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

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 $X_2$?

Answer 1

[Akira Agata]

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);

[xosro]

Thank you, Mr. Akira Agata Can I ask another question like this?

I have three random variables Y, X1 and X2 which X1 and X2 are independent variables and Y is an explanatory variable. I want to estimate a and b for the equation: Y=aX1+bX2 subject to a+b=1

Just, do not use:

Y=aX1+(1-a)X2

Because I have a sequence of independent variables X1, ..., Xn

Thanks if you guide me

[Akira Agata]

I'm a little bit confused your condition.

- Condition1: Y, X1 and X2 should be fitted to Y = a*X1 + b*X2.

- Condition2: a + b should be 1.

- Condition3: Y = a*X1 + (1-a)*X2 should NOT be used.

If my understainding is correct, these 3 conditions can not be satisfied at the same time. So, please tell us more details on your assumed condition.

[Torsten]

Use lsqlin.

Best wishes

Torsten.

[Walter Roberson]

I think the idea is that there is a sequence of X values and associated coefficients, and the sum of the coefficients must be 1.

[Torsten]

Yes, I think it's a linear fit of

l1*X1+l2*X2+...+ln*Xn

against

Y

under the constraint that

li >= 0 
sum(li) = 1

[xosro]

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.

[Torsten]

You shouldn't do that since it makes your problem nonlinear in the l's.

You should work with

Y = l1*X1+...+ln*Xn

sum(li) = 1
li >= 0

and use lsqlin to estimate the li's.

Best wishes

Torsten.