Using common samples to numerically estimate pairwise equality of three random variables

Question

Let $X,Y,Z$ be three discrete random variables which I can numerically sample. I need to numerically estimate the probability that $X=Y$ and the probability that $X=Z$. I would like to know whether the algorithm that I describe below is suited to this goal. In particular, I would like to know whether it is acceptable to use the same samples from $X$ to estimate both quantities.

My current algorithm generates $n$ independent samples of each random variable. That is, I generate three vectors: $$(x_1,\ldots,x_n)\\ (y_1,\ldots,y_n)\\ (z_1,\ldots,z_n)$$ Associated to these vectors, I define two counting variables as follows, $$c_{XY}:=\sum_{i=1}^n\displaystyle\delta_{x_iy_i}\qquad c_{XZ}:=\sum_{i=1}^n\delta_{x_iz_i} $$ where $\delta_{ij}$ is the Kronecker delta. Finally, I estimate my probabilities in the following way: $$P(X=Y)\approx \frac{c_{XY}}{n}\qquad P(X=Z)\approx \frac{c_{XZ}}{n}$$.

I would greatly appreciate any feedback.

Answer 1

Your two estimators $c_{XY}/n$ and $c_{XZ}/n$ are unbiased estimators of $P(X=Y)$ and $P(X=Z)$ respectively. They are not independent, however. Whether that is "acceptable" might depend on what you're using them for.