Hansen, Gupta and Azzalini have distribution density distribution for skew student's t, and from the density and distribution of Hansen (1994) he introduce conditional variance to the log likelihood which from the pdf there are no substitution or assumption made.
Is your question about the assumptions? All t-tests assume normality and independence of data, no outliers, equal population variance, and adequate sample size. Otherwise, for skewed data, use a non-parametric test. But parametric tests can still perform well with continuous data that are non-normal if you satisfy the sample size :)
@Foued Yes using skewed student t distribution is condition to the asymmetry of the data. My question is from the density distribution of Hansen construction of skewed student t distribution which from the his log likelihood function, he added a positive 0.5in(sigma). so how comes that the plus 0.5in(sigma) come into play?. I need better explanation of his density and distribution function and that of the log likelihood because i try it and was not given mine the same. so i dont known whether i miss something
@Foued am asking for the inference of the density function and the cumulative function of the skewed student t distribution. i have try to obtained the cumulative but am not getting Hansen output of the cdf of skewed student t distribution on the topic " Dynamic copula modelling for Value at Risk
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