TITLE:
Effective Truncation of a Student’s t-Distribution by Truncation of the Chi Distribution in a Chi-Normal Mixture
AUTHORS:
Daniel T. Cassidy
KEYWORDS:
Asset Pricing; Student’s t-Distribution; Cauchy; Truncation; Moments; Kurtosis
JOURNAL NAME:
Open Journal of Statistics,
Vol.2 No.5,
December
19,
2012
ABSTRACT: A Student’s t-distribution is obtained from a weighted average over the standard deviation of a normal distribution, σ, when 1/σ is distributed as chi. Left truncation at q of the chi distribution in the mixing integral leads to an effectively truncated Student’s t-distribution with tails that decay as exp (-q2t2). The effect of truncation of the chi distribution in a chi-normal mixture is investigated and expressions for the pdf, the variance, and the kurtosis of the t-like distribution that arises from the mixture of a left-truncated chi and a normal distribution are given for selected degrees of freedom 5. This work has value in pricing financial assets, in understanding the Student’s t--distribution, in statistical inference, and in analysis of data.