normal distributionorder-statisticsprobability
Suppose that $X_1, \ldots, X_n$ are mutlivariate normal with equal correlation
$\rho$ and each of them are marginally
distributed as $N(0,1)$. Let $X_{(1)}, \ldots, X_{(n)}$
be the corresponding order statistics. The distribution of
$X_{(1)}$ and $X_{(n)}$ are easily found. What about the
distribution of the other order statistics? Can anyone
give some reference on this?
Thank you. Hanna
There are many papers addressing such questions.
A good starting place is probably:
Pyke R. (1965), Spacings
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 27, No. 3 (1965), pp. 395-449
(It has a lot on the continuous case. Many papers refer to this paper, including some that do more with the discrete case.)
You should be able to read it online:
http://www.jstor.org/discover/10.2307/2345793
(for me it says 'read online free' without me being logged into any institutional access)
For continuous uniform distributions, the answers are easy. For discrete distributions, accurate answers are much harder, though if the discrete uniform takes many different values, the continuous calculation can sometimes be a reasonable approximation.
You appear to understand the steps involved. I am not sure if this is an assignment or exercise, so it may not be appropriate to show workings anyway, but am happy to sketch out the approach ...
Given: $X \sim \text{Uniform}(0,1)$ with pdf $f(x)$:
Then, the joint pdf of the $r^{\text{th}}$ and $s^{\text{th}}$ order statistics is say $g(x_r, x_s)$:
where I am using the OrderStat function from the mathStatica package for Mathematica to automate the calculation.
Given the joint pdf of the $r^{\text{th}}$ and $s^{\text{th}}$ order statistics, you seek their correlation:
All done. This should hopefully help in both forming the appropriate integrals, and checking your working.
Best Answer
As shown in Tong, Y. L. (1990). Multivariate normal distribution. Springer-Verlag., ch. 6, for the setup described in the question and for non-negative correlation coefficient $\rho\in [0,\;1)$, the distribution function (cdf) and the density of an order statistic $X_{(i)}$ are (where $\phi()$ and $\Phi()$ are the standard normal pdf and cdf)
$$G_{(i)}(x) = \int_{-\infty}^{\infty}F_{(i)}\left(\frac{x+\sqrt{\rho}z}{\sqrt{1-\rho}}\right)\phi(z)dz$$
and differentiating,
$$g_{(i)}(x) = \int_{-\infty}^{\infty}\frac 1{\sqrt{1-\rho}}f_{(i)}\left(\frac{x+\sqrt{\rho}z}{\sqrt{1-\rho}}\right)\phi(z)dz$$
where
$$f_{(i)}(y) = \frac{n!}{(i-1)!(n-i)!}[\Phi(y)]^{i-1}[\Phi(-y)]^{n-i}\phi(y)$$ and $$F_{(i)}(y) = \sum_{j=i}^n {n \choose j}[\Phi(y)]^{j}[\Phi(-y)]^{n-j}$$
i.e $f_{(i)}(y)$ and $F_{(i)}(y)$ are the pdf and cdf of the order statistic $(i)$ from an i.i.d. standard normal random sample.
For the corresponding results when the correlation coefficient is negative, the author refers to the book "Order Statistics", by H.A. David & H.N. Nagaraja ch. 5 (now in its 3d edition, 2003).