On the heavy-tailedness of Student’s t-statistic

Jonsson, Fredrik

Bernoulli, Tome 17 (2011) no. 1, p. 276-289 / Harvested from Project Euclid

Résumé

Let {X_i}_i≥1 be an i.i.d. sequence of random variables and define, for n≥2, ¶ $\hbox{\global\catcode`\@=4}T_{n}=\cases{n^{-1/2}\hat{\sigma}_{n}^{-1}S_{n},@\quad$\hat{\sigma}_{n}>0$,\cr 0,@\quad$\hat{\sigma}_{n}=0$,}$ $\qquad\mbox{with\ }S_{n}=\sum_{i=1}^{n}X_{i},$ $\hat{\sigma}^{2}_{n}=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-n^{-1}S_{n})^{2}.$ ¶ We investigate the connection between the distribution of an observation X_i and finiteness of E|T_n|^r for (n, r)∈ℕ_≥2×ℝ⁺. Moreover, assuming $T_{n}\stackrel {d}{\longrightarrow }T$ , we prove that for any r>0, lim _n→∞E|T_n|^r=E|T|^r<∞, provided there is an integer n₀ such that E|T_n0|^r is finite.

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Publié le : 2011-02-15
Classification:  finiteness of moments,  robustness,  Student’s t-statistic,  t-distributions,  t-test