TABLE OF CONTENTS In this chapter, we address the question What is a statistic ? In particular, we look at what role statistics play in scientific inference and give some common useful examples. We will examine their role in the two basic inference problems: hypothesis testing (the frequentist equivalent of model selection) and parameter estimation, with emphasis on the latter. Hypothesis testing will be dealt with in Chapter 7.
Recall that an important aspect of frequentist statistical inference is the process of drawing conclusions based on sample data drawn from the population (which is the collection of all possible samples). The concept of the population assumes that in principle, an infinite number of measurements (under identical conditions) are possible. Suppose X 1, X 2, ..., X n are n independent and identically distributed (IID) random variables that constitute a random sample from the population for which x 1, x 2, ..., x n is one realization. The population is assumed to have an intrinsic probability distribution (or density function) which, if known, would allow us to predict the likelihood of the sample x 1, x 2, ..., x n.
For example, suppose the random variable we are measuring is the time interval between successive decays of a radioactive sample. In this case, the population probability density function is a negative exponential (see Section 5.8.5), given by f( x ?) = [exp( ? x)/ ?)]/ ?. The...
