# Bias of Pearson correlation estimator of two Bernoulli variables

Suppose we have two correlated Bernoulli random variables, $$X_j$$ and $$X_k$$, whose marginal success probabilities are $$p_j$$ and $$p_k$$, and their correlation is charaterized by parameter $$D$$ such that the joint probability of two successes is $$p_jp_k+D$$, one success and failure is $$p_j(1-p_k)-D$$ or $$(1-p_j)p_k-D$$ and two failures is $$(1-p_j)(1-p_k)+D$$. The Pearson correlation between $$X_j$$ and $$X_k$$ will be denoted by $$r_{jk}$$. Here $$p_j,p_k,D,r_{jk}$$ are all unknown population values.

Now I have $$N$$ observations of $$X_j$$ and $$X_k$$. The $$i$$-th observation will be denoted by $$(X_{ij},X_{ik})$$. First I estimated $$p_j$$ and $$p_k$$ by $$\tilde{p}_j=(1/N)\sum_{i=1}^N X_{ij}$$ and $$\tilde{p}_k=(1/N)\sum_{i=1}^N X_{ik}$$. Then I standardize $$X_j$$ and $$X_k$$ by $$Y_j=\dfrac{X_{ij}-\tilde{p}_j}{\sqrt{\tilde{p}_j(1-\tilde{p}_j)}}$$ and $$Y_k=\dfrac{X_{ik}-\tilde{p}_k}{\sqrt{\tilde{p}_k(1-\tilde{p}_k)}}$$.

Now I estimate $$r_{jk}$$ using the estimator $$\tilde{r}_{jk}=(1/N)\sum_{i=1}^NY_{ij}Y_{ik}$$.

My questions are: Is the estimator $$\tilde{r}_{jk}$$ biased? If yes, what is the relation between $$\tilde{r}_{jk}$$ and $$r_{jk}$$? (What is the bias in expression?)

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