Question

Let \(R_{1}, \ldots, R_{n}\) be a binomial random sample with parameters \(m\) and \(0<\pi<1\), where \(m\) is known. Find a complete minimal sufficient statistic for \(\pi\) and hence find the minimum variance unbiased estimator of \(\pi(1-\pi)\).

Short answer

The complete minimal sufficient statistic is \( T = \sum_{i=1}^{n} R_i \), and the MVUE for \( \pi(1-\pi) \) is \( \hat{\pi}(1-\hat{\pi}) = \frac{T}{nm} \left( 1 - \frac{T}{nm} \right) \).

Step-by-step solution

Understanding the Binomial Random Sample

A binomial random sample consists of observations that are independently and identically distributed, each following a binomial distribution with parameters (m, \( \pi \)). This means each \( R_{i} \) represents the number of successes in \( m \) Bernoulli trials with the probability of success being \( \pi \).

Determine the Joint Probability Mass Function

The joint probability mass function (PMF) for the random sample \( R_1, R_2, \ldots, R_n \) can be expressed as the product of individual PMFs: \[ P(R_{1}, \, R_{2}, \, \ldots, \, R_{n} \, | \, \pi) = \prod_{i=1}^{n} \binom{m}{R_i} \pi^{R_i} (1-\pi)^{m-R_i} \].

Sufficiency of the Statistic

According to the factorization theorem, a statistic is sufficient for a parameter if the joint PMF can be factored into a product of two functions, one of which depends solely on the sample and the parameter. The joint PMF can be rewritten, factoring out the \( \pi \)-dependent part: \[ \prod_{i=1}^{n} \binom{m}{R_i} \cdot \pi^{\sum_{i=1}^{n} R_i} (1-\pi)^{nm-\sum_{i=1}^{n} R_i} \]. Thus, the statistic \( T = \sum_{i=1}^{n} R_i \) is sufficient for \( \pi \).

Completeness of the Sufficient Statistic

A statistic is complete if no non-zero function of it has an expectation of zero for all \( \pi \). Since \( T = \sum_{i=1}^{n} R_i \) follows a Binomial distribution with parameters \( mn \) and \( \pi \), it is both a complete and a sufficient statistic for \( \pi \) because the binomial distribution satisfies the completeness property due to its exponential family form.

Finding the Unbiased Estimator for \(\pi(1-\pi)\)

To find the minimum variance unbiased estimator (MVUE) for \( \pi (1-\pi) \), find an unbiased estimator based on the complete sufficient statistic. Since the expectation \( E(T) = nm\pi \), it follows \( \hat{\pi} = \frac{T}{nm} \) is an unbiased estimator for \( \pi \). Then, \( \hat{\pi}(1-\hat{\pi}) = \frac{T}{nm} \left( 1 - \frac{T}{nm} \right) \) is the MVUE for \( \pi(1-\pi) \).

Key concepts

Binomial Distribution

The binomial distribution is a foundational concept in statistics that models the number of successes in a fixed number of independent binary experiments or trials — each with the same probability of success. In the context of our exercise, you have a series of binomial samples, each characterized by parameters: the number of trials \( m \) and the probability of success \( \pi \).
This distribution is widely used to answer questions pertaining to binary outcomes, such as flipping a coin or checking whether a light bulb is defective.

• Each trial is independent, meaning the outcome of one trial does not affect another.
• The probability \( \pi \) remains constant throughout all trials.

Understanding the binomial distribution is crucial, as it provides the basis for identifying complete and sufficient statistics for parameter estimation. Here, the sum \( T = \sum R_i \) of binomial random variables is key to drawing inferences about the underlying probability \( \pi \).

Minimum Variance Unbiased Estimator

The concept of a Minimum Variance Unbiased Estimator (MVUE) revolves around finding an estimator that not only provides unbiased predictions of the true parameter value but also exhibits the smallest possible variance among all unbiased estimators.
It's an essential principle in statistical estimation, ensuring that our estimates are as close to the actual parameter as possible while maintaining reliability.
For our problem, the task was to find an MVUE for \( \pi(1-\pi) \). Starting with the unbiased estimator for \( \pi \), \( \hat{\pi} = \frac{T}{nm} \), the MVUE of \( \pi(1-\pi) \) is then \( \hat{\pi}(1-\hat{\pi}) = \frac{T}{nm} \left( 1 - \frac{T}{nm} \right) \).This form ensures that we're balancing precision with variability in our estimation approach, all derived from the complete sufficient statistic found earlier.

Factorization Theorem

The Factorization Theorem provides a powerful method to determine whether a statistic is sufficient.
It states that if you can factor the joint probability mass function (PMF) or probability density function (PDF) of a sample into two parts — one dependent only on the sample and another on the parameter of interest — then you have found a sufficient statistic.
In the exercise, the joint PMF was expressed as \[ P(R_{1}, \, R_{2}, \, \ldots, \, R_{n} \, | \, \pi) = \prod_{i=1}^{n} \binom{m}{R_i} \pi^{R_i} (1-\pi)^{m-R_i} \]and was factored as:\[ \prod_{i=1}^{n} \binom{m}{R_i} \cdot \pi^{\sum R_i} (1-\pi)^{nm-\sum R_i} \]showing that \( T = \sum R_i \) is indeed sufficient for \( \pi \).
By reformulating the PMF, we clearly delineate the parameter-specific section, making the underlying parameter's information captured in \( T \) sufficient for analysis.

Exponential Family

The exponential family of distributions is a broad class of probability distributions, often characterized by their simple, canonical form, which provides mathematical tractability.
Distributions that belong to this family include the normal, binomial, Poisson, and others. These distributions are particularly useful because they have nice statistical properties, such as sufficiency and completeness, which aid in statistical inference.

• The binomial distribution, seen in our exercise, belongs to the exponential family.
• Its parameters can be estimated using the methods derived from the properties of exponential family distributions.

Completeness, as used in the exercise, is a consequence of the binomial distribution being part of this family.
This means any unbiased estimator based on the statistic \( T = \sum R_i \) will exhibit minimal variance. Recognizing the binomial distribution as part of this broader family allows us to apply advanced theorems for efficient parameter estimation.