Effective Truncation of a Student’s t-Distribution by Truncation of the Chi Distribution in a Chi-Normal Mixture

Daniel T. Cassidy

Department of Engineering Physics, McMaster University, Hamilton, Canada.

DOI: 10.4236/ojs.2012.25067                

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.

Keywords

Asset Pricing; Student’s t-Distribution; Cauchy; Truncation; Moments; Kurtosis

Conflicts of Interest

The authors declare no conflicts of interest.

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