Question

What is the sufficient statistic for \(\theta\) if the sample arises from a beta distribution in which \(\alpha=\beta=\theta>0 ?\)

Short answer

The sufficient statistic for \(\theta\) is \(T(X) = (\prod_{i=1}^{n} x_i, \prod_{i=1}^{n} (1 - x_i))\).

Step-by-step solution

Write down the PDF knowing it's a Beta distribution

Since the random variables are distributed according to a beta distribution where \(\alpha = \beta = \theta>0\), each one has a probability density function given by \[ f(x_i|\theta) = \frac{x_i^{\theta - 1} (1 - x_i)^{\theta - 1}} {B(\theta, \theta)} \] where \(i=1,2,...,n\) for a data sample of size \(n\).

Express joint PDF

The joint PDF for multiple observations is just the product of the individual PDFs. Write this out as follows: \[ f(x_1,x_2,...,x_n|\theta) = \prod_{i=1}^{n} \frac{x_i^{\theta - 1} (1 - x_i)^{\theta - 1}} {B(\theta, \theta)} \]

Simplify joint PDF

By performing some basic simplification, rewrite that equation: \[ f(x_1,x_2,...,x_n|\theta) = \frac{1}{B(\theta, \theta)^n} \prod_{i=1}^{n} x_i^{\theta - 1} (1 - x_i)^{\theta - 1} \]

Determine if there's a sufficient statistic

If there's a sufficient statistic, it will be a function of the data \(x_1,x_2,...,x_n\) that enters the equation 'separately' from the parameter \(\theta\). From the equation left after simplification, we see that such components are \(\prod_{i=1}^{n} x_i\), which is the product of all \(x\) observations, and \(\prod_{i=1}^{n} (1 - x_i)\), which is the product of all \(1 - x\) observations.

Formulate sufficient statistic

Based on the insights from the previous steps, the sufficient statistic for \(\theta\) is given by the expressions \(\prod_{i=1}^{n} x_i\) and \(\prod_{i=1}^{n} (1 - x_i)\). We can write this succinctly as \(T(X) = (\prod_{i=1}^{n} x_i, \prod_{i=1}^{n} (1 - x_i))\).

Key concepts

Beta Distribution

The beta distribution is a versatile family of continuous probability distributions defined on the interval [0, 1], which is particularly useful for modelling the behavior of random variables limited to percentages, proportions, or probabilities. The beta distribution is characterized by two shape parameters, usually denoted as \(\alpha\) and \(\beta\), which dictate its skewness and determine the inclination of the distribution curve.

When the shape parameters are equal (\(\alpha=\beta\)), the distribution is symmetric around the midpoint of its support. This specific case of the beta distribution sees a lot of applications, as it serves as a conjugate prior distribution in Bayesian inference for binomial proportions. For example, if a random variable follows a symmetric beta distribution where both shape parameters equal \(\theta\), its probability density function (PDF) would be particularly balanced and typically bell-shaped, representing a scenario where the observed data has no inherent bias towards 0 or 1, but is equally likely to be anywhere within the interval.

Probability Density Function (PDF)

The probability density function (PDF) is central to understanding continuous probability distributions. It is a function that describes the relative likelihood for a random variable to take on a given value. The PDF for a beta distribution, when the shape parameters are equal, i.e., \(\alpha = \beta = \theta\), is given by:
\[ f(x|\theta) = \frac{x^{\theta - 1} (1 - x)^{\theta - 1}}{B(\theta, \theta)} \]
Here, \(B(\theta, \theta)\) is the beta function, which acts as a normalizing constant to ensure that the total area under the PDF curve equals one, satisfying the probability requirement. The beta function is itself an integral that generalizes the factorial function to non-integer values. Understanding the PDF is pivotal to a variety of statistical applications, including determining probabilities of intervals and finding expected values.

Joint Probability Density Function

The joint probability density function (joint PDF) plays an instrumental role when dealing with multiple random variables, which is common in statistical analysis involving samples rather than individual data points. In the case of the beta distribution, the joint PDF for a sample of size \(n\) is the product of the individual PDFs for each observation. The joint PDF is:
\[ f(x_1,x_2,...,x_n|\theta) = \prod_{i=1}^{n} \frac{x_i^{\theta - 1} (1 - x_i)^{\theta - 1}}{B(\theta, \theta)} \]
By multiplying the PDFs of individual observations, the joint PDF accounts for their collective behavior, which is essential for evaluating the likelihood of a sample. Simplifying the joint PDF, we consider the repetitiveness of the beta function term which is raised to the power of the sample size, \(n\), and combine the products of the powers of \(x_i\) and \((1- x_i)\), yielding a simplified form that is more practical for further analysis.

Statistical Inference

Statistical inference refers to the process of making judgments about a population based on a sample drawn from it. Central to statistical inference is the concept of a sufficient statistic, which is a summary of the data that contains all the information necessary to compute any estimate of the parameter of interest.

In our scenario with the beta distribution, a sufficient statistic for \(\theta\) would be a function of the data that captures all the information relevant for estimating \(\theta\). Through the simplification of the joint probability density function, we identify such functions as the product of all \(x\) observations and the product of all \((1 - x)\) observations. Formally, we denote the sufficient statistic for \(\theta\) as:
\[ T(X) = (\prod_{i=1}^{n} x_i, \prod_{i=1}^{n} (1 - x_i)) \]
This compact representation enables efficient estimation of \(\theta\) without the need to rely on the entire data set, thus simplifying the inference process. In essence, the sufficient statistic encapsulates the essence of the data with regard to the parameters of the distribution we're examining.