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
4.2: Sufficiency and Completeness
4.2 Sufficiency and Completeness
Let X 1, X 2,..., be a sequence of i.i.d. random variables having common pdf f(x; ?). We wish to estimate ? by some function d( X 1,..., X i), while using a stopping rule which is closed (that is, for every ?, P( N ? n) ?1 as n ? ?, although not necessarily uniformly in ?).
The sample space is E 1+ E 2+..., where E i is contained in R i and consists of those points ( X 1,..., X i) which serve as stopping points. Again, N denotes the random number of observations taken. Throughout, we assume that the relevant conditional probabilities exist.
Let T n= T( X 1,..., X n) be a sufficient statistic for the joint density of X 1,..., X n.
Definition 4.2.1
The sequence ( T 1, T 2,...) is called a sufficient sequence for the sequential model. Then we have the following result of E. Fay (see Lehmann (1950)).
Result 4.2.1
(E.Fay). If, for each n, T n= T (X 1,..., X n) is a sufficient statistic for ? in the fixed sample X 1,..., X n, then (T N , N) is a sufficient statistic for ? in the sequential case.
Proof
This theorem was proved...
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