Location and Scale Parameter Modelling of the t-Distribution
(A heteroscedastic t model)
The location and scale parameters of the t-distribution can be modelled in a similar manner to the Gaussian. In partiucular, the modelling of the scale parameter allows for heterogeneous errors in the response whilst maintianing a robust approach. This is a new area of research.
This model can be also be extended in a number of ways. The conditional distribution of the response is Gaussian and therefore a conditional sufficient statistic can be found for the location parameter. This allows methodology for an approximate REML approach to estimation of the location and scale parameters of the t-distribution. This is also a subject for current research and contained in my forthcoming thesis.
Currently there is only one paper on this topic and is given below. A PhD dissertation titled, "Scale Parameter Modelling of the t-Distribution" is almost completed and includes maximum likelihood and restricted maximum likelihood estimation procedures.
Taylor, J. D & Verbyla, A. P. (2004). Joint modelling of the location and scale parameters of the t-distribution. Statistical Modelling,4, 91-112.
(Note: This paper contains an error associated with the orthogonal transformation that is currently being corrected)
There is an R software mini library "hett" (heteroscedastic t) for the estimation and summary of the parameters for the heteroscedastic t. Currently it contains functions to estimate the location, scale and degrees of freedom parameters by maximum likelihood and summarize with appropriate standard errors. There is also documentation associated with the important functions in the package. The original code was written in S-Plus but since has been ported to R and developed further.
The paper above reveals the similarity between the Gaussian and the t for
the estimation of the location and scale parameters under heteroscedasticity.
The more general heteroscedastic exponential family of distributions has also
been heavily researched and many links to papers can be found on the well
maintained website of Gordon
The website maintained by Adelchi Azzalini has
software and details of the skew normal and skew t-distributions.This new class of
distribution allows for possible skewness and, for the skew t, provides a robust approach to modelling
data. More details are available at the website. Comparisons of the skew
t and the heteroscedastic t
can be found in the paper cited above.
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