Neyman construction

Assume ${\displaystyle X_{1},X_{2},...X_{n}}$ are random variables with joint pdf ${\displaystyle f(x_{1},x_{2},...x_{n}|\theta _{1},\theta _{2},...,\theta _{k})}$, which depends on k unknown parameters. For convenience, let ${\displaystyle \Theta }$ be the sample space defined by the n random variables and subsequently define a sample point in the sample space as ${\displaystyle X=(X_{1},X_{2},...X_{n})}$
Neyman originally proposed defining two functions ${\displaystyle L(x)}$ and ${\displaystyle U(x)}$ such that for any sample point,${\displaystyle X}$,


Given an observation, ${\displaystyle X^{'}}$, the probability that ${\displaystyle \theta _{1}}$ lies between ${\displaystyle L(X^{'})}$ and ${\displaystyle U(X^{'})}$ is defined as ${\displaystyle P(L(X^{'})\leq \theta _{1}\leq U(X^{'})|X^{'})}$ with probability of ${\displaystyle 0}$ or ${\displaystyle 1}$. These calculated probabilities fail to draw meaningful inference about ${\displaystyle \theta _{1}}$ 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.^[1] For example if ${\displaystyle \theta _{1}=5}$, then ${\displaystyle P(2\leq 5\leq 10)=1}$. Likewise, if ${\displaystyle \theta _{1}=11}$, then ${\displaystyle P(2\leq 11\leq 10)=0}$


As Neyman describes in his 1937 paper, suppose that we consider all points in the sample space, that is, ${\displaystyle \forall X\in \Theta }$, which are a system of random variables defined by the joint pdf described above. Since ${\displaystyle L}$ and ${\displaystyle U}$ are functions of ${\displaystyle X}$ they too are random variables and one can examine the meaning of the following probability statement:



That is, ${\displaystyle P(L(X)\leq \theta _{1}^{'}\leq U(X)|\theta _{1}^{'})=C}$ where ${\displaystyle 0\leq C\leq 1}$ and ${\displaystyle L(X)}$ and ${\displaystyle U(X)}$ are the upper and lower confidence limits for ${\displaystyle \theta _{1}}$^[1]