Approximating the Tail Probability of the t-Distribution : A Bayesian Approach

ABSTRACT: A Bayesian technique is used to approximate the tail probability of the t-distribution. A set of upper and lower bounds are obtained for this probability. Based on their simplicity and accuracy, these bounds are very adequate to use. Some members of these bounds are compared to some existing approximations. The possibility of using this new procedure for some other distributions is explored.

has a t-distribution.This statistic is very useful in the construction of tests and confidence intervals of θ .For a brief recent description of this distribution and its properties, see Stuart and Ord (1994) and Johnson, et al. (1995).
The importance of approximating the tail probability of this distribution is due to the fact that this probability is frequently used in constructing confidence intervals or in finding the pvalues of some statistical tests.There has been intensive work on approximating the tdistribution which produced approximations of very high accuracy, though some times very complicated.Fisher (1935) gave a direct expansion of the probability density function and hence of ) ; )as a series in , where v is the degree of freedom.Elfving (1955) suggested the following approximation: and Φ φ are respectively, the cumulative distribution and the density function of the standard normal distribution.Cucconi (1962) where is a number with tail area (probability Pinkham and Wilk (1954) suggested the use of the expansion: , and is the remainder term.

m R
Abu-Dayyeh and Ahmed (1993) considered a similar problem.They provided an upper bound for Mill ' s ratio Al-Saleh (1994) initiated a Bayesian approach to approximate Mill ' s ratio and hence to approximate the tail probability of the standard normal distribution.He obtained a sequence of upper bounds and a sequence of lower bounds of , each converges to . ) (x R As mentioned by Johnson et al. (1995), the available tables of the t-distribution are more than sufficient for almost all applications.However, a major concern raised by the above authors, is how to quickly evaluate the tail probability.It is well known that the t-distribution converges to the normal distribution as goes to infinity.Thus, for v , if is the distribution function of the t-distribution then . However this approximation is not so accurate for small .Recently, Li and Moor (1999) suggested the approximation of by , where λ is a shrinkage factor and its value is given by This approximation has a much simpler form and very accurate when compared with many of the approximations listed in Johnson et al. (1995).A weak point of this approximation is that it is written in terms of Φ , which has no closed form and has to be obtained from tables.For other approximations of the t-distribution see Johnson et al. (1995) and Gleason (2000).
In this paper we use the Bayesian approach introduced by Al-Saleh (1994) to obtain new approximations of the tail probability of the t-distributions.These approximations, which turned out to be of a simple form, can give very accurate values.The possibility of applying this approach to some other distributions is discussed.

Derivation of the Bounds
Assume that X is a random variable, which has a t-density with parameters θ and .

Then the density of
, where is a positive number and v x is any real number.Assume further that the median θ is otherwise.Then the posterior density of θ for given x can be written as: where stands for the t density with parameter (.) f ν and zero median and stands for the corresponding cumulative distribution.The main object of this paper is to approximate the tail probability, 1 .

F ) (x F −
Now, if ν =1, the distribution function of the t-distribution is the same as that of the Cauchy which has a closed form.For , the posterior expected value of is finite for v and is given by: for even values of , for odd values of .k Now, the last integral can be written as: Integrating by parts, it can be shown that for Thus, using (3), upper and lower bounds can be obtained for the tail probability of the standard t-distribution, i.e. for the quantity For even, i i µ can be written as Thus, for k with and v < 1 Hence, for k even we have: To see how accurate and U are, the two bounds have been obtained for some values of and .For =10 and k =3, 4, the two consecutive bounds are: For =15, the two bounds are: v And for ν =20, we have: Here, For a given and suitable , we take the average of the two bounds as an approximation of the tail probability 1 , i.e. for even and for odd we have k  Elfving (1955), Pinkham and Wilk (1954), and Li and Moor (1999)

Other Applications of the Technique
The Bayesian approach, which is used in this paper to approximate the t-distribution, was used by the author to approximate the normal distribution.An inspection of the procedure reveals that it can be applied to some other distributions.
If X has a density that is symmetric around zero and if we let that Now, depending on the functional form of , it may be possible to obtain a recursive formula for ) (x f ) (x i µ like the one in equation ( 3).We believe that some distributions such as the lognormal, non-central t and other location types-distribution can benefit from this procedure.Another useful application of the procedure is for estimating the cumulative distribution of the bivariate normal and other bivariate distributions.

Concluding Remarks
There has been considerable work on the possible approximations of the tail probability of the t-distribution.Simplicity as well as accuracy are important factors in assessing the value of an approximation.In this paper, we use a Bayesian approach to provide a set of upper and lower bounds of this probability; the set consists of [ ] 1 − v members.Any member of the set or a combination of members can serve as an approximation.Taking the average of two consecutive lower and upper bounds can be a good choice.It turns out that this approach is a suitable one in providing simple and accurate approximations and can be used for similar problems.Unlike many other approximations, the current procedure doesn't depend on ). x ( Φ

Table 2 :
denoted by 1 Values α for selected values of x when =10.It can be concluded from this table that the values of are even more accurate than v ) x ( * 5 α 3 α .