Effective Truncation of a Student’s <i>t</i>-Distribution by Truncation of the Chi Distribution in a Chi-Normal Mixture
- 1 Department of Engineering Physics, McMaster University, Hamilton, Canada
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 (-q 2 t 2 ). 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.
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