Neyman construction

Neyman construction is a frequentist method to construct an interval at a confidence level formula_1 such that if we repeat the experiment many times the interval will contain the true value of some parameter a fraction formula_2 of the time. It is named after Jerzy Neyman.
Theory.
Assume formula_3 are random variables with joint pdf formula_4, which depends on k unknown parameters. For convenience, let formula_5 be the sample space defined by the n random variables and subsequentially define a sample point in the sample space as formula_6
Neyman originally proposed defining two functions formula_7 and formula_8 such that for any sample point,formula_9,
Given an observation, formula_12, the probability that formula_13 lies between formula_14 and formula_15 is defined as formula_16 with probability of formula_17 or formula_18. These calculated probabilities fail to draw meaningful inference about formula_13 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 formula_20, then formula_21. Likewise, if formula_22, then formula_23
As Neyman describes in his 1937 paper, suppose that we consider all points in the sample space, that is, formula_11, which are a system of random variables defined by the joint pdf described above. Since formula_25 and formula_26 are functions of formula_9 they too are random variables and one can examine the meaning of the following probability statement:
That is,formula_41 where formula_42 where formula_31 and formula_32 the upper and lower confidence limits for formula_13
Coverage probability.
The coverage probability, formula_46, for Neyman construction is the frequency of experiments that the confidence interval contains the actual value of interest. Generally, the coverage probability is set to a formula_47 confidence. For Neyman construction, the coverage probability is set to some value formula_46 where formula_49. This value formula_46 tells how confidently that the true value is contained in the interval.
Implementation.
A Neyman construction can be carried out by performing multiple experiments that construct data sets corresponding to a given value of the parameter. The experiments are fitted with conventional methods, and the space of fitted parameter values constitutes the band which the confidence interval can be selected from.
Classic Example.
Suppose formula_9~formula_52, where formula_53 and formula_54 are unknown constants where we wish to estimate formula_53. We can define (2) single value functions, formula_25 and formula_26, defined by the process above such that given a pre-specified confidence level ,formula_46, and random sample formula_59=(formula_60)
Another Example.
formula_68 are iid random variables, and let formula_69. Suppose formula_70. Now to construct a confidence interval with formula_71 level of confidence. We know formula_72 is sufficient for formula_73. So, 
This produces a formula_77 confidence interval for formula_73 where,