Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) ba random sample from a distribution with one of two pdfs. If \(\theta=1\), then \(f(x ; \theta=1)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2},-\infty

Short answer

The MLE of \(\theta\) will depend on the specific \(X_i\) values observed. If \(L_1(\theta = 1) > L_2(\theta = 2)\) for these values, the MLE will be \(\theta = 1\). Otherwise, the MLE will be \(\theta = 2\).

Step-by-step solution

Understand the maximum likelihood estimation method

The maximum likelihood estimation (MLE) method involves determining the value of a parameter that makes the observed data most probable. In other words, it maximizes the likelihood function \(L(\theta)\) which is the product of the pdfs of each observation \(X_i\).

Write down the likelihood function

Depending on the value of \(\theta\), we would have a different likelihood function. For \(\theta = 1\), the likelihood function \(L_1(\theta)\) is the product of \(f(x ; \theta=1)\) for all \(X_i\), that is \(L_1(\theta = 1) = \prod_{i=1}^{n}\frac{1}{\sqrt{2 \pi}} e^{-X_i^{2}/2}\). Similarly, for \(\theta = 2\), the likelihood function \(L_2(\theta)\) is the product of \(f(x ; \theta=2)\) for all \(X_i\), that is \(L_2(\theta = 2) = \prod_{i=1}^{n}\frac{1}{\pi(1 + X_i^2)}\).

Determine the MLE

To find the MLE, we need to determine which of the likelihood functions (corresponding to \(\theta = 1\) or \(\theta = 2\)) is greater, given our observed data \(X_1, X_2, ..., X_n\). If \(L_1(\theta = 1) > L_2(\theta = 2)\), then our estimate will be \(\theta = 1\), else it will be \(\theta = 2\). This determination can be made either through direct computation (if the \(X_i\) values are given) or analytically, on the basis of the forms of the likelihood functions.

Key concepts

Likelihood Function

In the context of maximum likelihood estimation, the likelihood function plays a pivotal role. It is a function of the parameter \( \theta \) given the observed data. To construct a likelihood function, we take the product of the probability density functions (pdfs) of all the observations. This product reveals how probable the observed data is, assuming a specific \( \theta \).

• If \( \theta = 1 \), the likelihood function \( L_1(\theta) \) is built from the normal distribution: \ \( L_1(\theta = 1) = \prod_{i=1}^{n} \frac{1}{\sqrt{2 \pi}} e^{-X_i^{2}/2} \).
• For \( \theta = 2 \), the likelihood function \( L_2(\theta) \) uses the Cauchy distribution: \ \( L_2(\theta = 2) = \prod_{i=1}^{n} \frac{1}{\pi(1 + X_i^2)} \).

The goal is to see which likelihood function is maximized given the data points we have. This involves calculating each function's value, given all samples, and determining which scenario (value of \( \theta \)) fits our observations best.

Probability Density Function

The probability density function (pdf) describes the likelihood of a random variable to take on a specific value. It's central to probability theory and essential in constructing the likelihood function. For the given problem, we have:

• A pdf for \( \theta = 1 \) which is the normal distribution: \ \( f(x; \theta=1) = \frac{1}{\sqrt{2 \pi}} e^{-x^{2}/2} \). This distribution is symmetrical around zero and its shape is bell-like.
• The second option is the pdf for \( \theta = 2 \) which is the Cauchy distribution: \ \( f(x; \theta=2) = \frac{1}{\pi(1+x^{2})} \). This distribution is wider and has heavier tails than the normal distribution, indicating more probability in the extremes.

These functions help us calculate the likelihood of observing our data given each possible \( \theta \). By comparing values derived from these pdfs, we can optimize the likelihood function.

Parameter Estimation

Parameter estimation involves determining the parameters of a statistical model that best explain the observed data. In maximum likelihood estimation, this equates to: identifying the parameter \( \theta \) that maximizes the likelihood function. Here’s a simplified view of how parameter estimation works for this exercise:

• First, calculate the likelihood for each possible \( \theta \) by plugging in the observed data into their respective likelihood functions.
• Then, compare the results: if \( L_1(\theta = 1) > L_2(\theta = 2) \), \( \theta = 1 \) is the maximum likelihood estimate. Otherwise, if \( L_2(\theta = 2) > L_1(\theta = 1) \), \( \theta = 2 \) is preferred.

This method relies on the assumption that the complex form of the observations stems from one of these parameter-driven processes. By optimizing the likelihood, we efficiently decide which parameter value best suits the data.