ABSTRACT
Suppose that a population probability mass function (pmf) or probability density function (pdf) is https://www.w3.org/1998/Math/MathML"> f ( x ; θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6374.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with https://www.w3.org/1998/Math/MathML"> x ∈ X ⊆ R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6375.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where https://www.w3.org/1998/Math/MathML"> θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6376.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is an unknown parameter, https://www.w3.org/1998/Math/MathML"> θ ∈ Θ ⊆ R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6377.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
A hypothesis is a statement under consideration about an unknown parameter https://www.w3.org/1998/Math/MathML"> θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6378.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
Jerzy Neyman and Egon S. Pearson discovered a fundamental approach to formulate a test for statistical hypotheses. The Neyman-Pearson collaboration (1928a,b) first emerged with formulations and constructions of tests through comparisons of likelihood functions which blossomed into other landmark papers (Neyman and Pearson, 1933a, b).
Suppose that one has to choose between two hypotheses: https://www.w3.org/1998/Math/MathML"> H 0 : θ ∈ Θ 0 vs. H 1 : θ ∈ Θ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6379.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where https://www.w3.org/1998/Math/MathML"> Θ 0 ⊂ Θ , Θ 1 ⊂ Θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6380.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and https://www.w3.org/1998/Math/MathML"> Θ 0 ∩ Θ 1 = ∅ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6381.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , an empty set. Based on the evidence collected via random samples https://www.w3.org/1998/Math/MathML"> X 1 , … , X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6382.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , a statistical problem is to select one of the hypotheses which seems more reasonable. The basic question is this: given the sample evidence, if one must decide in favor of https://www.w3.org/1998/Math/MathML"> H 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6383.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> or https://www.w3.org/1998/Math/MathML"> H 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6384.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , which hypothesis is it going to be?
We refer to https://www.w3.org/1998/Math/MathML"> H 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6385.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> H 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6386.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> respectively as the null and alternative hypotheses. A hypothesis is called simple if it pinpoints a specific pmf or pdf. That is, https://www.w3.org/1998/Math/MathML"> H 0 : θ = θ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6387.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> H 1 : θ = θ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6388.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , with https://www.w3.org/1998/Math/MathML"> θ 0 ≠ θ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6389.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> known, would be both simple. A hypothesis which is not simple is called composite, for example, https://www.w3.org/1998/Math/MathML"> H 0 : θ > θ 0 , H 0 : θ < θ 0 , H 1 : θ ≥ θ 1 , H 1 : θ ≤ θ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6390.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are composite hypotheses. A hypothesis, https://www.w3.org/1998/Math/MathML"> H 1 : θ ≠ θ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6391.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , is called a two-sided hypothesis. In Section 8.2, we formulate two types of errors in decision making. Section 8.3 develops a most powerful (MP) test for choosing between simple null vs. simple alternative hypotheses. In Section 8.4, the idea of a uniformly most powerful (UMP) test is pursued when https://www.w3.org/1998/Math/MathML"> H 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6392.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is simple but https://www.w3.org/1998/Math/MathML"> H 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq6393.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is one-sided. Section 8.5 gives examples of two-sided tests and briefly touches upon unbiased and uniformly most powerful unbiased (UMPU) tests.
