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I have a random sample of size n that are distributed as Bernoulli random variables with parameter $p$. Given

$$v=\frac{p}{1-p}$$ we are asked to find the the MLE of $v$, say $\hat{v}$. Further, we are told that an unbiased estimator of $v$ does not exist, so using $\hat{v}$, we are asked to find the bias of $v$. Here's what I've done so far:

Note that $$v=\frac{p}{1-p}=\frac{1}{1-p} -1$$ $$\implies v+1=\frac{1}{1-p}$$ $$\implies 1-\frac{1}{1+v}=p$$ $$\implies p=\frac{v}{1+v}$$ $$\implies 1-p=\frac{1}{1+v}$$ Now, let $L(p|\boldsymbol{x})=\Pi_{i=1}^n f_i(x_i|p)$ denote the likelihood function of $p$. Then have $$L(p|\boldsymbol{x})=p^t(1-p)^{n-t}$$ $$\implies L(p|\boldsymbol{x})=(\frac{v}{1+v})^t(\frac{1}{1+v})^{n-t}$$ $$\implies L(p|\boldsymbol{x})=v^t(\frac{1}{1+v})^n$$ where $t=\sum X_i$. Now let $log(L(p|\boldsymbol{x}))=l(p|\boldsymbol{x})$. Then we have $$l(p|\boldsymbol{x})=t\ln(v) - n\ln(1+v)$$ $$\implies \frac{d\;l(p|\boldsymbol{x})}{dv}=\frac{t}{v}-\frac{n}{1+v}$$ Setting the derivative equal to 0, we obtain $$\frac{t}{v}-\frac{n}{1+v}=0$$ $$\implies t(1+v) - nv = 0$$ $$\implies t = v(n-t)$$ $$\implies \hat{v}=\frac{t}{n-t}$$ Taking the second derivative of our log-likelihood with respect to $v$ yields $\hat{v}=\frac{t}{n-t}$ as our MLE. Now, Bias is defined for any estimator of a parameter $\tau(\theta)$, say $W(\boldsymbol{X})$, to be $$E[W(\boldsymbol{X})]-\tau(\theta)$$ In our case, we're dealing with odds ratio. My question: how do we find the expectation of $\hat{v}$? I've tried using approximation formulas but it leads me to obtain zero bias.

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This looks like a bit of a trick question. There is a non-zero chance that $\sum X_i=n$, in which case $\frac{t}{n-t}$ is infinite, so the expectation (and thus bias) is also infinite.

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  • $\begingroup$ I apologize for taking so long to give this a check mark. It's been busy few weeks. You are absolutely right, sir. We look at the actual expectation, plug in n, and boom we diverge to infinity. I was working waaaaay too hard at this point and very much overthought this question. $\endgroup$ – Savage Henry Oct 7 '15 at 2:54
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You can use a taylor expansion of $v=v(p)$ at $p_{0}$, i.e.,$$v(p)=v(p_{0})+\dot{v}(p_{0})(p-p_{0})+\frac{\dot{v}(p_{0})}{2}(p-p_{0})^{2}+\mathcal{R}_{3}$$ When we take $p$ at the MLE $\hat{p}=\frac{\sum x_{i}}{n}$, $\hat{v}=v(\hat{p})$ (by invariability of MLE), we have $$\mathbf{E}(\hat{v})-v(p_{0})=\frac{1}{(1-p_{0})^{3}}\mathbf{Var}(\hat{p})$$ Obviously an unbiased estimator of $v$ does not exist

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