User:Carlos A Coelho/Near-exact distributions

Near-exact Distributions are asymptotic distributions developed under a new concept of approximating distributions. Based on a decomposition (i.e., a factorization or a split in two or more terms) of the characteristic function (c.f.) of the statistic under study, or of the c.f. of its logarithm. These are asymptotic distributions which lay much closer to the exact distribution than common asymptotic distributions.

One has to obtain first a convenient decomposition of the c.f. and then identify the component part or parts which yield a manageable distribution and are to be left unchanged and the part or parts which not yielding a manageable distribution, have to be replaced by an asymptotic approximation which corresponds to the c.f. of a manageable distribution. All this has to be done in such a way that the resulting c.f. to yield a manageable distribution, from which p-values and quantiles are easy to compute.

By using this procedure we are able to obtain very well-fitting near-exact distributions or approximations even in situations where common asymptotic distributions are not easy to be developed or they do not perform well. Actually, it is even possible to obtain near-exact distributions even for statistics for which there are no asymptotic distributions developped. This is so, because we only need to obtain good asymptotic approximations for a part of the original c.f..

If the asymptotic replacement is adequatelly chosen, the resulting near-exact distributions, in case they refer to test statistics used in Multivariate Analysis or Multivariate Statistics, may become asymptotic not only for increasing sample sizes but also for increasing number of variables used, or even yet for increasing number of matrices or vectors being tested.

Although the whole process may seem a bit complicated, the criterious choice of the part of the original c.f. to be left unchanged, together with a criterious choice for the asymptotic replacement(s) for the part of the original c.f. to be replaced, may lead to an overal quite simple process and to very well-fitting near-exact distributions.

So far, near-exact distributions have been developed for a wide range of statistics, namely likelihood ratio test statistics used in Multivariate Analysis, whose exact distributions have highly complicated structures.

A Simple Example
Let us suppose we have a statistic, call it $$\Lambda\!$$, which has the same distribution as that of the product of a number $$p$$ of independent r.v.'s $$Y_j$$ $$(j=1,\dots,p)$$. Let $$Y_j\!$$ have a $Beta(a_j,b/2)$ distribution. Let us further suppose that both $$p$$ and $$b$$ are odd integers.

Then, since $$ \Lambda $$ has the same distribution as a product of independent r.v.'s, a good way to address its distribution is to consider the r.v. $$W=-\log\,\Lambda$$ and try to work through the c.f. of $$W\!$$. (The reason why we consider the minus sign in the definition of $$W\!$$ is only to bring the p.d.f. (probability density function) of $$W$$ onto the positive real semi-axis instead of leaving it on the negative one.)

Actually, if we take $$\Phi^{}_W(t)$$ to be the c.f. of $$W\!$$, we may write

where the last equality is possible to write as a consequence of the independence of the r.v.'s $$Y_j\!$$ $$(j=1,\dots,p)$$. In ($$) above, $$E\left(Y_j^{-it}\right)$$ is indeed what is called the Mellin transform of the r.v. $$Y_j\!$$ and its expression may be easily obtained for example from the expression of the $$h$$-th moment of a Beta distributed r.v..

In order to keep this first example really simple, let us take $$p=3$$, $$b=5$$ and $$a_j=a-j/2$$, for some real $$a>3/2$$, so that we have

\begin{array}{rcl} \Phi^{}_W(t) & = & \prod_{j=1}^3 \frac{\Gamma(a-j/2+5/2)\,\Gamma(a-j/2-{\rm i}t)}{\Gamma(a-j/2)\,\Gamma(a-j/2+5/2-{\rm i}t)}\\[7pt] & = & \prod_{j=1}^3 \frac{\Gamma(a-j/2+2)\,\Gamma(a-j/2-{\rm i}t)}{\Gamma(a-j/2)\,\Gamma(a-j/2+2-{\rm i}t)}\,\frac{\Gamma(a-j/2+5/2)\,\Gamma(a-j/2+2-{\rm i}t)}{\Gamma(a-j/2+2)\,\Gamma(a-j/2+5/2-{\rm i}t)} \end{array} $$ ... (to be completed)