Rao inequality was extended to the sequential case by Wolfowitz (1947). Towards this we need the following lemma pertaining to a random sum of i.i.d. random variable.
Lemma 4.3.1
Let S N= X 1+ X 2+...+ X N where the X i are i.i.d.
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If E(N)< ? and EX< ?, we have E(S N )=E(N)E(X).
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If E(X)=0, E( X 2)< ?, and E(N)< ?, E(
)= E(N)E( X 2) . -
If E[g(X)]=E[h(X)]=0, E [ g 2 (X)]< ?, E[ h 2 (X)]< ? and P(X= 0)<1, then
(4.3.1) 
Proof
Notice that (i) and (ii) follow from Theorem 2.4.2. (iii) has been established by Lehmann (1950) under a different set of sufficient conditions. Consider
where Y i=1 if N ? i and zero otherwise. So, left hand expression
since for i< j, Y j=1 implies that Y i=1.
In order to interchange expectation and infinite summation, the series shouldbe absolutely integrable. That is
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