Stat 5101 Lecture Notes - School of Statistics

7.3. SAMPLING DISTRIBUTIONS OF SAMPLE MOMENTS 201Theorem 7.19. If X has a Student t distribution with ν degrees of freedom,then moments of order k exist if and only if k2and X ∼ t(ν), thenvar(X) =νν−2 .The proof is a homework problem (7-5).Another important property of the t distribution is given in the followingtheorem, which we state without proof since it involves the Stirling approximationfor the gamma function, which we have not developed, although we willprove a weaker form of the second statement of the theorem in the next chapterafter we have developed some more tools.Theorem 7.21. For every x ∈ Rf ν (x) → φ(x),as ν →∞,where φ is the standard normal density, andt(ν)D−→ N (0, 1),as ν →∞.Comparison of the t(1) density to the standard Cauchy density given byequation (1) on p. 191 in Lindgren shows they are the same (it is obvious thatthe part depending on x is the same, hence the normalizing constants must bethe same if both integrate to one, but in fact we already know that Γ( 1 2 )=√ πalso shows the normalizing constants are equal). Thus t(1) is another namefor the standard Cauchy distribution. The theorem above says we can thinkof t(∞) as another name for the standard normal distribution. Tables of the tdistribution, including Tables IIIa and IIIb in the Appendix of Lindgren includethe normal distribution labeled as ∞ degrees of freedom. Thus the t family ofdistributions provides lots of examples between the best behaved distributionof those we’ve studied, which is the normal, and the worst behaved, which isthe Cauchy. In particular, the t(2) distribution has a mean but no variance,hence the sample mean of i. i. d. t(2) random variables obeys the LLN but notthe CLT. For ν>2, The t(ν) distribution has both mean and variance, hencethe sample mean of i. i. d. t(ν) random variables obeys both LLN and CLT,but the t(ν) distribution is much more heavy-tailed than other distributions wehave previously considered.