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I am working on the following problem from a statistics qualifying exam. I have attempted progress at the 3 parts but do not feel certain, or satisfied, with my work. I am not sure if $\textit{I}$ am making errors or if there are typos in the problem to begin with. Here is the set-up:

Suppose $X$ and $Y$ are independent exponentially distributed random variables with scale parameters $\lambda$ and $\mu$ respectively, i.e. $f(x;\lambda)=\frac{1}{\lambda}\mathrm{exp}(-x/\lambda)$ and similarly for $Y$. Let $Z=\mathrm{min}(X,Y)$, and $W=1$ when $Z=X$ and $W=0$ otherwise.

I need to do the following: (1) Find the joint distribution of $Z$ and $W$. (2) Prove that $Z$ and $W$ are independent. (3) In the case that $(X,Y)$ are not observable but instead $(Z_1,W_1),\ldots,(Z_n,W_n)$ are observed (i.e. in a random sample of size n$\geq$2), find the maximum likelihood estimates of $(\lambda,\mu)$ in terms of $\bar{Z}$ and $\bar{W}$, the sample averages. Then determine if $\hat{\lambda}_{MLE}^{-1}$ is unbiased for $\lambda^{-1}$.

I have determined the following:

(i) $Z$ is exponentially distributed with scale parameter $\frac{\lambda\mu}{\lambda+\mu}$.

(ii) $W=\mathcal{1}_{X\leq Y}$.

(iii) The joint p.d.f. of $X$ and $Y$ is $f(x,y;\lambda,\mu)=\frac{1}{\lambda\mu}\mathrm{exp}(-x/\lambda-y/\mu)$.

For the joint distribution, I have computed

$\mathbb{P}(Z\leq t,W=1)=\int_{0}^{t}\int_{0}^{y}f(x,y;\lambda,\mu)\,\mathrm{d}x\,\mathrm{d}y=1-\mathrm{exp}(-t/\mu)+\frac{\lambda}{\lambda+\mu}(\mathrm{exp}(-t(\frac{1}{\lambda}+\frac{1}{\mu}))-1)$, $\mathbb{P}(Z\leq t,W=0)=1-\mathrm{exp}(-t/\lambda)+\frac{\mu}{\lambda+\mu}(\mathrm{exp}(-t(\frac{1}{\lambda}+\frac{1}{\mu}))-1)$.

However, this leads me to the conclusion that $Z$ and $W$ are not independent since I can compute $\mathbb{P}(Z\leq t)=1-\mathrm{exp}(-t(\frac{1}{\lambda}+\frac{1}{\mu}))$ and $\mathbb{P}(W=1)=\frac{\mu}{\lambda+\mu}$. (That is, multiplying things out I find $\mathbb{P}(Z\leq t,W=1)\neq\mathbb{P}(Z\leq t)\mathbb{P}(W=1))$. I suspect I have not properly computed the joint distribution but am unsure how to check whether it is correct.

Secondarily, when it comes to finding the M.L.E., I know that if I observe $\bar{W}$, then $n\bar{W}$ times we have $X\leq Y$ (so $Z=X$) and the other $n(1-\bar{W})$ times we have $Y\leq X$ (so $Z=Y$). Using this, I write the likelihood function in terms of $\bar{Z}$ and $\bar{W}$ as

$L(\lambda,\mu;\bar{Z},\bar{W})=(\lambda\mu)^{-n}\mathrm{exp}(-n^2\bar{Z}\bar{W}/\lambda)\mathrm{exp}(-n^2\bar{Z}(1-\bar{W})/\mu)$.

From this I derived that $\hat{\lambda}_{MLE}=n\bar{Z}\bar{W}$ and $\hat{\mu}_{MLE}=n\bar{Z}(1-\bar{W})$. But now I am certainly unsure how to compute $\mathbb{E}\hat{\lambda}_{MLE}^{-1}$. (If it is a typo, I can compute $\mathbb{E}\hat{\lambda}_{MLE}$.)

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I think you are correct saying your problem is that you may have miscalcuated the joint distribution. I would change the order of integration and the limits to

$$\mathbb P(Z < t, W=1) = \mathbb P(X < t, Y > X) = \int_{x=0}^{t}\int_{y=x}^{\infty}f(x,y;\lambda,\mu)\,\mathrm{d}y\,\mathrm{d}x = \frac{\mu}{\lambda+\mu}\left(1 - e^{-t\left(\frac1\lambda+\frac1\mu\right)}\right)$$

and similarly

$$\mathbb P(Z < t, W=0) = \mathbb P(Y < t, X > Y) = \int_{y=0}^{t}\int_{x=y}^{\infty}f(x,y;\lambda,\mu)\,\mathrm{d}x\,\mathrm{d}y = \frac{\lambda}{\lambda+\mu}\left(1 - e^{-t\left(\frac1\lambda+\frac1\mu\right)}\right)$$

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