This thesis considers two distinct aspects of the empirical Bayes decision problem. The first aspect considered is the problem or point estimation and hypothesis testing. The second aspect considered is that of estimating the prior distribution and then the estimation of posterior distribution and confidence intervals.
In the first aspect considered we assume that there exists an unobservable parameter space 𝔏={λ} on which is defined a prior distribution G(λ). For any action a from a class A there is a loss, L(a,λ) ≥ 0, which we incur when we take action a and the true parameter is λ. There exists an observable random vector X̰=(X₁,...X<sub>k</sub>), k ≥ 1, which admits of a sufficient statistic T=T(X̰) for λ. The conditional density function (c.d.f.) of T is f(t(λ). We assume that there exists a decision function δ<sub>ɢ</sub>(t) from a class D (δεD) implies that δ(t)εA for all t) such that the expected loss,
R(δ,G) = ∫∫L(δ(t),λ) f(t|λ)dtdG(λ),
is minimized. This minimizing decision function is called a Bayes decision function and the associated minimum expected loss is called the Bayes risk R(G). We assume that there exists a sequence or independent identically distributed random vectors <(X₁,...,X<sub>k</sub>,λ)<sub>n</sub>> (or <(T,λ)<sub>n</sub> >) with each element distributed independently of and identically to (X₁,...,X<sub>𝗄</sub>,λ) (or (T,λ). The problem is to construct sequential decision functions, δ<sub>n</sub>(t;t₁,t₂,...,t<sub>n</sub>)=δ<sub>n</sub>(t), which are asymptotically optimal (a.o.), that is which satisfy
lim<sub>n→∞</sub> R(δ<sub>n</sub>(T),G) = R(G).
We extend a theorem or Robbins (Ann. Math. Statist. 35,1-20) to provide a simple method for the construction or a.o. point estimators of λ with a squared-error loss function when f(t|λ) is continuous. We extend the results or Samuel (Ann. Math. Statist., 34, 1370-1385) to provide a.o. tests of certain parametric hypotheses with loss functions which are polynomials in λ. The c.d.f.'s which are considered are all continuous and include some or those of the exponential class and some whose range depends upon the parameter. This latter result is applied to the problem or the one-sided truncation of density functions.
The usefulness of all or these results is predicated upon the tact that the forms or the Bayes decision functions can be determined in such a way that the construction or the analogous a.o. empirical Bayes decision functions is simple. Two manipulative techniques, which provide the desired forms of the Bayes decision function, are introduced. These techniques are applied to several examples, and a.o. decision functions are defined.
To estimate the prior distribution we assume that there exists a sequence of independent identically distributed random vectors <(T,λ)<sub>n</sub>>) each distributed according to the joint density function J(t,λ)=G(λ)F(t|λ). The sequence <λ<sub>n</sub>> of <(T,λ)<sub>n</sub>> is unobservable. G(λ) belongs to a subclass g of a class G<sub>p</sub>(g) and F(t|λ) belongs to a class F. G<sub>p</sub>(g) is determined by the conditions: (a) G(λ) is absolutely continuous with with respect to Lebesgum measure; (b) its density function, g(λ), is determined completely by a continuous function of its first p moments, p ≥ 2; (c) the first p moments are finite; (d) the subclass g contains those density functions which are determined by a particular known continuous function. The class F is determined by the condition that there exist known functions h<sub>𝗸</sub>(.), k=1,...,p, such that E[h<sub>𝗸</sub>(T)|λ] = λᵏ. Under these conditions we construct an estimate, Gn(λ), of G(λ) such that
lim<sub>n→∞</sub> E[(G<sub>n</sub>(λ) - G(λ))²] = 0, a.e.λ.
Estimates of the posterior distribution and of confidence intervals are constructed using G<sub>n</sub>(λ). / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/87331 |
| Date | January 1965 |
| Creators | Rutherford, John Ross |
| Contributors | Statistics, Statistics |
| Publisher | Virginia Polytechnic Institute |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | iii, 108 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20305210 |
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