Help: how to estimate this A

[stony]

Suppose real valued data X1, X2, ..., Xn are from a mixture distribution f(x) = (f1(x) + f2(x))/2, the component distributions f1(x) and f2(x) meet the following conditions:

f1(x) = f2(-x);
f1(x) = f2(x)exp(x/A);

can we estimate A based on X1, X2, ..., Xn? Many thanks for any hints

[BGM]

Seemingly you have a functional equation in :



Suppose this identity only holds for , then a trivial solution is the exponential distribution



and the resulting mixture distribution is a Laplace distribution.

In that case the maximum likelihood estimator for is just the half of the sample mean of the absolute values of the data.

[stony]

Thanks for your discussion. I understand this. But what if f1(x) and f2(x) are other distributions? The identity may still holds but the ML estimator derived for the Laplace distribution may not work. Is there any way to derive without assuming the specific component distributions?

[Dason]

Is this homework or do you have a real application for this?