Let $Z_i=\theta+\frac{X_i}{Y_i}$ where random vector $(X_i,Y_i)$ distributed uniformly in unit circle with center at $(0,0)$. I need to construct consistent estimator for parameter $\theta$ for sample $Z_1,..,Z_n$ I haven’t got an idea how to do this. The solution should Be logical in some sense. I believe that probability density function for this random vector is $\frac{1}{\pi}$ whenever my pair is in the circle and zero else. So from this pdf I can evaluate pdf of $X$ and $Y$ by integrating. And I don’t know what should I do after
Consistent estimator. Statistics
probabilitystatistics
Best Answer
Let $W = (W_1,W_2) \sim N(0,I_2)$, that is, the two coordinates are independent $N(0,1)$. Then, $W / \|W\|$ is uniformly distributed on the unit circle. Multiplying by a proper scalar random variable $R$, we can make $R(W/\|W\|)$ uniformly distributed in the unit sphere. That is, $(X,Y)$ will have the same distribution as $R(W/\|W\|)$ and hence $X/Y$ will have the same distribution as $$ \frac{RW_1/\|W\|}{R W_2/ \|W\|} = \frac{W_1}{W_2} \sim \text{??}(0,1) $$ The ?? is a well-known distribution. So, you are dealing with a heavy tailed location family. A robust estimator of the mean, such as the median, can give you a consistent estimator. (There are other choices). You can try to prove the median is consistent. It would be for any continuous location family.