Examples and Problems in Mathematical Statistics

CHAPTER 6

Confidence and Tolerance Intervals

PART I: THEORY

6.1 GENERAL INTRODUCTION

When θ is an unknown parameter and an estimator is applied, the precision of the estimator can be stated in terms of its sampling distribution. With the aid of the sampling distribution of an estimator we can determine the probability that the estimator θ lies within a prescribed interval around the true value of the parameter θ. Such a probability is called confidence (or coverage) probability. Conversely, for a preassigned confidence level, we can determine an interval whose limits depend on the observed sample values, and whose coverage probability is not smaller than the prescribed confidence level, for all θ. Such an interval is called a confidence interval. In the simple example of estimating the parameters of a normal distribution N(μ, σ2), a minimal sufficient statistic for a sample of size n is (n, ). We wish to determine an interval (μ (n, ), (n, )) such that

(6.1.1)

for all μ, σ. The prescribed ...