# Math Help - Distribution for the max of two (correlated) standard normal variables

1. ## Distribution for the max of two (correlated) standard normal variables

Dear all,

I have been struggling with this apparently hard question (my background is on medical sciences):

There are two standard normal variables

X1 ~N (0,1)
X2~ N(0,1)

with correlation = rho.

I would like to obtain the probability density function for

X3 = max(abs(X1),abs(X2))

where max = maximum and
abs = absolute value

I have spent considerable time on this during the last few weeks, and could not find an adequate reference to solve this issue.

If you have any tips (references, solutions, websites) I really appreciate it.

All the best,

Alonso

2. ## Re: Distribution for the max of two (correlated) standard normal variables

Originally Posted by Alonso
Dear all,

I have been struggling with this apparently hard question (my background is on medical sciences):

There are two standard normal variables

X1 ~N (0,1)
X2~ N(0,1)

with correlation = rho.

I would like to obtain the probability density function for

X3 = max(abs(X1),abs(X2))

where max = maximum and
abs = absolute value

I have spent considerable time on this during the last few weeks, and could not find an adequate reference to solve this issue.

If you have any tips (references, solutions, websites) I really appreciate it.

All the best,

Alonso
This citation appears to be to what you need:

Exact distribution of the max/min of two Gaussian random variables

If rho were a known value I would generate an empirical distribution for the maximum by generating a large random sample of the RVs and using the sample of maximums of these as as my empirical distribution.

CB

3. ## Re: Distribution for the max of two (correlated) standard normal variables

Many thanks, CB!

I believe that the distribution I am looking for is the distribution of the max of two half-normal distributions, since my objective is to select the max(abs(X1),abs(X2)), regardless of the signal.

the empirical approach you suggest is very interesting for usual scenarios, but in my case I am working on the tail of the distribution, with alpha levels equal or smaller to 10^-8. So, empirical distributions are not efficient (i.e. time-consuming).

Thanks again!

Alonso

4. ## Re: Distribution for the max of two (correlated) standard normal variables

Originally Posted by Alonso
Many thanks, CB!

I believe that the distribution I am looking for is the distribution of the max of two half-normal distributions, since my objective is to select the max(abs(X1),abs(X2)), regardless of the signal.

the empirical approach you suggest is very interesting for usual scenarios, but in my case I am working on the tail of the distribution, with alpha levels equal or smaller to 10^-8. So, empirical distributions are not efficient (i.e. time-consuming).

Thanks again!

Alonso
If you are interested in the distribution that far our then things may well be simpler since the probability of both absolute values being very far from $1$ is small since you are effectively dealing with the probability that $P(|X1|>a \text{ or }|X2|>a)$ That is for large $a$

$P(\max(|X1|,|X2|)>a)\approx P( (|X1|>a) \text{ or }(|X2|>a) )$

and there are a couple of other approximations I could think of that might work.

CB

5. ## Re: Distribution for the max of two (correlated) standard normal variables

yes, CB!, the approximation you suggestion work fine for several scenarios. Thank you very much. I was told to seek for the max(X1^2, X2^2), which is the max of two chi-square statistics with 1 df. If you have any further suggestions, please, let me know.
Thanks again!

6. ## Re: Distribution for the max of two (correlated) standard normal variables

Originally Posted by Alonso
yes, CB!, the approximation you suggestion work fine for several scenarios. Thank you very much. I was told to seek for the max(X1^2, X2^2), which is the max of two chi-square statistics with 1 df. If you have any further suggestions, please, let me know.
Thanks again!
Originally Posted by CaptainBlack
If you are interested in the distribution that far our then things may well be simpler since the probability of both absolute values being very far from $1$ is small since you are effectively dealing with the probability that $P(|X1|>a \text{ or }|X2|>a)$ That is for large $a$

$P(\max(|X1|,|X2|)>a)\approx P( (|X1|>a) \text{ or }(|X2|>a) )$

and there are a couple of other approximations I could think of that might work.

CB
Originally Posted by Alonso
yes, CB!, the approximation you suggestion work fine for several scenarios. Thank you very much. I was told to seek for the max(X1^2, X2^2), which is the max of two chi-square statistics with 1 df. If you have any further suggestions, please, let me know.
Thanks again!
Thinking about the correlation may mean that we have to refine this a bit:

$P(\max(|X_1|,|X_2|)>a)= P( (|X_1|>a) \text{ or }(|X_2|>a) ) \\ \\ \phantom{xxxxxxxxxxxxxxxxxxxxxxx} -P(|X_1|>a\text{ and }|X_2|>a))$

But it is the latter tha is tricky to calculate?

But that does mean that we can say:

$P(\max(|X_1|,|X_2|)>a)\le P( (|X_1|>a) \text{ or }(|X_2|>a) )$

(It is also the correlation that might be the problem with $\max(X_1^2, X_2^2)$ )

CB

7. ## Re: Distribution for the max of two (correlated) standard normal variables

Hm, that's true. I believe that that approximation may be suitable in some instances, because the correlation is very low. Your comments did help me a lot, and I thank you very much for your time, CB.