how can i estimate parameters of two independent gamma distributed variables with one same parameter in matlab?

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xosro 2018-3-8

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评论： Torsten 2018-3-9

Suppose we have two independent random variables $X_1$ and $X_2$ where $X_1 \sim (\alpha_1, \beta)$ and $X_2 \sim (\alpha_2, \beta)$. how can i estimate three parameters $(\alpha_1, \alpha_2, \beta)$ from given data for $X_1$ and $X_2$?

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Akira Agata 2018-3-9

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I think one possible way to do this is to use maximum likelihood estimation method, like:

% Sample data
X1 = gamrnd(2,10,1000,1);
X2 = gamrnd(5,10,1000,1);
% Fit each data to Gamma distribution
pd1 = fitdist(X1,'Gamma');
pd2 = fitdist(X2,'Gamma');
% Assume beta = (beta1 + beta2)/2, and estimate alpha1 and alpha2
beta = (pd1.b + pd2.b)/2;
alpha1 = mle(X1,'pdf',@(X1,alpha1) gampdf(X1,alpha1,beta),'start',pd1.a);
alpha2 = mle(X2,'pdf',@(X2,alpha2) gampdf(X2,alpha2,beta),'start',pd2.a);

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xosro 2018-3-9

编辑：xosro 2018-3-9

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I'm sorry. As it is mentioned above (Walter Roberson and Torsten), conditions 1 and 2 are correct. I was trying to describe the question in the form of simple, But it is possible that b is replaced with (1-a) but this is not used in general form (l1*X1+l2*X2+...+ln*Xn).

I may use the following form:

y=l(1)/sum(l)*X1+...+l(n)/sum(l)*Xn

I should try this.