Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a Poisson distribution with parameter \(\theta, 0<\theta<\infty .\) Let \(Y=\sum_{1}^{n} X_{i}\) and let \(\mathcal{L}[\theta, \delta(y)]=[\theta-\delta(y)]^{2}\). If we restrict our considerations to decision functions of the form \(\delta(y)=b+y / n\), where \(b\) does not depend on \(y\), show that \(R(\theta, \delta)=b^{2}+\theta / n .\) What decision function of this form yields a uniformly smaller risk than every other decision function of this form? With this solution, say \(\delta\), and \(0<\theta<\infty\), determine \(\max _{\theta} R(\theta, \delta)\) if it exists.

Short answer

The decision function that minimizes the potential for error is \( \delta(y) = \frac{y}{n} \). There's no maximum risk \( \max_{\theta} R(\theta, \delta) \) as it decreases with increasing \( \theta \).

Step-by-step solution

Express the Average Risk Function

Rewrite the risk function \( R(\theta, \delta) = b^{2} + \frac{\theta}{n} \). This function represents the risk of making an error in our parameter estimation in terms of the unknown parameter \( \theta \) and decision function \( \delta(y) = b + \frac{y}{n} \).

Differentiate the Average Risk Function for Minimizing Potential for Error

To find the decision function that minimizes potential for error, differentiate \( R(\theta, \delta) \) with respect to \( b \) and set to 0, hence \( \frac{dR(\theta, \delta)}{db} = 0 \) deriving the equation yields \( 2b = 0 \). Solving for \( b \) we get \( b = 0 \). Therefore, the decision function \( \delta(y) = \frac{y}{n} \) yields a uniformly smaller risk as compared to the decision functions of the form \( \delta(y) = b + \frac{y}{n} \).

Find Maximum Risk with Optimal Decision function

Substitute the decision function \( \delta(y) = \frac{y}{n} \) into the risk function getting \( R(\theta, \delta) = b^{2} + \frac{\theta}{n} \), there's no trace of \( b \) so it simplifies as \( R(\theta, \delta) = \frac{\theta}{n} \). Since \( R(\theta, \delta) = \frac{\theta}{n} \) is a decreasing function in \( \theta \), it does not possess a Maximum value.

Key concepts

Random Sample

A random sample is a set of observations or data points drawn from a larger population, where each observation has an equal chance of being selected. In the context of a Poisson distribution, a random sample might consist of counts or occurrences of a certain event over a given period.
For example, if you are studying the number of cars passing through a toll booth in one hour, each hour's count would represent a random sample from the distribution of all possible hourly counts.
Understanding random samples is essential because it allows statisticians to make inferences about the population from which the sample is drawn. In the given exercise, the random sample consists of variables \(X_1, X_2, \ldots, X_n\), each representing a Poisson-distributed count with the common parameter \(\theta\).

• Each observation \(X_i\) follows the Poisson distribution independently.
• The sum \(Y = \sum_{i=1}^{n} X_i\) is the combined total of these observations, crucial for parameter estimation.

Risk Function

The risk function is a measure of the expected loss associated with a decision procedure. In statistical terms, it assesses how off our estimates might be compared to reality. In the exercise, the risk function is given as \( R(\theta, \delta) = b^2 + \frac{\theta}{n} \).
This risk function evaluates the potential error in estimating the parameter \(\theta\) using a decision function \(\delta(y)\). The risk function is crucial because it provides a metric for comparing different decision functions based on their expected performance.

• The term \(b^2\) represents the error due to a constant offset in the decision function.
• The term \(\frac{\theta}{n}\) represents the expected error due to the randomness in the sample observations.

Understanding the risk function allows statisticians to minimize potential errors by selecting an optimal decision function.

Decision Function

A decision function in statistics is a rule or formula used to make estimations or predictions about a population parameter based on sample data. In the exercise, the decision function is defined as \(\delta(y) = b + \frac{y}{n}\), where \(b\) is a constant not depending on \(y\) and \(y\) is the sum of the random sample.
Decision functions are pivotal in ensuring that estimates are as close to the true parameter as possible, reducing the risk of error.

Finding the Optimal Decision Function

The goal in statistical decision theory is often to find the decision function that minimizes risk. For the exercise, this involves differentiating the risk function \( R(\theta, \delta) \) concerning \(b\) and solving to find \(b = 0\). This yields the decision function \( \delta(y) = \frac{y}{n} \) as the optimal choice, meaning it provides the smallest expected error or risk.

Parameter Estimation

Parameter estimation is the process of using sample data to estimate the values of parameters in a statistical model. In the Poisson distribution context, it involves determining the parameter \(\theta\) that best describes the data.
In the provided exercise, the sum \(Y = \sum_{i=1}^{n} X_i \) is used in the decision function to estimate \(\theta\). By employing the form \(\delta(y) = \frac{y}{n}\), we utilize the sample sum to derive an estimate of \(\theta\). This technique leverages the power of

• Simple calculations: Using \(\frac{y}{n}\) simplifies the estimate to the sample mean, a standard approach in Poisson distribution.
• Optimization: The choice \(\delta(y) = \frac{y}{n}\) is derived as an optimal form to reduce estimation errors, as shown by minimizing the risk function.

Ultimately, parameter estimation aims to positively identify the most accurate representation of \(\theta\), enabling predictions and analyses based on the Poisson model.