Results 1 to 13 of 13

Math Help - Convolution

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    8

    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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    Quote Originally Posted by Iskatel View Post
    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?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    8
    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?!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    Quote Originally Posted by Iskatel View Post
    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:

    M_Y = M_X \times M_Z
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    8
    OK, thank you for having a look at it.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    I just noticed the densities of X and Z are known. Then you can definitely use MGF. Simply factorize E(e^{Yt}) into the product of the two known MGF's and your parameters will be apparent.

    Of course I'm assuming the generating functions exist...
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Apr 2010
    Posts
    8
    Quote Originally Posted by Anonymous1 View Post
    I just noticed the densities of X and Z are known. Then you can definitely use MGF. Simply factorize E(e^{Yt}) 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.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    Quote Originally Posted by Iskatel View Post
    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 Y? 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 f_Y is integrable...
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Apr 2010
    Posts
    8
    Quote Originally Posted by Anonymous1 View Post
    What information do you have about Y? 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 f_Y 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?!
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    Quote Originally Posted by Iskatel View Post
    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...
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Apr 2010
    Posts
    8
    Quote Originally Posted by Anonymous1 View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Super Member Anonymous1's Avatar
    Joined
    Nov 2009
    From
    Big Red, NY
    Posts
    517
    Thanks
    1
    Quote Originally Posted by Iskatel View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Newbie
    Joined
    Apr 2010
    Posts
    8
    Quote Originally Posted by Anonymous1 View Post
    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).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. convolution
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 25th 2010, 04:38 AM
  2. Convolution(sum) of X1~U(0,1) and X2~exp(2)
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 7th 2010, 11:57 AM
  3. Convolution
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 18th 2009, 10:07 AM
  4. Help with convolution??
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 14th 2009, 03:56 PM
  5. Convolution
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: May 29th 2007, 05:32 AM

Search Tags


/mathhelpforum @mathhelpforum