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
4.7: Interval and Point Estimates for the Mean
4.7 Interval and Point Estimates for the Mean
4.7.1 Interval Estimation for the Mean
In Section 4.6, we discussed fixed-width confidence intervals for a general parameter, in this section we shall present the large-sample fixed-width sequential confidence intervals for the population mean. The main results can be followed by assuming certain convergence theorems, the understanding of which requires some knowledge of measure theory.
The basic asymptotics of Chow and Robbins (1965) will be given as lemmas and their main results and N das' (1969) results will be stated as theorems.
Lemma 4.7.1
(Chow and Robbins, 1965). Let Y n( n=l, 2,...) be any sequence of random variables such that Y n >0 a.s. (almost surely) lim n ?? Y n= 1 a.s., let f(n) be any sequence of constants such that
and for each t>0 define
| (4.7.1) |  |
Then N is well-defined and non-decreasing as a function of t,
| (4.7.2) |  |
and
| (4.7.3) |  |
Proof
(4.7.2) can easily be verified. In order to prove (4.7.3) we observe that for N>1, Y N< f(N)/t<[ f(N)/ f(N ?1)] Y N ?1, from which (4.7.3) follows as t ? ?.
Lemma 4.7.2
(Chow and Robbins, 1965). If the assumptions of Lemma 4.7.1 are satisfied and if E(sup n Y n)< ?, then
| (4.7.4) |  |
Proof
Let Z=sup n Y n; then E( Z)< ?. Choose m such that f( n)/ f(
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