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Let $(X_1, Y_1), (X_2, Y_2), \dots , (X_n, Y_n)$ be a random sample of size $n$ from the continuous distribution with joint pdf $f_{X, Y}(x, y|\theta) = \frac{1}{\theta y}e^{-\frac{x}{\theta y}}I(x)_{(0,\infty)}I(y)_{(0, 1)}I(\theta)_{(0,\infty)}$.

(1) Find a complete and sufficient statistic for $\theta$.

(2) Find the maximum likelihood estimator for $\theta$.

(3) Find the maximum likelihood estimator for $P(X < Y )$.

(4) Let $V = \frac{X}{\theta}$ and $W = Y$. Show the joint distribution of (V;W) does not depend on $\theta$.

(5) De fine the statistic $S((X_1, Y_1), \dots, (X_n, Y_n)) = \frac{\sum^n_{i=1}X_iY_i}{\sum^n_{j=1}X_j}$. Show that $S((X_1, Y_1), \dots, (X_n, Y_n))$ is an ancillary statistic for the model $f_{X, Y}(x, y|\theta)$, taking note that I cannot claim I have a scale parameter family when I have a joint distribution. Thus, to show that S is ancillary, one has to show that its distribution does not depend on $\theta$.

For (1), I am showing sufficient by using Factorization Theorem, but I am having a bit of trouble with the completeness.

For (2), I think I am getting the wrong information trying to take the partial with respect to x and y and setting them equal to $0$ and then trying to find the determinant greater than $0$

For (3), (4) and (5) I am very confused on where to even begin.

Any assistance is greatly appreciated.

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1 Answer 1

I believe for Part (1), you can just show that it's an exponential family, and then by the exponential family theorem, then the $\sum\limits_{i=1}^nt_{1}(X)$ is your complete and sufficient statistic.

For your part (2), you have to take the natural log of your likelihood function, and take the partial with respect to your parameter, then solve for your parameter, and that is your MLE of $\theta$. You should get $\frac{1}{n}*\sum\limits_{i=1}^n\frac{X_{i}}{Y_{i}}$

For part (3), you have to find a g($\theta$) = $Pr(X<Y)$, More complete answer here

For part (4), do a transformation, by taking your $g^{-1}(x)$ and your $g^{-1}(y)$ and plugging what you get into the original pdf and taking the determinant of the Jacobian matrix (which is just $\theta$), and cancelling, and you get a pdf of V's and W's that is free of $\theta$.

For part (5), see the link above for a more concise and complete response for your part (5).

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