Hello, I need an advice about order statistics. I need to find analytical approximations or bounds (so table/numeric approximations are out of the scope) of the variance (and mean, but especially variance) of the sample maximum from a nonstandard Normal distribution. Also extreme values theory is out of the scope. What I've found in the classic book by David and Nagaraja (2003, par. 4) seems to be general and intractable, maybe something more specific involving Normal distribution could be more easy to deal with. Do you think it is possible to find something better, considering that this book seems an exhaustive review of this framework?
Thank you for your help.
Ref:
David, H. A., Nagaraja, H. N. (2003) Order Statistcs. New York: John Wiley
Dear Manuel,
I don't quite understand your question but you might find this reference useful:
Dawson, R. (2011) How significant is a boxplot outlier? Journal of Statistics Education, 19(2), www.amstat.org/publications/jse/v19n2/dawson.pdf.
Hi Peter, thank you for your answer, but it's not exactly what I need.
If X1,X2,...Xn are i.i.d. random variables distributed as a Normal with mean a and variance b2, and X(1)
Dear Manuel,
If you accept a free distribution, the variance of X(n) could be infinite. But if you assume known distribution it could have finite variance, for example for finite populations. The interest of distribution of infinite populations is more theoretic than practical. The reasons can be read in this book (in Spanish):
http://www.lulu.com/shop/mariano-ruiz-espejo/fundamentos-de-la-inferencia-estad%C3%ADstica-objetiva/paperback/product-21821572.html
Dear Manuel,
I don't remember the formula for the variance of the sample maximum of the STANDARD normal distribution.
IF you have that at hand, the solution is simple:
The connection between standard and nonstandard normal distributions is given by a linear transformation. Using your notation with a and b for mean and standard deviation,
X_i = b*Y_i + a
where Y_i are standard normal distributed. Naturally, X_(n) = b*Y_(n)+a and therefore
Var(X_(n)) = b^2 * Var(Y_(n)).
So if you know the variance Var(Y_(n)) in the standard normal case, you get the nonstandard case from this formula.
I hope this helps.
Cheers, ilja.
Hello, Manuel,
I think by ruling out extreme value theory, you're ruling out the very thing you need most. By expanding the asymptotic form for the distribution of the maximum, EVT gives asymptotically Var(X(n)) = pi2/(12 log n) for the standard normal distribution, which is accurate to within about 5% for n>10. Multiply it by your b2, and you have an estimate for your variance. A passable asymptotic estimate for E(X(n)) is (2 log n)1/2[1 - log(4 pi log n)/(4 log n)], but this approximation underestimates the deviation by more than 5% for n < 200. Multiply by your scale b and add your mean a, and you have an estimate for the expectation value.
Ref: Kendall's Advanced Theory of Statistics, Vol. I, Sixth Ed. (1994).
The answer depends on the distribution of the iid random variables X1, X2, ..., Xn.
I too think you rule out extreme value theory to your detriment. I don't know your reference on order statistics but I usually start by checking Johnson, Kotz, and Balakrishnan, Continuous Univariate Distributions for an informed if now dated knowledge survey. They also provide comprehensive tools in the initial chapters. I like Beirlant, Goegebeur, Segers and Teugels, Statistics of Extremes fora nice balance between theory and application.
( I did grow up with David's first edition of Order Statistics, and presume the decades have put much new material into the 3rd edition.)
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