Approximation of t distribution is essential for finding the p-value in a computer program (while testing the hypothesis about the means). Is it enough to have three decimal point accuracy of the approximation? How many decimals should be correctly approximated by a function? Finally, how to approximate the cdf of t-distribution efficiently? Please suggest some thoughts on the same. Thank you.
Maybe this helps: http://siba-ese.unisalento.it/index.php/ejasa/article/download/10990/10805
If the approximation is good enough depends on your needs. How much may the empirical error rates deviate from the nominal ones? This will also depend on the chosen alpha. An approximation may work good enough for low alphas but not for high, and vice versa. So first you must clearly define your requirements and then you can check if an approximation will be satisfying.
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"Every probability and statistics class explains that the Student-t distribution is approximately normal, and everybody says the approximation is good when ν, the number of degrees of freedom in the t distribution, is 30 or larger."
J. Cook takes a closer examination :
This page looks a little more carefully into the error in this approximation.
http://www.johndcook.com/normal_approx_to_t.html
http://www.johndcook.com/relative_error_normal_approx.html
otherwise, the following reference
http://arxiv.org/pdf/math/0508604.pdf
shows that the saddlepoint expansion is valid without any moment restriction and concludes :
The Student’s t-statistic is one of the most commonly used statistics in inference. We have derived a saddlepoint approximation for the Student’s t-statistic under no moment condition. The key results are summarized as follows.
1. The saddlepoint approximation provides extremely accurate approximations to the distribution of the Student’s t-statistic. The approximation is particularly useful in calculating small probabilities in the tail areas, which are often of great interest in practice.
2. The saddlepoint approximation holds under no moment condition. This makes the application of the saddlepoint approximation very broad. This is significant for the user since one can use the approximation without having to worry about whether or not the result is valid.
3. The Student’s t-statistic is very robust against possible outliers.
For those reasons, the saddlepoint approximation of the Student’s t-statistic should always be used in practice whenever possible.
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Ya! It is well known that the normal approximation works well when the df is large. but how can we improve the approximation accuracy using the normalization (eg. Wallace,1959; Hill, 1970, 1975; Bailey 1980; Li and de moor, 1999; Gleason, 2000) for small df say 3 to 30. Is it enough a three decimal point accuracy to the cdf of t distribution with a simple formula(with out using normal cdf or its approximation)? Is there any scope to develop such functions?
Several fortran routines are available -
At CMU, http://lib.stat.cmu.edu/general/ look for cdflib
At NIST, http://www.nist.gov/itl/sed/datapac.cfm look for tcdf
I've also included an inverse CDF, so you can check one against the other. I found both of these on the web.
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