Thread: Convolution

Hi all,
I am trying to figure the following out:
Let Z,X be random variables and Y = X + Z;
For a known f(X) and g(Z), and a given data set of Y it is required to find the parameters of f and g functions.
I wonder if this is at all possible. If f and g are Normal then the distribution of Y is also Normal with the sum of means and variances of f and g. In this case I presume it is not possible to derive the mean of X and Z separately. Is this true? and if it is is this true for any arbitrary pair of functions f and g.

With thanks,
R2

Originally Posted by Iskatel

Hi all,
I am trying to figure the following out:
Let Z,X be random variables and Y = X + Z;
For a known f(X) and g(Z), and a given data set of Y it is required to find the parameters of f and g functions.
I wonder if this is at all possible. If f and g are Normal then the distribution of Y is also Normal with the sum of means and variances of f and g. In this case I presume it is not possible to derive the mean of X and Z separately. Is this true? and if it is is this true for any arbitrary pair of functions f and g.

With thanks,
R2

Are Z and X independent?

Yes, X and Z are independent. Say, f - lognormal and g - Weibull or gamma. Is it possible to find unique parameters of these two distributions based on the observations of Y, say, using MLE?!

Originally Posted by Iskatel

Yes, X and Z are independent. Say, f - lognormal and g - Weibull or gamma. Is it possible to find unique parameters of these two distributions based on the observations of Y, say, using MLE?!

Honestly, I'm not sure...

The only thing I can think of using to do this is:

OK, thank you for having a look at it.

I just noticed the densities of X and Z are known. Then you can definitely use MGF. Simply factorize into the product of the two known MGF's and your parameters will be apparent.

Of course I'm assuming the generating functions exist...

Originally Posted by Anonymous1

I just noticed the densities of X and Z are known. Then you can definitely use MGF. Simply factorize into the product of the two known MGF's and your parameters will be apparent.

Of course I'm assuming the generating functions exist...

Thank you for your suggestion. I tried using Weibull for both X and Z. It looks like it is not possible to find the parameters of f and g uniquely based on observations of Y. More like trying to solve 5 = x + z for x and z.

Originally Posted by Iskatel

Thank you for your suggestion. I tried using Weibull for both X and Z. It looks like it is not possible to find the parameters of f and g uniquely based on observations of Y. More like trying to solve 5 = x + z for x and z.

What information do you have about Can you fit the data If so, the above task should be easy enough.

Try it, I don't see why it wouldn't work. You know, as long as is integrable...

Originally Posted by Anonymous1

What information do you have about Can you fit the data If so, the above task should be easy enough.

Try it, I don't see why it wouldn't work. You know, as long as is integrable...

The information on Y is in the form of a set of observation, say: Y = {24, 15, 78 ...}.
To be honest I am not entirely sure how to fit the data on M[Y]=M[X]M[Z]. And my main issue with this is if the convolution of two Normal distributions is also Normal then you would fit the data using MLE for f(y)-Normal(m[Y],Var[Y]). Since m[Y]=m[X]+m[Z] there is no way to deduce unique m[X] or m[Z], isn't it?! Same for the variance. Hence, for other distributions similar problem may hold true?!

Originally Posted by Iskatel

The information on Y is in the form of a set of observation, say: Y = {24, 15, 78 ...}.
To be honest I am not entirely sure how to fit the data on M[Y]=M[X]M[Z]. And my main issue with this is if the convolution of two Normal distributions is also Normal then you would fit the data using MLE for f(y)-Normal(m[Y],Var[Y]). Since m[Y]=m[X]+m[Z] there is no way to deduce unique m[X] or m[Z], isn't it?! Same for the variance. Hence, for other distributions similar problem may hold true?!

You know X and Z to be normal? Then clearly, we can fit Y with a normal distribution. Determine its mean and variance empirically...

Originally Posted by Anonymous1

You know X and Z to be normal? Then clearly, we can fit Y with a normal distribution. Determine its mean and variance empirically...

Surely we can fit the data to Y as we know it is Normal because X and Z are Normal. And we will get f(y) which is a Normal distribution with mean - mean[Y] and variance-var[Y]. However, I want to find the means and variances of Z and X.
Thanks for trying to help.

Originally Posted by Iskatel

Surely we can fit the data to Y as we know it is Normal because X and Z are Normal. And we will get f(y) which is a Normal distribution with mean - mean[Y] and variance-var[Y]. However, I want to find the means and variances of Z and X.
Thanks for trying to help.

Okay I see the issue now...

Well you have a bound on the sum of your means, and you can standardize one of them to draw some inference on the other...

Obviously there are an infinite number of possible combinations for the parameters on X and Z, though.

Originally Posted by Anonymous1

Okay I see the issue now...

Well you have a bound on the sum of your means, and you can standardize one of them to draw some inference on the other...

Obviously there are an infinite number of possible combinations for the parameters on X and Z, though.

My main question was whether this generalises to all combinations of different forms of f(X) and f(Z). Say if the resulting f(Y) is bimodal, can we find unique parameters for f(X) and f(Z).