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Chapter 6: Problem 5

Let \(Y_{1}

Short Answer

Expert verified
The maximum likelihood estimator for \(\theta\) is \(\hat{\theta} = \frac{y_1 + y_n}{2}\) and for \(\rho\) is \(\hat{\rho} = \frac{y_n - y_1}{2}\). Only \(\hat{\theta}\) is an unbiased estimator, while \(\hat{\rho}\) is biased.

Step by step solution

01

Formulate the likelihood function

First, let's write down the density function for a uniform distribution: \(f(y|\theta, \rho) = \frac{1}{2\rho}, \text{ for } \theta - \rho \leq y \leq \theta + \rho\). The joint density is the product of the individual densities, i.e., \(f(y_1, y_2, ..., y_n | \theta, \rho) = \prod_{i=1}^{n} f(y_i | \theta, \rho)\), noting that \(y_1 = \min(y_i)\) and \(y_n = \max(y_i)\).
02

Solve for maximum likelihood estimates

Next, solve the log-likelihood function for maximum likelihood estimates of \(\theta\) and \(\rho\). Since \(\theta\) lies in the interval of \(y_1\) and \(y_n\), the MLE of \(\theta\) is their average, i.e., \(\hat{\theta} = \frac{y_1 + y_n}{2}\). For \(\rho\), maximize the log-likelihood (after using constraints) to get \(\hat{\rho} = \frac{y_n - y_1}{2}\).
03

Determine if the estimates are unbiased

The estimates are unbiased if the expected values of \(\hat{\theta}\) and \(\hat{\rho}\) are equal to \(\theta\) and \(\rho\) respectively. After evaluating the expected values, one can find that \(E(\hat{\theta}) = \theta\) and \(E(\hat{\rho}) \neq \rho\). Therefore, \(\hat{\theta}\) is an unbiased estimator for \(\theta\), whereas \(\hat{\rho}\) is a biased estimator for \(\rho\).

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Most popular questions from this chapter

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 the beta distribution with \(\alpha=\beta=\theta\) and \(\Omega=\\{\theta: \theta=1,2\\} .\) Show that the likelihood ratio test statistic \(\Lambda\) for testing \(H_{0}: \theta=1\) versus \(H_{1}: \theta=2\) is a function of the statistic \(W=\) \(\sum_{i=1}^{n} \log X_{i}+\sum_{i=1}^{n} \log \left(1-X_{i}\right)\)

Let \(X_{1}, X_{2}\), and \(X_{3}\) have a multinomial distribution in which \(n=25, k=4\), and the unknown probabilities are \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\), respectively. Here we can, for convenience, let \(X_{4}=25-X_{1}-X_{2}-X_{3}\) and \(\theta_{4}=1-\theta_{1}-\theta_{2}-\theta_{3} .\) If the observed values of the random variables are \(x_{1}=4, x_{2}=11\), and \(x_{3}=7\), find the maximum likelihood estimates of \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\).

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 \(0 \leq p_{1} \leq p_{2} \leq 1\)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(\Gamma(\alpha=3, \beta=\theta)\) distribution, \(0<\theta<\infty\). Determine the mle of \(\theta\).
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