Question

Let \(Y_{4}\) be the largest order statistic of a sample of size \(n=4\) from a distribution with uniform pdf \(f(x ; \theta)=1 / \theta, 0

Short answer

The Bayesian estimator \( \delta(Y_{4}) \) of \( \theta \) using the loss function \( |\delta(Y_{4})-\theta| \) is \( \frac{Y_4}{3} \).

Step-by-step solution

Recognize Prior and Likelihood Distributions

Firstly, observe the given prior distribution \( g(\theta) = \frac{2}{\theta^3} \) and the likelihood function which represent the distribution of the sample (uniform distribution) \( f(x ; \theta) = \frac{1}{\theta} \).

Calculate Posterior Distribution

By Bayes theorem the posterior distribution is proportional to the product of likelihood and the prior distribution. Therefore, \( f(\theta|Y_{4}) \propto f(Y_{4}|\theta)g(\theta) \). Substitute for \( f(Y_{4}|\theta) \) as the probability density function of the largest order statistic \( Y_4 \) of a sample of size 4 which is \( f(Y_{4}|\theta) = 4(\frac{Y_4}{\theta})^3\frac{1}{\theta} \). Hence, \( f(\theta|Y_{4}) \propto 4(\frac{Y_4}{\theta})^3\frac{1}{\theta} \cdot \frac{2}{\theta^3} = \frac{8Y_4^3}{\theta^7} \). This is the Kernel of an inverted gamma distribution with parameters \( \alpha = 4 \), \( \beta = Y_4 \), thus we can say that \( \theta | Y_4 \sim IG(4, Y_4) \).

Find the Bayesian Estimator

The Bayesian estimator considering the loss function \( L(\theta, \delta (Y_{4})) = |\delta(Y_{4})-\theta| \), i.e., the Bayes risk is obtained by minimizing the posterior expected loss. In this case, the Bayes estimator will be \( E[\theta|Y_{4}] \) (posterior mean of \( \theta \)). The expectation of an inverted gamma distribution, \( E[\theta] = \frac{\beta}{\alpha - 1} \), then the Bayesian estimator will be \( \delta(Y_{4}) = \frac{Y_4}{4 - 1} = \frac{Y_4}{3} \).

Key concepts

Uniform Distribution

When a distribution assigns equal probability to all values within a limited range, it is referred to as a uniform distribution. In simpler terms, each outcome within the specified range has an equal chance of occurring. This is often represented mathematically in the form of a rectangle, hence it is sometimes called the rectangular distribution.

The probability density function (pdf) for a continuous uniform distribution from 0 to \theta is given by: \( f(x; \theta) = \frac{1}{\theta} \) for \( 0 < x < \theta \), and zero otherwise. In the given exercise, the uniform distribution helps determine the behavior of the sample data from which the largest order statistic is drawn.

Order Statistic

Order statistics provide specific values from a sample data set after it has been ordered from smallest to largest. \( Y_{4} \), in our context, is the largest order statistic from a sample of size 4. This means \( Y_{4} \) is the fourth largest value, or simply put, the maximum value in our sample. For a uniform distribution, the probability density function for the largest value is given by \( 4\left(\frac{Y_4}{\theta}\right)^3\frac{1}{\theta} \). This function is crucial for finding the posterior distribution in the Bayesian framework.

Loss Function

A loss function, in the context of statistical estimation, is a way to measure the cost of errors in estimation. It compares the estimated value with the true parameter value to provide a numerical representation of the 'loss' incurred due to the difference. In this problem, the loss function given is an absolute-error loss function, defined as \( L(\theta, \delta(Y_{4})) = |\delta(Y_{4})-\theta| \). This loss function penalizes deviations without considering direction, meaning that underestimating or overestimating the parameter by the same amount results in the same penalty.

Prior Distribution

The prior distribution in Bayesian statistics represents our knowledge or beliefs about an unknown parameter before considering the current data. It is the framework for incorporating past experience or subjective judgments. For a parameter \( \theta \), the prior distribution in our exercise is specified as \( g(\theta) = \frac{2}{\theta^3} \) for \( 1 < \theta < \infty \), and zero elsewhere. This particular prior suggests that smaller values of \( \theta \) are considered more probable before observing the data.

Posterior Distribution

The posterior distribution is the updated belief about the unknown parameter after taking the observed data into account. It combines the prior distribution with the likelihood of the observed data. Using Bayes' theorem, the posterior distribution for \( \theta \) given the observed order statistic \( Y_{4} \), is proportional to the product of the prior and the likelihood. The resulting posterior distribution in the exercise happens to be an inverted gamma distribution with specific parameters derived from the data and the prior distribution.

Bayes Theorem

Bayes' theorem is the cornerstone of Bayesian statistics, providing the mathematical rule for updating probabilities. It describes the probability of an event based on prior knowledge of conditions that might relate to the event. Symbolically, the theorem is expressed as \( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \), where \( P(A|B) \) is the posterior probability, \( P(A) \) is the prior probability, \( P(B) \) is the marginal probability, and \( P(B|A) \) is the likelihood. In the context of the exercise, Bayes' theorem enables the calculation of the posterior distribution from the given prior and the likelihood function.

Expected Loss

The expected loss in Bayesian estimation is the expected value of the loss function with respect to the posterior distribution. It quantifies the average 'penalty' for employing a particular estimator. Minimizing the expected loss leads to the best estimator for the given loss function and the observed data. Since we are dealing with the absolute-error loss function in this problem, the optimal Bayesian estimator is the one that minimizes this expected absolute difference between the estimator and the true parameter.

Inverted Gamma Distribution

The inverted gamma distribution is often used in Bayesian analysis, typically as the posterior distribution in certain scenarios. It is closely related to the gamma distribution but 'inverted' in a sense. The probability density function of an inverted gamma distribution with shape parameter \( \alpha \) and scale parameter \( \beta \) is given by \( f(\theta; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}\theta^{-\alpha-1}e^{-\frac{\beta}{\theta}} \) for \( \theta > 0 \). In the given exercise, the posterior distribution of \( \theta \) given the largest order statistic \( Y_{4} \) is identified to be an inverted gamma distribution with \( \alpha = 4 \) and \( \beta = Y_{4} \).