Question

Consider two Bernoulli distributions with unknown parameters \(p_{1}\) and \(p_{2}\). If \(Y\) and \(Z\) equal the numbers of successes in two independent random samples, each of size \(n\), from the respective distributions, determine the mles of \(p_{1}\) and \(p_{2}\) if we know that \(0 \leq p_{1} \leq p_{2} \leq 1\)

Short answer

The mles of \(p_{1}\) and \(p_{2}\) when \(0 \leq p_{1} \leq p_{2} \leq 1\) are \(\hat{p_{1}} = min(Y, Z)/n\), and \(\hat{p_{2}} = max(Y, Z)/n\).

Step-by-step solution

Understand the Maximum Likelihood Estimation (MLE)

The Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a model. It works by maximizing the likelihood function which is the probability of the observed data given the parameters.

Apply MLE to the Given Bernoulli Distributions

To apply MLE, consider each Bernoulli distribution separately at first. For each distribution, the likelihood function is given by \(L(p_{i}|Y) = {n \choose Y} p_{i}^Y (1-p_{i})^{n-Y}\), where \(i \in {1,2}\), n is the number of trials and Y is the observed number of successes. The mle of \(p_{i}\) is obtained by taking the derivative of the log likelihood function with respect to \(p_{i}\), setting it to zero and solving for \(p_{i}\). That is, \( \hat{p_{i}} = Y/n \). This is under the assumption that \(0 \leq p_{i} \leq 1\).

Integrate the Condition $0 \leq p_{1} \leq p_{2} \leq 1$ Into the Problem

With the condition $0 \leq p_{1} \leq p_{2} \leq 1$, solutions will be subjected to this inequality. If the maximum likelihood estimate of \(p_{1}\), \(\hat{p_{1}} = Y/n\), is greater than the maximum likelihood estimate of \(p_{2}\), \(\hat{p_{2}} = Z/n\), then this falls outside the restriction and one would have to adjust for this by setting \(p_{1}\) to equal \(p_{2}\), that is, the lower bound \(p_{2}\), hence becoming \(p_{2} \leq \hat{p_{2}} \leq 1\). Thus, the mle of \(p_{1}\) becomes \(\hat{p_{1}} = min(Y, Z)/n\), and the mle of \(p_{2}\) becomes \(\hat{p_{2}} = max(Y, Z)/n\).

Key concepts

Bernoulli Distributions

Bernoulli distributions are a fundamental concept in probability theory and statistics, representing processes that have exactly two possible outcomes, often termed as 'success' and 'failure'. Examples include flipping a coin (heads or tails) or checking if a light bulb works (on or off). For a Bernoulli distribution, there is a single parameter, denoted as \( p \), which is the probability of success on each trial.

The distribution is mathematically expressed as:

• \( P(success) = p \)
• \( P(failure) = 1 - p \)

This simple distribution forms the building block for more complex models and is instrumental in teaching fundamental concepts of variability and probability.

Statistical Method

Statistical methods are a collection of techniques used for collecting, analyzing, interpreting, and presenting empirical data. In the context of Bernoulli distributions or any other statistical model, these methods help identify patterns, test hypotheses, and make predictions based on sample data.

Maximum Likelihood Estimation (MLE) is one such statistical method prized for its utility in estimating unknown parameters of a probability distribution. By selecting the parameter values that make the observed data most probable, MLE provides a way to model the underlying distribution of data points, fitting our understanding of the phenomena being studied to the actual observed outcomes.

Parameter Estimation

Parameter estimation is the process of using sample data to estimate the parameters of the probability distribution that generated the data. In a Bernoulli distribution, the parameter \( p \) represents the probability of occurrence of the event of interest. To estimate this probability, techniques such as the Maximum Likelihood Estimation (MLE) are employed.

The premise is to find the value of \( p \) that would make the observed data most likely to occur. With Bernoulli trials, where each trial is independent, the likelihood of observing a specific set of outcomes has a direct mathematical expression. Through MLE, we obtain a 'point estimate', offering a single best guess for the parameter sought after. In the case of the Bernoulli trials in our exercise, the parameters \(p_1\) and \(p_2\) are estimated using the formula \( \hat{p_i} = Y/n \), where \(Y\) is the number of successes and \(n\) is the total number of trials.

Likelihood Function

The likelihood function is a key element in the process of parameter estimation through Maximum Likelihood Estimation. It represents the probability of the observed sample data given a set of parameters. For Bernoulli distributions, the likelihood of observing \(Y\) successes out of \(n\) trials when the probability of success is \(p\) is given by the binomial formula:

\( L(p|Y) = {n \choose Y} p^Y (1-p)^{n-Y} \).

MLE seeks the value of \(p\) that maximizes this function. In practice, we often work with the logarithm of the likelihood function, as it simplifies the calculations (turns products into sums) and reaches its maximum at the same parameter values as the original likelihood function. By taking the derivative of the log-likelihood with respect to \(p\), setting it to zero, and solving, we're able to find the estimate that makes the observed data most probable. This methodical approach reinforces the robustness of statistical methods in making informed inferences from sample data.