, Volume 69, Issue 2, pp 175-183

First online:

Examples of non-normal observations for which a studentized sample mean has a student’s T-distribution

  • Nitis MukhopadhyayAffiliated withDepartment of Statistics, University of Connecticut Storrs Email author 
  • , Bhargab ChattopadhyayAffiliated withDepartment of Statistics, University of Connecticut Storrs

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Statistical methodologies and their practice often rely upon tests and confidence interval procedures based on Studentized sample means of independent observations from a normal parent population and their Student’s t distributions. This is specially so when the sample size n is small. An unmistakable impression one is left with, whether implied or not, is that such exact Student’s t distributions may not be valid when the observations are dependent or non-normal. We show that one cannot discard the possibility of an exact Student’s t distribution for a Studentized sample mean, or its suitable multiple, simply because the observations may be dependent or non-normal. In arriving at this conclusion, we have uncovered a very interesting and seemingly unknown feature (Theorem 2.1) of an n-dimensional multivariate t distribution with equi-correlation p, arbitrary degree of freedom v, and arbitrary n.

Key Words

Dependent data Equi-correlated t Multi-modal data Multivariate Cauchy Multivariate F Multivariate t Non-normal data Peaked data