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If $X \sim Binomial (n_1, p_1)$ and $Y \sim Binomial(n_2, p_2)$ and $X,Y$ are independent, how do I find the MLE for $\phi = p_1 - p_2$?

I tried finding the joint distribution for $X,Y$ but, I didn't see how that would shed light on $\phi$ as it is the difference between the two success probabilities. I'm stuck on how to approach this problem, and would appreciate any pointers.

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1 Answer 1

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Since 𝑋 and π‘Œ are independent, the difference between the MLEs of 𝑝1 and 𝑝2 should be the MLE of πœ™.

Let y1 and y2 denote the numbers of "successes" for 𝑋 and π‘Œ. The MLEs of 𝑝1 and 𝑝2 are y1/𝑛1 and y2/n2 respectively.

Thus, the MLE of πœ™ is y1/n1 - y2/n2.

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    $\begingroup$ Do you have a citation for this statement: "Since 𝑋 and π‘Œ are independent, the difference between the MLEs of 𝑝1 and 𝑝2 should be the MLE of πœ™"? $\endgroup$
    – Krakus
    Apr 24, 2020 at 12:29
  • $\begingroup$ We can use the plug-in estimator property of the MLE. So if the MLE of X is x/n1 and for Y is y/n2, then the MLE of X-Y is x/n1 - x/n2 $\endgroup$
    – Will
    Apr 24, 2020 at 14:49

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