Neyman construction, named after Jerzy Spława-Neyman, is a frequentist method to construct an interval at a confidence level C , {\displaystyle C,\,} such that if we repeat the experiment many times the interval will contain the true value of some parameter a fraction C {\displaystyle C\,} of the time.
Theory Assume X 1 , X 2 , . . . X n {\displaystyle X_{1},X_{2},...X_{n}} are random variables with joint pdf f ( x 1 , x 2 , . . . x n | θ 1 , θ 2 , . . . , θ k ) {\displaystyle f(x_{1},x_{2},...x_{n}|\theta _{1},\theta _{2},...,\theta _{k})} , which depends on k unknown parameters. For convenience, let X {\displaystyle {\mathcal {X}}} be the sample space defined by the n random variables and subsequently define a sample point in the sample space as X = ( X 1 , X 2 , . . . X n ) {\displaystyle X=(X_{1},X_{2},...X_{n})}
Neyman originally proposed defining two functions L ( x ) {\displaystyle L(x)} and U ( x ) {\displaystyle U(x)} such that for any sample point, X {\displaystyle X} ,
L ( X ) ≤ U ( X ) {\displaystyle L(X)\leq U(X)} ∀ X ∈ X {\displaystyle \forall X\in {\mathcal {X}}}
L and U are single valued and defined. Given an observation, X ′ {\displaystyle X^{'}} , the probability that θ 1 {\displaystyle \theta _{1}} lies between L ( X ′ ) {\displaystyle L(X^{'})} and U ( X ′ ) {\displaystyle U(X^{'})} is defined as P ( L ( X ′ ) ≤ θ 1 ≤ U ( X ′ ) | X ′ ) {\displaystyle P(L(X^{'})\leq \theta _{1}\leq U(X^{'})|X^{'})} with probability of 0 {\displaystyle 0} or 1 {\displaystyle 1} . These calculated probabilities fail to draw meaningful inference about θ 1 {\displaystyle \theta _{1}} since the probability is simply zero or unity. Furthermore, under the frequentist construct the model parameters are unknown constants and not permitted to be random variables. For example if θ 1 = 5 {\displaystyle \theta _{1}=5} , then P ( 2 ≤ 5 ≤ 10 ) = 1 {\displaystyle P(2\leq 5\leq 10)=1} . Likewise, if θ 1 = 11 {\displaystyle \theta _{1}=11} , then P ( 2 ≤ 11 ≤ 10 ) = 0 {\displaystyle P(2\leq 11\leq 10)=0}
As Neyman describes in his 1937 paper, suppose that we consider all points in the sample space, that is, ∀ X ∈ X {\displaystyle \forall X\in {\mathcal {X}}} , which are a system of random variables defined by the joint pdf described above. Since L {\displaystyle L} and U {\displaystyle U} are functions of X {\displaystyle X} they too are random variables and one can examine the meaning of the following probability statement:
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