Statistical Inference Based on the likelihood

Adelchi Azzalini

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9780412606502
• eBook ISBN: 9781351414463
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

The Likelihood plays a key role in both introducing general notions of statistical theory, and in developing specific methods. This book introduces likelihood-based statistical theory and related methods from a classical viewpoint, and demonstrates how the main body of currently used statistical techniques can be generated from a few key concepts, in particular the likelihood.Focusing on those methods, which have both a solid theoretical background and practical relevance, the author gives formal justification of the methods used and provides numerical examples with real data.

CHAPTER 1

Introduction and overview

1.1 Statistical inference

This book is about the theory of statistical inference, or at least about a considerable part of it. To explain the meaning of the term ‘statistical inference’, let us start with what an encyclopedia says about it.

Statistics

This can be defined as the body of methods concerned with the abstraction of synthetic information from observed data, with the purpose of characterizing those aspects of a phenomenon of interest which are relevant to particular aims Thus, statistics has a wide range of applications in studying all phenomena where it is supposed that some erratic factors are present in addition to some systematic factors whose effects are to be highlighted; so it turns out that the outcome of the latter factors, which could have been described by some ‘mathematical law’, is overlapped by a component which transforms such a law into ‘statistical regularity’

To achieve the aim of examining the essential characteristics of a phenomenon, statistical theory makes wide use of probability theory techniques, especially so when the available observations do not cover all possible instances of the phenomenon under considerations In such a case, ‘statistical inference’ problems arise, where the aim is to infer the characteristics of the whole from an observed portion of it

(A Naddeo, 1963)*

Therefore, in order to properly carry out a statistical investigation, it is necessary to establish exactly what is the phenomenon under consideration, and to state explicitly which are the observable characteristics, called variables, relevant for our purposes. We must also specify the subset of instances which we want to focus on. For instance, suppose that the variable of interest is the height of people. We might consider it in total generality (for the entire human population) or focus on a certain geographic context (such as a nation) and/or on an certain time. This set of instances is called the population, and the single instances are called individuals or subjects; sometimes the terms cases and units are used. This derivation of this terminology is historical, statistics having originally developed alongside and in interaction with demography; nowadays, the terms ‘individual’ and ‘population’ do not necessarily refer to human beings, when they are used in a technical sense.

In some investigations, the entire population is examined, and this is called census. In other cases, only a sample, i e a subset of the population, is examined, but the target is still to investigate the properties of the whole population. Statistical inference constitutes the operation through which information provided by the sample is used to draw conclusions about the characteristics of the population. This sort of procedure is a step forward with respect to a purely descriptive statistical approach, which simply attempts to synthesize the more relevant features of the observed data The present text is about the theory and the methods which direct the statistical inference operation.

To illustrate the concepts just introduced, let us introduce a very simple practical example. An industry which produces hydraulic pumps purchases from different suppliers many components necessary for its production. In particular, plastic gaskets used to join mechanical elements are supplied by a chemical company in batches of 5000

Obviously, the buyer needs to evaluate the quality of the gaskets supplied in order to eliminate, or at least substantially reduce, the possibility that a faulty gasket is used. Since the cost of repairing a pump found to be faulty is far higher than the cost of the gasket itself, it would be desirable to test the gaskets by putting them to work at appropriate water pressure for a short time, before adopting them for production. On the other hand, also the time required to set up and perform the test of the gaskets represents a cost

One way out is to examine not all the gaskets supplied but only a few, 50 say, and use the information provided by these tests to evaluate the number of faulty gaskets in the entire batch, and to decide about possibly returning the batch to the supplier if it is unsatisfactory. In doing so, we have to take into account that the subset tested will not in general contain faulty gaskets in exactly the same ratio as the batch.

In this example, we can regard the batch of 5000 gaskets as the population under investigation, and each gasket as an individual. At this point in time, our interest in the individual is confined to one specific aspect, namely whether it is ‘conforming’ or ‘not conforming’ to the specifications The observed characteristics of the sample elements are not of direct interest, except as a means of making inferences about the characteristics of the population as a whole

Let us now discuss in detail why we often examine only a sample instead of the whole population, which is apparently a preferable approach since it would avoid any indeterminacy in our conclusions (when carried out with full accuracy).

• The inspection cost of the entire population may be excessive, either because the number of subjects is large or because the individual inspection cost is high. Even when economic resources would allow a census of the population, the damage caused by a sample-type study may well not exceed the census cost.

• Since a complete investigation of the population often takes a long time, this may easily conflict with promptness requirements. For instance, the general census of the human population of a nation is performed at very long intervals (usually every ten years), and the results of these investigations are published much later. On the other hand, there are many economic and social problems, such as those related to cost of living or unemployment, for which substantially delayed information is not acceptable, since governments and agencies must take their decisions far more promptly.

• In very many cases, there is a virtual population which is effectively infinite; this situation occurs when instances of the phenomenon under study can be replicated as many times as we like. Suppose, for instance, that we want to study the capacity of a drug in lowering blood pressure in human beings; the relevant population is then formed by all people to whom the drug could possibly be given, i e the entire human population, present and future. Clearly, only a sample study is feasible here. Note that most scientific and technological experiments fall in this situation.

• In some cases, the inspection of the sample units destroys the units themselves. For example, the reliability analysis of a batch of light bulbs involves keeping them on for a rather long time, possibly until they wear out. Therefore, at the end of the inspection, the light bulbs are of degraded quality, if not dead. If the population is finite, clearly a census of the population is ruled out, unless the existence of the population after testing is irrelevant.

1.2 Sampling

It is evident that the sample has to represent, as far as possible, the population characteristics, in order to allow extension of the features of the sample to the population. This requirement is sometimes referred to by saying that the sample must be representative.

To illustrate this point, suppose for example that members of an amateur scientific society wish to carry out a survey of the attitudes and behaviour of the inhabitants of their city concerning racial problems. Initially, the members...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction and overview
• 2 Likelihood
• 3 Maximum likelihood estimation
• 4 Hypothesis testing
• 5 Linear models
• 6 Generalized linear models
• Appendix: Complements of probability theory
• Main abbreviations and symbols
• Answers to selected exercises
• Essential bibliography
• References
• Author index
• Subject index