Neyman construction

Neyman construction is a frequentist method to construct an interval at a confidence level such that if we repeat the experiment many times the interval will contain the true value of some parameter a fraction of the time. It is named after Jerzy Neyman.

TheoryEdit

Assume   are random variables with joint pdf  , which depends on k unknown parameters. For convenience, let   be the sample space defined by the n random variables and subsequentially define a sample point in the sample space as  
Neyman originally proposed defining two functions   and   such that for any sample point, ,

  •    
  • L and U are single valued and defined.

Given an observation,  , the probability that   lies between   and   is defined as   with probability of   or  . These calculated probabilities fail to draw meaningful inference about   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  , then  . Likewise, if  , then  

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

Under the frequentist construct the model parameters are unknown constants and not permitted to be random variables. Considering all the sample points in the sample space as random variables defined the joint pdf above, that is all   it can be shown that   and   are functions of random variables and hence random variables. Therefore one can look at the probability of   and   for some  . If   is the true value of  , we can define   and   such that the probability   and   is equal to pre-specified confidence level .

That is,  where   where   and   the upper and lower confidence limits for  [1]

Coverage probabilityEdit

The coverage probability,  , 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   confidence. For Neyman construction, the coverage probability is set to some value   where  . This value   tells how confidently that the true value is contained in the interval.

ImplementationEdit

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 ExampleEdit

 
Plot of 50 confidence intervals from 50 samples generated from a normal distribution.

Suppose  ~ , where   and   are unknown constants where we wish to estimate  . We can define (2) single value functions,   and  , defined by the process above such that given a pre-specified confidence level , , and random sample  =( )

 
 
where   ,  
and   follows a t distribution with (n-1) degrees of freedom.  ~t 

[2]

Another ExampleEdit

  are iid random variables, and let  . Suppose  . Now to construct a confidence interval with   level of confidence. We know   is sufficient for  . So,

 
 
 

This produces a   confidence interval for   where,

 
 .

[3]

See alsoEdit

ReferencesEdit

  1. ^ a b Neyman, J. (1937). "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 236 (767): 333–380. doi:10.1098/rsta.1937.0005. JSTOR 91337.
  2. ^ Rao, C. Radhakrishna (13 April 1973). Linear Statistical Inference and its Applications: Second Editon. John Wiley & Sons. pp. 470–472. ISBN 9780471708230.
  3. ^ Samaniego, Francisco J. (2014-01-14). Stochastic Modeling and Mathematical Statistics. Chapman and Hall/CRC. p. 347. ISBN 9781466560468.