Summary
(Logan, Mallows, Rice & Shepp 1973) analyse the limit probability distribution of the statistic \({S_n}(p) = \sum\nolimits_{i - 1} {{X_i}/{{\left( {\sum\nolimits_{i = 1} {{{\left| {{X_j}} \right|}^p}} } \right)}^{1/p}}} \) as n → ∞, when X{ini} is in the domain of attraction of a stable law with stability index α. By simulations, we provide quantiles of the usual critical levels of the finite-sample distributions of the Student\rss t-statistic as \({S_n}(p){\left[ {\left( {n - 1} \right)/\left( {n - S_n^2(p)} \right)} \right]^{1/2}}\) with p = 2. The response surface method is used to provide approximate quantiles of the finite-sample distributions of the Student’s t-statistic.
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Notes
1 L(z) is a slowly varying function as z → ∞, if for every constant\(c > 0,\,\,{\lim _{x \to \infty }}{{L(cz)} \over {L\left( z \right)}}\) exists and is equal to 1. See (Ibragimov & Linnik 1971) (p 394) for more details on slowly varying functions.
2 The case β = 1, called as totally skewed to the right, is also of interest for applications in option pricing.
3 The α-stable random variables are generated with the algorithm of Weron (Weron 1996). This algorithm is analytically identical with the stable random number generator written by J.H. McCulloch which is available on McCulloch’s website, http://economics.sbs.ohio-state.edu/jhm/jhm.html
4 Response surface methodology has been used in various statistical and econometric applications, see (Myers, Khuri & Carter, Jr. 1989) for more on this topic.
5 Alternatively, the method of (Peizer & Pratt 1968) may be used to approximate the quantiles of the t-distribution
$$2 t_{\xi}(n)=\sqrt{n\left[\exp \left\{\left(n-\frac{6}{5}\right)\left(\frac{z(\xi)}{n-\frac{2}{3}-\frac{1}{10 n}}\right)^{2}\right\}-1\right]},$$where z(ξ) is the ξ-quantile of the standard normal distribution.
References
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Kim, JR. Finite-sample distributions of self-normalised sums. Computational Statistics 18, 493–504 (2003). https://doi.org/10.1007/BF03354612
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DOI: https://doi.org/10.1007/BF03354612