Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of size \(n\) from a geometric distribution that has \(\operatorname{pmf} f(x ; \theta)=(1-\theta)^{x} \theta, x=0,1,2, \ldots, 0<\theta<1\), zero elsewhere. Show that \(\sum_{1}^{n} X_{i}\) is a sufficient statistic for \(\theta\).

Short answer

\(\sum_{i=1}^{n} X_i\) is a sufficient statistic for \(\theta\).

Step-by-step solution

Joint Probability Mass Function

The joint probability mass function of \(n\) independent random variables, \(X_1, X_2, ..., X_n\) is the product of their individual pmfs. So, it is given by \[f(x_1, x_2, ..., x_n; \theta) = \prod_{i=1}^{n}f(x_i; \theta) = \prod_{i=1}^{n}(1 - \theta)^{x_i} \theta = (1-\theta)^{\sum_{i=1}^n x_i} \theta^{n}\]

Factorize the Joint pmf

According to the factorization theorem, a statistic \(T(X)\) is said to be a sufficient statistic for \(\theta\) if the joint pmf can be factorized as a product of two functions such that the first one depends only on \(T(X)\) and \(\theta\) and the second one only on the sample. We try to factorize the joint pmf obtained in step 1. We write the joint pmf as \[f(x_1, x_2, ..., x_n; \theta) = (1-\theta)^{\sum_{i=1}^n x_i} \theta^{n} = g(\sum_{i=1}^n x_i, \theta) \cdot h(x_1, ..., x_n)\] where \(g(\sum_{i=1}^n x_i, \theta) = (1-\theta)^{\sum_{i=1}^n x_i} \theta^{n}\) and \(h(x_1, ..., x_n) = 1\) (a function independent of \(\theta\))

Apply the Factorization Theorem

By factorization theorem, since we have been able to express the joint pdf as a product of two functions such that one of them depends on \(\theta\) and the statistic \(T(X) = \sum_{i=1}^{n} X_i\) and the other function does not depend on \(\theta\), we can conclude that \(T(X) = \sum_{i=1}^{n} X_i\) is a sufficient statistic for \(\theta\).

Key concepts

Probability Mass Function

A Probability Mass Function (PMF) refers to a function that gives us the probability that a discrete random variable is exactly equal to some value. To put it simply, while dealing with discrete outcomes, like the number of heads observed when flipping a coin multiple times, the PMF can tell us the likelihood of getting a certain number of heads.For instance, consider a game where you flip a coin and the PMF can answer questions like, 'What is the probability of the coin landing heads up exactly three times?' In our geometric distribution scenario, the PMF, denoted as \( f(x; \theta) \), represents the probability of observing the first success on the \(x\)-th trial when the probability of success on each trial is \(\theta\).

Geometric Distribution

Imagine you're repeatedly playing a game where you have a chance to win each time you play, but it's not certain when you'll win. This scenario is modeled by the geometric distribution. It describes the number of trials needed for the first success in a series of independent trials, each with the same probability of success.

Characteristics of Geometric Distribution

• Memorylessness: Future probabilities don't depend on past results; it 'forgets' all previous failures once a success occurs.
• Discrete nature: It deals with distinct outcomes, usually whole numbers, indicating the count of trials.
• Constant success probability: Each trial has a fixed probability \( \theta \) of success.

It's crucial for a student to understand that the PMF related to geometric distribution has a particular form, \( f(x; \theta) = (1 - \theta)^x \theta \), and this form aids in defining the distribution's characteristics.

Factorization Theorem

Diving into the world of statistics, you might come across the term 'sufficient statistic.' It's essentially a statistic that holds all the necessary information about a sample to make inferences about a population parameter. The factorization theorem is a tool that statisticians use to identify such sufficient statistics.The theorem states that a statistic is sufficient for a parameter if the joint probability mass function of the data can be broken down into two parts: one part depends solely on the sample and the parameter, while the other part does not depend upon the parameter. This is exactly what we see in the exercise, where the joint PMF is factored into a part depending on the sum of the sample values and \(\theta\), and another part that's independent of \(\theta\).

How the Factorization Theorem Helps

• It simplifies data analysis by reducing the data to a sufficient statistic.
• It enables us to focus on the most informative part of the data regarding the parameter of interest.
• It is fundamental for proving and understanding properties of estimators in statistics.

The theorem's elegance lies in its ability to confirm whether we've captured all the essential information in a compact form, making statistical analysis much more streamlined and focused.