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Crossposting link: https://math.stackexchange.com/questions/3312349/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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  • $\begingroup$ Please don't cross-post questions. Choose one site that best fits your question. If you later decide it belongs elsewhere (e.g. if it's off-topic), flag it for migration. $\endgroup$ – Glen_b -Reinstate Monica Aug 4 '19 at 0:41

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