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8

Tests of Hypotheses

8.1 Introduction

Suppose that a population pmf or pdf is given by f(x;θ) where x ∈ χ ⊆ ℜ and
θ is an unknown parameter which belongs to a parameter space Θ ⊆ ℜ.
Definition 8.1.1 A hypothesis is a statement about the unknown param-
eter θ.
In a problem the parameter θ may refer to the population mean and the
experimenter may hypothesize that θ ≥ 100 and like to examine the plausibil-
ity of such a hypothesis after gathering the sample evidence. But, a formula-
tion of testing the plausibility or otherwise of a single hypothesis leads
to some conceptual difficulties. Jerzy Neyman and Egon S. Pearson discov-
ered fundamental approaches to test statistical hypotheses. The Neyman-
Pearson collaboration first emerged (1928a,b) with formulations and con-
structions of tests through comparisons of likelihood functions. These blos-
somed into two landmark papers of Neyman and Pearson (1933a,b).
Neyman and Pearson formulated the problem of testing of hypotheses as
follows. Suppose one is contemplating to choose between two plausible hy-
potheses.

where Θ ⊂ Θ, Θ ⊂ Θ and Θ ∩ Θ = ϕ, the empty set. Based on the evidence
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collected from random samples X , ..., X , obtained from the relevant popu-
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lation under consideration, the statistical problem is to select one hypoth-
esis which seems more reasonable in comparison with the other. The experi-
menter may, for example, decide to opt for H compared with H based on
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the sample evidence. But we should not interpret this decision to indicate that
H is thus proven to be true. An experimenter’s final decision to opt for the
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hypothesis H (or H ) will simply indicate which hypothesis appears more
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favorable based on the collected sample evidence. In reality it is possible,
however, that neither H nor H is actually true. The basic question is this:
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Given the sample evidence, if one must decide in favor of H or H , which
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hypothesis is it going to be? This chapter will build the methods of such
decision-making.
We customarily refer to H and H respectively as the null and alterna-
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tive hypothesis. The null hypothesis H or the alternative hypothesis H is
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