Maximum of normal random variables
Puneet Gupta
How do I find the distrbution of maximum of n normal random variables
?
1) If they are independent, then it is easy to find the c.d.f of the
resultant max (F(Max <D) =F(d1<D, d2<D..). For the case of gaussian
distribution, I dont know of any closed form expression for this.
Essentially, I want to estimate the mean and variance of the MAx
distribution. Is the above distribution approximable by normal
distribution ?
2) What if the random variables are correlated ?
Thanks
--Puneet
dave fournier
Puneet Gupta wrote:
Hi,
Similar problems arise in multivariate probit discrete choice models.
The method used to esitmate the integrals is monte carlo of a particular
kind.
The components of the desired (trunacated) multivariate normal
distribution
are generated sequentially and the usual choleski decomposition of the
covariance matrix is used. In the field the method is known as the GHK
algorithm.
Dave
>
> Thanks
>
> --Puneet
--
Dave Fournier, Otter Research Ltd
PO Box 2040, Sidney, B.C. V8L 3S3, Canada
250-655-3364
email: ot...@otter-rsch.com http://otter-rsch.com
Herman Rubin
>Hi,
>How do I find the distrbution of maximum of n normal random variables
>?
>1) If they are independent, then it is easy to find the c.d.f of the
>resultant max (F(Max <D) =F(d1<D, d2<D..). For the case of gaussian
>distribution, I dont know of any closed form expression for this.
>Essentially, I want to estimate the mean and variance of the MAx
>distribution. Is the above distribution approximable by normal
>distribution ?
In the case of identical distributions, there is a limit
form which is not normal. I know of no computation of the
mean and variance for more than 2.
For different means and variances, it is approximately
normal only if the largest is dominant.
>2) What if the random variables are correlated ?
In general, it is more difficult. However, if they all
have the same mean and variance, and the same positive
correlation, one can write di = z + qi, where z and the
q's are independent normal random variables. Then for
large numbers of variables, the maximum of the qi will
be closer to a constant, and the variation will be
dominated by that of z.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Robert Israel
Herman Rubin <hru...@stat.purdue.edu> wrote:
>In article <w91re8v19b0z@legacy>, Puneet Gupta <pun...@ucsd.edu> wrote:
>>How do I find the distrbution of maximum of n normal random variables
>>?
>>1) If they are independent, then it is easy to find the c.d.f of the
>>resultant max (F(Max <D) =F(d1<D, d2<D..). For the case of gaussian
>>distribution, I dont know of any closed form expression for this.
>>Essentially, I want to estimate the mean and variance of the MAx
>>distribution. Is the above distribution approximable by normal
>>distribution ?
>In the case of identical distributions, there is a limit
>form which is not normal. I know of no computation of the
>mean and variance for more than 2.
Consider n iid r.v.'s with density f(x) and cdf F(x). The density for
their maximum is n f(x) F(x)^(n-1), so the mean is
mu(n) = int_{-infinity}^infinity n x f(x) F(x)^(n-1) dx
and the variance is
V(n) = int_{-infinity}^infinity n x^2 f(x) F(x)^(n-1) dx - mu^2.
In the case of the standard normal r.v., f(x) = 1/sqrt(2*Pi)*exp(-x^2/2),
Maple comes up with closed forms for only
mu(2) = 1/sqrt(Pi), V(2) = 1 - 1/Pi, mu(3) = 3/(2 sqrt(Pi))
but I believe V(3) = 1 - (9 - 2 sqrt(3))/(4 Pi) (at least, it matches
quite well numerically).
But of course all are easily evaluated in floating point:
n mu(n) V(n)
2, 0.56418958354776, 0.68169011381621
3, 0.84628437532164, 0.55946720379737
4, 1.0293753730039, 0.4917152368747
5, 1.1629644736405, 0.4475340690208
6, 1.2672063606115, 0.4159271089832
7, 1.3521783756069, 0.3919177761268
8, 1.4236003060453, 0.3728971432867
9, 1.4850131622092, 0.3573533263579
10, 1.5387527308352, 0.3443438232607
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2