Page 464 - Probability and Statistical Inference

9

Confidence Interval Estimation

9.1 Introduction

As the name confidence interval suggests, we will now explore methods to
estimate an unknown parameter θ ∈ Θ ⊆ ℜ with the help of an interval. That
is, we would construct two statistics T (X), T (X) based on the data X and
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propose the interval (T (X), T (X)) as the final estimator of θ. Sometimes the
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lower bound T (X) may coincide with –∞ and in that case the associated
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interval (–∞, T (X)) will be called an upper confidence interval for θ. On the
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other hand, sometimes the upper bound T (X) may coincide with ∞ and in
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that case the associated interval (T (X), ∞) will be called a lower confidence
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interval for θ.
We may add that the concepts of the “fiducial distribution” and “fiducial
intervals” originated with Fisher (1930) which led to persistent and substan-
tial philosophical arguments. Among others, J. Neyman came down hard on
Fisher on philosophical grounds and proceeded to give the foundation of the
theory of confidence intervals. This culminated in Neyman’s (1935b,1937)
two path–breaking papers. After 1937, neither Neyman nor Fisher swayed
from their respective philosophical stance. However, in the 1961 article, Sil-
ver jubilee of my dispute with Fisher, Neyman was a little kinder to Fisher in
his exposition. It may not be entirely out of place to note that Fisher died on
July 29, 1962. The articles of Buehler (1980), Lane (1980) and Wallace (1980)
gave important perspectives of fiducial probability and distribution.

We prefer to have both the lower and upper end points of the
confidence interval J, namely T (X) and T (X), depend
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exclusively on the (minimal) sufficient statistics for θ.
Now, let us examine how we should measure the quality of a proposed
confidence interval. We start with one fundamental concept defined as fol-
lows.
Definition 9.1.1 The coverage probability associated with a confidence
interval J = (T (X), T (X)) for the unknown parameter θ is measured by
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