Applied Statistics

4Confidence Estimations – One‐ and Two‐Sample Problems

4.1 Introduction

In confidence estimation, we construct random regions in the parameter space so that this region covers the unknown parameter with a given probability, the confidence coefficient. In this book we consider special regions, namely intervals called confidence intervals. We then speak about interval estimation. We will see that there are analogies to the test theory concerning the optimality of confidence intervals, which we exploit to simplify many considerations.

A confidence interval is a function of the random sample; that is to say, it is also random in the sense of depending on chance, hence we speak of a random interval. However, once calculated based on observations, the interval has, of course, non‐random bounds. Sometimes only one of the bounds is of interest; the other is then fixed – this concerns the estimator as well as the estimate itself. At least one boundary of a confidence interval must be random; if both boundaries are random, the interval is two‐sided, and if only one boundary is random, it is one‐sided.

We make no difference between a random interval and its realisation with real boundaries but speak always about confidence intervals. What we mean in special cases will be easy to understand. When we speak about the expected length of a confidence interval, we of course mean a random interval.

Definition 4.1

Let Y = (y₁, y₂, … , y_n)^T be a random sample with realisations Y ε {Y}, whose ...