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Magazine Chapter 10: Problem 10
Let \(\left(k_{1}, k_{2}, \ldots, k_{t}\right)\) be the vector of sample observations representing a multinomial random variable with parameters \(n, p_{1}, p_{2}, \ldots\), and \(p_{t}\). Show that the maximum likelihood estimate for \(p_{i}\) is \(k_{i} / n, i=1,2, \ldots, t\).
Short Answer
Expert verified
The maximum likelihood estimate for \(p_{i}\) in a multinomial distribution is \(\hat{p_{i}} = \frac{k_{i}}{n}\)
Step by step solution
01
Understanding the Parameters and Distribution
Here we understand that a multinomial distribution typically describes an experiment where each outcome can fall into one of multiple categories and each observation is independent of the others. The parameters are the probabilities \(p_{i}\) that an observation falls into the i-th category, and we know that their sum equal to 1. \(k_{i}\) refers to the number of times the i-th outcome is observed in the sample.
02
Formulate the Likelihood Function
Given \(n\) independent observations, the probability of the observations is the product of the individual probabilities. In mathematical terms, this is represented by the multinomial probability mass function, which is: \[L(p_{1}, p_{2}, \ldots, p_{t}) = \frac{n!}{k_{1}! k_{2}! \ldots k_{t}!} p_{1}^{k_{1}} p_{2}^{k_{2}} \ldots p_{t}^{k_{t}} \] Note that \(L\) stands for the likelihood of the observations given the parameters.
03
Calculate the Log-Likelihood
Calculate the natural logarithm of the likelihood function (log-likelihood), because it simplifies the derivative. The log-likelihood function is: \[l = ln(L) = ln\left(n!\right) - \sum_{i=1}^{t} ln\left(k_{i}!\right) + \sum_{i=1}^{t} k_{i} ln\left(p_{i}\right) \]
04
Maximize the Log-Likelihood
In order to find the maximum likelihood estimate, differentiate the log-likelihood function with respect to each \(p_{i}\) and set it to zero. These derivatives are: \[\frac{d l}{d p_{i}} = \frac{k_{i}}{p_{i}} - \frac{n}{1 - \sum_{j=1, j鈮爄}^{t} p_{j}}\] Since it's given that \(\sum_{i=1}^t p_{i} = 1\), by setting the equation to zero and solving it, it is derived that the estimator for \(p_{i}\) is \(\hat{p_{i}} = \frac{k_{i}}{n}\)
05
Explain the Result
The result means that the maximum likelihood estimate for \(p_{i}\) in a multinomial distribution is just the proportion of times the i-th category appears within the sample.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Multinomial Distribution
The multinomial distribution is a generalization of the binomial distribution. It models the outcome of a random experiment where each trial can result in one of several categories and the probability of each category is constant. This distribution is particularly useful when dealing with categorical data where each observation can fall into one of several categories.
For example, if you were rolling a dice, there are six potential outcomes. If you rolled the dice multiple times, keeping track of how many times each number appears, you're essentially dealing with a multinomial distribution. In the context of the provided exercise, the vector \(k_{1}, k_{2}, \textellipsis, k_{t}\) represents the frequency of each outcome in a series of \(n\) trials, and \(p_{1}, p_{2}, \textellipsis, p_{t}\) represent the probabilities of each of the \(t\) outcomes.
For example, if you were rolling a dice, there are six potential outcomes. If you rolled the dice multiple times, keeping track of how many times each number appears, you're essentially dealing with a multinomial distribution. In the context of the provided exercise, the vector \(k_{1}, k_{2}, \textellipsis, k_{t}\) represents the frequency of each outcome in a series of \(n\) trials, and \(p_{1}, p_{2}, \textellipsis, p_{t}\) represent the probabilities of each of the \(t\) outcomes.
Likelihood Function
The likelihood function is a fundamental concept in statistics, used to estimate the parameters of a statistical model. In essence, it provides a measure of how well a set of parameters explains the observed data. For multinomial distributions, the likelihood function is the product of the probabilities for all observed outcomes, given the parameters.
The likelihood function depends on the probability mass function, which calculates the probability of observing a particular set of outcomes. As seen in the given exercise, the function \(L(p_{1}, p_{2}, \textellipsis, p_{t})\) represents the likelihood of observing the sample data given the probability parameters of the model. It's important to note that we aim to find the parameter values that maximize this function, as these are the most likely parameters given the sample data.
The likelihood function depends on the probability mass function, which calculates the probability of observing a particular set of outcomes. As seen in the given exercise, the function \(L(p_{1}, p_{2}, \textellipsis, p_{t})\) represents the likelihood of observing the sample data given the probability parameters of the model. It's important to note that we aim to find the parameter values that maximize this function, as these are the most likely parameters given the sample data.
Log-Likelihood
Calculating the likelihood function can often result in very small numbers, due to the product of probabilities. To deal with this, statisticians often work with the natural logarithm of the likelihood function, which transforms the product into a sum, making the calculations more manageable. This transformation is what we call the log-likelihood.
The log-likelihood function simplifies the maximization process because taking the derivative of a sum is more straightforward than the derivative of a product. In the exercise, the log-likelihood function \(l\) is obtained by taking the natural logarithm of the likelihood function \(L\). Maximizing \(l\) rather than \(L\) leads to the same parameter estimates because the logarithm is a monotonically increasing function鈥攖hat is, if \(L\) increases, so does \(l\), and vice versa.
The log-likelihood function simplifies the maximization process because taking the derivative of a sum is more straightforward than the derivative of a product. In the exercise, the log-likelihood function \(l\) is obtained by taking the natural logarithm of the likelihood function \(L\). Maximizing \(l\) rather than \(L\) leads to the same parameter estimates because the logarithm is a monotonically increasing function鈥攖hat is, if \(L\) increases, so does \(l\), and vice versa.
Probability Mass Function
The probability mass function (PMF) assigns a probability to each possible outcome of a discrete random variable. In a multinomial distribution, the PMF describes the probability of any particular combination of numbers of successes for the categories. Mathematically, the PMF is used within the likelihood function to provide a formula for the likelihood of observing a given set of outcomes based on certain parameters.
In the context of our exercise, the multinomial PMF is represented by a function of the probabilities \(p_{1}, p_{2}, \textellipsis, p_{t}\) and the number of times the corresponding outcome is observed, noted as \(k_{1}, k_{2}, \textellipsis, k_{t}\). This PMF is critical for calculating the likelihood function, which is then maximized to find the most suitable parameters that describe the data.
In the context of our exercise, the multinomial PMF is represented by a function of the probabilities \(p_{1}, p_{2}, \textellipsis, p_{t}\) and the number of times the corresponding outcome is observed, noted as \(k_{1}, k_{2}, \textellipsis, k_{t}\). This PMF is critical for calculating the likelihood function, which is then maximized to find the most suitable parameters that describe the data.
