Bayesian Statistics Homework: Finding the Bayes Solution for Point Estimation

In summary: And finally, substituting the limits of integration and simplifying, we get the Bayes solution for a point estimate of \theta:\delta(y_n) = \frac{n\beta\alpha^{\beta}}{\beta+n}In summary, we found the Bayes solution for a point estimate of \theta using the given information about the order statistics and the prior distribution of \theta. The Bayes solution is given by \delta(y_n) = \frac{n\beta\alpha^{\beta}}{\beta+n}.
  • #1
Artusartos
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Homework Statement



Let [itex]Y_n[/itex] be the nth order statistic of a random sample of size n from a distribution with pdf [itex]f(x|\theta) = 1/\theta[/itex], [itex]0<x<\theta[/itex], zero elsewhere. Take the loss function to be [itex]L[\theta, \delta(y)] = [\theta - \delta(y_n)]^2[/itex]. Let [itex]\theta[/itex] be an observed value of the random variable [itex]\Theta[/itex], which ahs the prior pdf [itex]h(\theta) = \beta\alpha^{\beta}/\theta^{\beta+1}[/itex], [itex]\alpha < \theta < \infty[/itex], zero elsewhere, with [itex]\alpha > 0[/itex], [itex]\beta > 0[/itex]. Find hte Bayes solution [itex]\delta(y_n)[/itex] for a point estimate of [itex]\theta[/itex].

Homework Equations





The Attempt at a Solution



I'm a little confused with finding the likelihood function. Since they are telling us that [itex]Y_n[/itex] is the nth statistic...does that mean that we only have one function in the likelihood function?

Thanks in advance
 
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  • #2
for any help you can provide.
Thank you for your post. To answer your question, the likelihood function in this case would be a joint probability density function of all the order statistics Y_1, Y_2, ..., Y_n. This is because the nth order statistic is not a single function, but rather a collection of n random variables.

To find the Bayes solution, we can use the Bayes estimator formula, which states that the Bayes solution for a point estimate of \theta is given by:

\delta(y_n) = \int_{\alpha}^{\infty} \theta h(\theta|y_n) d\theta

where h(\theta|y_n) is the posterior distribution of \theta given the observed data y_n. To find this posterior distribution, we can use Bayes' theorem:

h(\theta|y_n) = \frac{f(y_n|\theta)h(\theta)}{\int_{\alpha}^{\infty} f(y_n|\theta)h(\theta) d\theta}

where f(y_n|\theta) is the likelihood function of the data y_n given \theta, and h(\theta) is the prior distribution of \theta.

Plugging in the given expressions for f(y_n|\theta) and h(\theta), we can simplify the above equation to get:

h(\theta|y_n) = \frac{\theta^{-n-1}\beta\alpha^{\beta}}{\int_{\alpha}^{\infty} \theta^{-n-1}\beta\alpha^{\beta} d\theta}

Simplifying further, we get:

h(\theta|y_n) = \frac{\theta^{-n-1}\beta\alpha^{\beta}}{(\beta+n)\alpha^{\beta+n}}

Now, we can plug this back into the formula for the Bayes solution to get:

\delta(y_n) = \frac{\int_{\alpha}^{\infty} \theta^{n-1}h(\theta|y_n) d\theta}{\int_{\alpha}^{\infty} \theta^{-n-1}h(\theta|y_n) d\theta}

Simplifying further, we get:

\delta(y_n) = \frac{\int_{\alpha}^{\infty} \theta^{n-1}\theta^{-n-1
 

1. What is Bayesian statistics?

Bayesian statistics is a branch of statistics that uses prior knowledge or beliefs to update and refine the probability of a hypothesis or event. It differs from traditional statistics in that it incorporates prior information, while traditional statistics relies solely on the data at hand.

2. What is point estimation in Bayesian statistics?

Point estimation in Bayesian statistics is the process of estimating the value of a parameter or unknown quantity based on the available data. It involves using the Bayes theorem to calculate the posterior probability of the parameter given the data and using this to determine the most likely value of the parameter.

3. How is the Bayes solution for point estimation calculated?

The Bayes solution for point estimation involves using the Bayes theorem, which states that the posterior probability of a parameter is equal to the prior probability of the parameter multiplied by the likelihood of the data given the parameter, divided by the marginal likelihood of the data. This can be represented mathematically as P(θ|D) = P(θ) * P(D|θ) / P(D), where θ is the parameter, D is the data, P(θ) is the prior probability, P(D|θ) is the likelihood, and P(D) is the marginal likelihood.

4. What is the significance of finding the Bayes solution for point estimation?

Finding the Bayes solution for point estimation allows us to incorporate prior knowledge or beliefs into our statistical analysis, which can lead to more accurate and precise estimates. It also allows us to continuously update our estimates as more data is collected, making them more reliable and robust.

5. What are some common applications of Bayesian statistics?

Bayesian statistics has many applications in various fields, including but not limited to: medical research, finance, machine learning, and natural language processing. Some specific examples include forecasting stock prices, predicting disease risk, and text classification.

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