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let $X_{1}$, $X_{2}$, $X_3$, $X_{4}$ be iid bernoulli rvs with $\mathbb{P}(0)=0.5$, $\mathbb{P}(1)=0.5$.

$Y_{1} = X_{1}+X_{2}+X_{3}$ and $Y_{2}=X_{1}+X_{2}+X_{4}$

$Y_{1}$, $Y_{2}$ are dependent binomial rvs by definition. I need to find max likelihood estimate of $X_{1}$ given I have observed $Y_{1}$ and $Y_{2}$.

how should I go about it?

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up vote 2 down vote accepted

There are two different cases.

If the observed values differ by $1$, this implies known values for $X_3$ and $X_4$, and the lower of the two observed values is the value of $X_1+X_2$. If this is $0$ or $2$, it implies the value $0$ or $1$, respectively, for $X_1$; if it is $1$, both values of $X_1$ are equally likely.

On the other hand, if the observed values are equal, this implies that the values of $X_3$ and $X_4$ are equal. Given that they are equal, the problem effectively becomes finding the maximum likelihood estimate for $X_1$ given $X_1+X_2+X_3$. If the observed value is $0$ or $3$, that determines $X_1$ with value $0$ or $1$, respectively. If the observed value is $1$ or $2$, then the value of $X_1$ is $0$ or $1$, respectively, with probability $2/3$, and $1$ or $0$, respectively, with probability $1/3$, so the maximum likelihood estimate in this case is $0$ or $1$, respectively.

In summary, the maximum likelihood estimate for $X_1$ given the two observed values is given by

$$ \begin{array}{c|cccc} &0&1&2&3\\\hline 0&0&0&-&-\\ 1&0&0&=&-\\ 2&-&=&1&1&\\ 3&-&-&1&1 \end{array} $$

where $-$ indicates a case that cannot occur and $=$ indicates that both values of $X_1$ are equally likely.

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The usual method of finding mle involves solving score equations.how can we formulate this problem in terms of likelihood function. This would require the determination of pdf for Y1 and Y2 expressed as a function of X1(parameter that needs to be estimated) –  xplore29 Nov 8 '12 at 23:36
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@xplore29: I'm not sure in which context this is the "usual" method. What I did is how I think this problem is most efficiently solved. If you're looking for a solution using one particular method, you should have mentioned that in the question. In that case I would suggest that you ask a new question (linking to this one) asking specifically how to obtain the maximum likelihood estimate using the method of your choice. (In fact you did the opposite here; you asked "how should I go about it?", not "I want to go about it in this particular manner, how can I do that?".) –  joriki Nov 9 '12 at 6:21
    
thankyou for guiding me.....I was actually trying to find a generalized solution. I ll post a more targeted question in new post and ll link it to this one. I appreciate your suggestions. –  xplore29 Nov 10 '12 at 4:17
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