ABSTRACT
As the name confidence interval suggests, we set out to explore methods to estimate 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/eq7330.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with the help of an interval. That is, we will determine two statistics https://www.w3.org/1998/Math/MathML"> T L ( X ) , T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7331.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> based on the data https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7332.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and propose the interval https://www.w3.org/1998/Math/MathML"> T L ( X ) , T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7333.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> as a final estimator of 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/eq7334.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . If https://www.w3.org/1998/Math/MathML"> T L ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7335.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is https://www.w3.org/1998/Math/MathML"> - ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7336.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the interval https://www.w3.org/1998/Math/MathML"> - ∞ , T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7337.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is called an upper confidence interval for 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/eq7338.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . But, if https://www.w3.org/1998/Math/MathML"> T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7339.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is https://www.w3.org/1998/Math/MathML"> ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7340.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the interval https://www.w3.org/1998/Math/MathML"> T L ( X ) , ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7341.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is called a lower confidence interval for 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/eq7342.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
We prefer reporting both lower and upper end points of a confidence interval https://www.w3.org/1998/Math/MathML"> J https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7343.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , namely https://www.w3.org/1998/Math/MathML"> T L ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7344.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML"> T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7345.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , that depend only on a (minimal) sufficient statistic for 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/eq7346.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
Coverage probability associated with the confidence interval, https://www.w3.org/1998/Math/MathML"> J = T L ( X ) , T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7347.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , for 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/eq7348.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is measured by: () https://www.w3.org/1998/Math/MathML"> P θ θ ∈ T L ( X ) , T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7349.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
The confidence coefficient associated with https://www.w3.org/1998/Math/MathML"> J https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7350.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is defined as: () https://www.w3.org/1998/Math/MathML"> m i n θ ∈ Θ P θ θ ∈ T L ( X ) , T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7351.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
However, the coverage probability, https://www.w3.org/1998/Math/MathML"> P θ θ ∈ T L ( X ) , T U ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7352.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , may not involve 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/eq7353.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in many standard applications. In those situations, the confidence coefficient will coincide with the coverage probability itself. Thus, we will use these phrases interchangeably.
Before we proceed any further, we add some historical perspectives. The concepts of both “fiducial distribution” and “fiducial intervals” originated with Fisher https://www.w3.org/1998/Math/MathML"> ( 1930 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7354.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , which led to persistent and substantial philosophical arguments. Among others, Neyman came down hard on Fisher on philosophical grounds and proceeded to give the foundation of the theory of confidence intervals. This culminated in Neyman’s https://www.w3.org/1998/Math/MathML"> ( 1935 b , 1937 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7355.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> two pathbreaking papers on this subject. After 1937, neither Neyman nor Fisher swayed from their respective philosophical stance. However, in the 1961 article, Silver jubilee of my dispute with Fisher, Neyman was kinder to Fisher in his exposition. It may not be out of place to note that Fisher died on July 29, 1962. Buehler (1980), Lane (1980), and Wallace (1980) gave important accounts of fiducial distributions. It looks like some researchers are recently reinventing Fisher’s fiducial arguments in the name of “implicit 190distributions.” One may refer to Mukhopadhyay (2006) for some pertinent comments.
Customarily, one first fixes a small preassigned number https://www.w3.org/1998/Math/MathML"> α ∈ ( 0,1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7356.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and looks for a confidence interval for 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/eq7357.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with confidence coefficient https://www.w3.org/1998/Math/MathML"> ( 1 - α ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7358.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We refer to such an interval as https://www.w3.org/1998/Math/MathML"> ( 1 - α ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7359.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> or https://www.w3.org/1998/Math/MathML"> 100 ( 1 - α ) % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7360.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> confidence interval.
Suppose https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7361.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is distributed as https://www.w3.org/1998/Math/MathML"> N ( μ , 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7362.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where https://www.w3.org/1998/Math/MathML"> μ ( ∈ R ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7363.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is unknown. Let https://www.w3.org/1998/Math/MathML"> T L ( X ) = X - 1.96 , T U ( X ) = X + 1.96 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7364.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , leading to () https://www.w3.org/1998/Math/MathML"> J = ( X - 1.96 , X + 1.96 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7365.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
The coverage probability is: https://www.w3.org/1998/Math/MathML"> P μ { X - 1.96 < μ < X + 1.96 } = P { | Z | < 1.96 } , where Z = X 1 - μ is distributed as N ( 0,1 ) if μ is the true population mean, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7366.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
which is 0.95 and it does not depend upon https://www.w3.org/1998/Math/MathML"> μ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7367.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . See Figure 9.1.1. Clearly, https://www.w3.org/1998/Math/MathML"> z 0.025 = 1.96 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7368.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> So, the confidence coefficient associated with https://www.w3.org/1998/Math/MathML"> J https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7369.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is 0.95. ▴ Standard normal pdf: the shaded area between <inline-formula> <alternatives> <mml:math display="inline" xmlns:mml="<a href="https://www.w3.org/1998/Math/MathML" target="_blank">https://www.w3.org/1998/Math/MathML</a>"> <mml:mo>-</mml:mo> <mml:msub> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mtext> </mml:mtext> <mml:mtext>and</mml:mtext> <mml:mtext> </mml:mtext> <mml:msub> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mtext> </mml:mtext> <mml:mtext>with</mml:mtext> <mml:mtext> </mml:mtext> <mml:msub> <mml:mrow> <mml:mi>z</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1.96</mml:mn> <mml:mtext> </mml:mtext> <mml:mtext>is</mml:mtext> <mml:mtext> </mml:mtext> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mtext> </mml:mtext> <mml:mtext>where</mml:mtext> <mml:mtext> </mml:mtext> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.05</mml:mn> <mml:mtext>.</mml:mtext> <mml:mtext> </mml:mtext> </mml:math> <inline-graphic xlink:href="<a href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7370.tif" target="_blank">https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/eq7370.tif</a>" xmlns:xlink="<a href="https://www.w3.org/1999/xlink" target="_blank">https://www.w3.org/1999/xlink</a>"/> </alternatives> </inline-formula> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429113741/fa82963e-6109-44c7-ad13-faec498214f4/content/fig9_1_1.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/>
The basic methodology involves what we call a pivotal approach and it is both flexible and versatile. In Section 9.2, we discuss one-sample problems. An example of simultaneous confidence intervals (Example 9.2.6) is included! We provide an interpretation of a confidence coefficient and a notion of accuracy measure. Section 9.3 includes some two-sample problems.
