/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Let \(X_{1}, X_{2}\) be a random... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 1

Let \(X_{1}, X_{2}\) be a random sample from a Cauchy distribution with pdf $$ f\left(x ; \theta_{1}, \theta_{2}\right)=\left(\frac{1}{\pi}\right) \frac{\theta_{2}}{\theta_{2}^{2}+\left(x-\theta_{1}\right)^{2}}, \quad-\infty

Short Answer

Expert verified
The solution involves applying Bayes' theorem to obtain the posterior pdf, evaluating the posterior pdf for given values, and determining the maximum posterior pdf. Computer programs such as Python or R can be utilized to find the point of maximum.

Step by step solution

01

Formulate the posterior pdf

To find the pdf, Bayes鈥� theorem can be used. Given that the sample from the Cauchy distribution has a likelihood \(f\left(x_{1},x_{2} |\theta_{1}, \theta_{2} \right)\) and the prior is \(h\left(\theta_{1}, \theta_{2} \right)\propto 1\), the posterior pdf can be given as: \[f\left(\theta_{1},\theta_{2}|x_{1},x_{2}\right)\propto f\left(x_{1},x_{2}|\theta_{1},\theta_{2}\right)h\left(\theta_{1},\theta_{2}\right)\]
02

Evaluate the posterior pdf

Now the resulting posterior pdf should be evaluated for the given data values (\(x_{1}=1, x_{2}=4\)) and parameter values (\(\theta_{1}=1,2,3,4\) and \(\theta_{2}=0.5,1.0,1.5,2.0\)). The parameter values which maximize the posterior are to be found.
03

Determine the maximum of the posterior pdf

The maximum of the posterior pdf should be determined from the evaluated values. To do this, look for the highest resulting pdf from the list of the evaluated values from the given parameters.
04

Computer program knowledge

In practice, finding the maximum of the posterior pdf can be done by the use of programming. Languages like Python, R, or software such as MATLAB can be used. Methods might include gradient descent or numerical optimization methods, but this will depend on the specific courses taken, and isn't a part of this particular problem's mathematical solution itself.

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Most popular questions from this chapter

The following amounts are bets on horses \(A, B, C, D\), and \(E\) to win. $$ \begin{array}{cr} \text { Horse } & \text { Amount } \\ \hline A & \$ 600,000 \\ B & \$ 200,000 \\ C & \$ 100,000 \\ D & \$ 75,000 \\ E & \$ 25,000 \\ \hline \text { Total } & \$ 1,000,000 \end{array} $$ Suppose the track wants to take \(20 \%\) off the top, namely, \(\$ 200,000\). Determine the payoff for winning with a \(\$ 2\) bet on each of the five horses. (In this exercise, we do not concern ourselves with "place" and "show.") Hint: Figure out what would be a fair payoff so that the track does not take any money (that is, the track's take is zero), and then compute \(80 \%\) of those payoffs.

In Example 11.2.2, let \(n=30, \alpha=10\), and \(\beta=5\), so that \(\delta(y)=(10+y) / 45\) is the Bayes estimate of \(\theta\). (a) If \(Y\) has a binomial distribution \(b(30, \theta)\), compute the risk \(E\left\\{[\theta-\delta(Y)]^{2}\right\\}\). (b) Find values of \(\theta\) for which the risk of part (a) is less than \(\theta(1-\theta) / 30\), the risk associated with the maximum likelihood estimator \(Y / n\) of \(\theta\).

Write the Bayes model of Exercise \(11.5 .2\) as $$ \begin{aligned} Y & \sim b(n, p), \quad 00 \end{aligned} $$ Set up the estimating equations for the mle of \(g(y \mid \theta)\), i.e., the first step to obtain the empirical Bayes estimator of \(p\). Simplify as much as possible.

11.3.7. Consider the Bayes model $$ X_{i} \mid \theta, i=1,2, \ldots, n \sim \text { iid with distribution } b(1, \theta), 0<\theta<1 $$ (a) Obtain the Jeffreys' prior for this model. (b) Assume squared-error loss and obtain the Bayes estimate of \(\theta\).

Consider the Bayes model \(X_{i} \mid \theta, i=1,2, \ldots, n \sim\) iid with distribution \(b(1, \theta), 0<\theta<1\) $$ \Theta \sim h(\theta)=1 $$ (a) Obtain the posterior pdf. (b) Assume squared-error loss and obtain the Bayes estimate of \(\theta\).
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