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Chapter 6: Problem 7

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

Expert verified
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

01

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 \(p_{1}^{Y}(1-p_{1})^{n-Y}p_{2}^{Z}(1-p_{2})^{n-Z}\)
02

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 \(Y log(p_{1}) + (n - Y)log(1 - p_{1}) + Z log(p_{2}) + (n - Z)log(1 - p_{2})\)
03

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 \(\hat{p} = max(\hat{p_{1}}, \hat{p_{2}})\)

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Most popular questions from this chapter

Consider a location model $$X_{i}=\theta+e_{i}, \quad i=1, \ldots, n$$ where \(e_{1}, e_{2}, \ldots, e_{n}\) are iid with pdf \(f(z)\). There is a nice geometric interpretation for estimating \(\theta .\) Let \(\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)^{\prime}\) and \(\mathbf{e}=\left(e_{1}, \ldots, e_{n}\right)^{\prime}\) be the vectors of observations and random error, respectively, and let \(\mu=\theta 1\) where 1 is a vector with all components equal to one. Let \(V\) be the subspace of vectors of the form \(\mu_{i}\) i.e, \(V=\\{\mathbf{v}: \mathbf{v}=a \mathbf{1}\), for some \(a \in R\\} .\) Then in vector notation we can write the model as $$\mathbf{X}=\boldsymbol{\mu}+\mathbf{e}, \quad \boldsymbol{\mu} \in V$$

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Bernoulli distribution with parameter \(p .\) If \(p\) is restricted so that we know that \(\frac{1}{2} \leq p \leq 1\), find the mle of this parameter.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\mu, \sigma^{2}\right)\). (a) If the constant \(b\) is defined by the equation \(\operatorname{Pr}(X \leq b)=0.90\), find the mle of \(b\). (b) If \(c\) is given constant, find the mle of \(\operatorname{Pr}(X \leq c)\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pmf \(p(x ; \theta)=\theta^{x}(1-\theta)^{1-x}, x=0,1\), where \(0<\theta<1 .\) We wish to test \(H_{0}: \theta=1 / 3\) versus \(H_{1}: \theta \neq 1 / 3\) (a) Find \(\Lambda\) and \(-2 \log \Lambda\). (b) Determine the Wald-type test. (c) What is Rao's score statistic?

A machine shop that manufactures toggle levers has both a day and a night shift. A toggle lever is defective if a standard nut cannot be screwed onto the threads. Let \(p_{1}\) and \(p_{2}\) be the proportion of defective levers among those manufactured by the day and night shifts, respectively. We shall test the null hypothesis, \(H_{0}: p_{1}=p_{2}\), against a two-sided alternative hypothesis based on two random samples, each of 1000 levers taken from the production of the respective shifts. Use the test statistic \(Z^{*}\) given in Example \(6.5 .3\). (a) Sketch a standard normal pdf illustrating the critical region having \(\alpha=0.05\). (b) If \(y_{1}=37\) and \(y_{2}=53\) defectives were observed for the day and night shifts, respectively, calculate the value of the test statistic and the approximate \(p-\) value (note that this is a two-sided test). Locate the calculated test statistic on your figure in Part (a) and state your conclusion. Obtain the approximate \(p\) -value of the test.
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