Question

Let \(Y_{1}, \ldots, Y_{n}\) be a random sample from the uniform distribution on \((0, \theta)\), and take as prior the Pareto density with parameters \(\beta\) and \(\lambda\), $$ \pi(\theta)=\beta \lambda^{\beta} \theta^{-\beta-1}, \quad \theta>\lambda, \quad \beta, \lambda>0 $$ (a) Find the prior distribution function and quantiles for \(\theta\), and hence give prior one- and two-sided credible intervals for \(\theta\). If \(\beta>1\), find the prior mean of \(\theta\). (b) Show that the posterior density of \(\theta\) is Pareto with parameters \(n+\beta\) and \(\max \left\\{Y_{1}, \ldots, Y_{u}, \lambda\right\\}\), and hence give posterior credible intervals and the posterior mean for \(\theta\). (c) Interpret \(\lambda\) and \(\beta\) in terms of a prior sample from the uniform density.

Short answer

The prior and posterior are Pareto distributions. Prior credible intervals are based on the Pareto model, while the posterior updates these based on sample maximum. The parameters \( \lambda \) and \( \beta \) represent threshold and belief in upper values, respectively.

Step-by-step solution

Understand the Prior Distribution

The given prior is a Pareto distribution with density \( \pi(\theta) = \beta \lambda^{\beta} \theta^{-\beta-1} \). A Pareto distribution has its distribution function \( F(\theta) = 1 - \left(\frac{\lambda}{\theta}\right)^\beta \) for \( \theta > \lambda \). This cumulative distribution helps in finding quantiles and credible intervals.

Calculate Prior Credible Intervals

For a one-sided credible interval, we solve \( F(\theta) = 0.95 \) or another suitable probability. Solving \( 1 - \left(\frac{\lambda}{\theta}\right)^\beta = \alpha \) gives: \( \theta_{\alpha} = \lambda (1-\alpha)^{-1/\beta} \). A two-sided (1-\alpha) credible interval is symmetrical around the median.

Find the Prior Mean of \( \theta \)

The expectation of a Pareto distribution is \( E(\theta) = \frac{\beta \lambda}{\beta - 1} \) for \( \beta > 1 \). If \( \beta \leq 1 \), the expectation is undefined, as the integral diverges.

Formulate the Posterior Distribution

With the sample from the uniform distribution, the likelihood is proportional to \( \theta^{-n} \) for \( \theta > \max(Y_1, ..., Y_n) \). Combining prior and likelihood, the posterior has density \( \theta^{-(n+\beta+1)} \) with the same Pareto form for \( \theta > \max(Y_1, ..., Y_n, \lambda) \). This shows the posterior remains Pareto.

Determine Posterior Parameters

The posterior distribution is Pareto with parameters \( n+\beta \) and \( \max(Y_1, ..., Y_n, \lambda) \). Using the form \( \theta^{-(n+\beta+1)} \), it indicates the parameter update and shift due to new data.

Posterior Credible Intervals and Mean

The posterior credible interval can be derived similarly to the prior by replacing parameters. The posterior mean is \( E(\theta | \text{data}) = \frac{(n+\beta) M}{(n+\beta-1)} \) if \( n+\beta > 1 \), where \( M = \max(Y_1, ..., Y_n, \lambda) \).

Interpret Parameters \( \lambda \) and \( \beta \)

\( \lambda \) acts like a minimum threshold from a prior point of view. \( \beta \) relates to the prior belief regarding the weight or `shape` of possible outcomes above this threshold, analogous to the scale of a sample.

Key concepts

Pareto Distribution

In Bayesian statistics, the Pareto distribution is often used as a prior distribution due to its simplicity and interpretability. It models phenomena where there is a lower bound and a "long tail"; in other words, situations where larger values are less probable but possible. A classic application is wealth distribution.
In mathematical terms, for the Pareto distribution with parameters \(\lambda\) and \(\beta\), the density function is given by:

• \(\pi(\theta) = \beta \lambda^{\beta} \theta^{-\beta-1}\)

This formulation implies a threshold level \(\lambda\) below which no values can occur, and \(\beta\) which impacts the shape of the distribution. The cumulative distribution function is:

• \(F(\theta) = 1 - \left(\frac{\lambda}{\theta}\right)^\beta\)

Prior and Posterior Distribution

A prior distribution reflects existing beliefs about a parameter before considering new data. For instance, using the Pareto distribution with parameters \(\lambda\) and \(\beta\) captures initial insight into the parameter \(\theta\). The two parameters signify a baseline assumption \(\lambda\), and a belief in the shape of the data captured by \(\beta\).
The posterior distribution combines the prior with new data. This exercise shows how, if a sample is drawn from the uniform distribution, the resulting posterior also follows a Pareto distribution, but with updated parameters. Specifically, given a uniform sample and Pareto prior, the posterior distribution has parameters \(n+\beta\) and \(\max(Y_1, \ldots, Y_n, \lambda)\). This highlights a neat characteristic: Pareto priors lead to Pareto posteriors, making them conjugate in this context.

Credible Intervals

Credible intervals provide a range in which an unknown parameter, like \(\theta\), likely falls, given the observed data. For the Pareto distribution, credible intervals can be calculated using its cumulative distribution function. A one-sided credible interval sets the cumulative probability, say \(\alpha = 0.95\), such that:

• \(1 - \left(\frac{\lambda}{\theta}\right)^\beta = \alpha\)
• Solve for \(\theta\) to obtain \(\theta_{\alpha} = \lambda (1-\alpha)^{-1/\beta}\).

A two-sided credible interval is symmetrically calculated around the median. Credible intervals offer intuitive insights into parameter uncertainty, differing from frequentist confidence intervals, as they directly incorporate prior beliefs.

Sample from Uniform Distribution

Sampling from the uniform distribution is a simple yet powerful concept in Bayesian statistics. When sampling from a uniform distribution \((0, \theta)\), every value within this range has an equal probability of being selected. This characteristic provides a straightforward way to model uncertainty when no specific outcome is known or favored.
In the exercise given, the uniform distribution serves as the likelihood component when combined with a Pareto prior. It implies that we have a series of observations \(Y_1, \ldots, Y_n\) below an unknown ceiling \(\theta\). These samples, within a Bayesian framework, allow for the updating of prior beliefs to form posterior distributions. The resulting posterior also retains a Pareto form, illustrating the efficiency of Bayesian updating with uniform samples.