Question

Consider two Bernoulli distributions with unknown parameters \(p_{1}\) and \(p_{2}\). If \(Y\) and \(Z\) equal the numbers of successes in two independent random samples, each of size \(n\), from the respective distributions, determine the mles of \(p_{1}\) and \(p_{2}\) if we know that \(0 \leq p_{1} \leq p_{2} \leq 1\)

Short answer

The maximum likelihood estimate (MLE) of \(p_{1}\) and \(p_{2}\) are given as \(\hat{p_{1}} = \frac{Y}{n}\) and \(\hat{p_{2}} = \frac{Z}{n}\) respectively. If \(p_{1}\) is more than \(p_{2}\) considering the given constraint, then both \(p_{1}\) and \(p_{2}\) should be the maximum value of \(\hat{p_{1}}\) and \(\hat{p_{2}}\). So, \(\hat{p} = max(\hat{p_{1}}, \hat{p_{2}})\).

Step-by-step solution

Write down the likelihood function

For Bernoulli distributions, the likelihood function for observing the given data sample is given by \(p^{y}(1-p)^{n-y}\) where \(y\) is the number of successes and \(n\) is the total sample size. The combined likelihood function for \(Y\) key and \(Z\) key becomes \(p_{1}^{Y}(1-p_{1})^{n-Y}p_{2}^{Z}(1-p_{2})^{n-Z}\)

Log transformation

To simplify the function, we take the logarithm of the likelihood function and obtain its log-likelihood. We ignore the constants, as they won't affect the position of the maximum, yielding \(Y log(p_{1}) + (n - Y)log(1 - p_{1}) + Z log(p_{2}) + (n - Z)log(1 - p_{2})\)

Solve for MLEs

By taking derivatives of this function with respect to \(p_{1}\) and \(p_{2}\) and equating it to zero, we can solve for \(p_{1}\) and \(p_{2}\). The equations obtained are: \[ Y = n p_{1} \] and \[ Z = n p_{2} \] So the MLEs of \(p_{1}\) and \(p_{2}\) are \(\hat{p_{1}} = \frac{Y}{n}\) and \(\hat{p_{2}} = \frac{Z}{n}\) respectively. But we must remember that \(0 \leq p_{1} \leq p_{2} \leq 1\). If value of \(p_{1}\) is more than \(p_{2}\), then we can't accept this result. In this case, both \(p_{1}\) and \(p_{2}\) should be take the same value which is \(\hat{p} = max(\hat{p_{1}}, \hat{p_{2}})\)