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Chapter 8: Problem 5

Suppose that \(X\) is a discrete random variable with \(P(X=1)=\theta\) and \(P(X=2)\) \(=1-\theta .\) Three independent observations of \(X\) are made: \(x_{1}=1, x_{2}=2, x_{3}=2\). a. Find the method of moments estimate of \(\theta\). b. What is the likelihood function? c. What is the maximum likelihood estimate of \(\theta ?\) d. If \(\Theta\) has a prior distribution that is uniform on \([0,1],\) what is its posterior density?

Short Answer

Expert verified
a. \(\theta = \frac{1}{3}\); b. \(\theta (1-\theta)^2\); c. \(\theta = \frac{1}{3}\); d. Beta(2, 3).

Step by step solution

01

Method of Moments Estimate

The method of moments estimate equates the sample mean to the theoretical mean. Given the observations \(x_1 = 1, x_2 = 2, x_3 = 2\), the sample mean is \(\bar{x} = \frac{1 + 2 + 2}{3} = \frac{5}{3}\). The theoretical mean for \(X\) is \(E(X) = 1 \cdot \theta + 2 \cdot (1-\theta) = 2 - \theta\). Equating these:\[ 2 - \theta = \frac{5}{3} \]Solving for \(\theta\), we get \(\theta = \frac{1}{3}\).
02

Likelihood Function

The likelihood function \(L(\theta)\) is the probability of the observed data given \(\theta\). For the observations \(x_1 = 1, x_2 = 2, x_3 = 2\), the likelihood is:\[ L(\theta) = P(X=1)^1 \cdot P(X=2)^2 = \theta \cdot (1-\theta)^2 \]
03

Maximum Likelihood Estimate

To find the maximum likelihood estimate of \(\theta\), we maximize the likelihood function. The log-likelihood is:\[ \log L(\theta) = \log(\theta) + 2\log(1-\theta) \]Differentiating with respect to \(\theta\) and setting to zero gives:\[ \frac{1}{\theta} - \frac{2}{1-\theta} = 0 \]Solving, we find \(\theta = \frac{1}{3}\).
04

Posterior Density

If \(\Theta\) has a prior distribution that is uniform on \([0,1]\), then the prior is \(f(\theta) = 1\). Using Bayes' theorem:\[ f(\theta | x) \propto L(\theta)f(\theta) = \theta \cdot (1-\theta)^2 \cdot 1 \]The posterior density is a Beta distribution, specifically Beta(2, 3):\[ f(\theta | x) = \frac{\theta (1-\theta)^2}{B(2,3)} \]where \(B(2,3)\) is the normalizing constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Method of Moments
The Method of Moments is a simple yet powerful way to estimate parameters by equating theoretical moments to sample moments.
Moments can be thought of as averages or expected values of random variables.
In our exercise, we calculated the sample mean from our observations of the random variable \(X\), which took on the values 1 and 2 in a set of three trials.
  • The sample mean, \( \bar{x} = \frac{5}{3} \), is found by averaging all the observed values.
  • The theoretical mean, or expected value, is given by \( E(X) = 1 \cdot \theta + 2 \cdot (1-\theta) = 2-\theta \).
  • By equating the sample mean to the theoretical mean, we solve the equation to find \( \theta = \frac{1}{3} \).
This estimation method leverages simplicity at the cost of not always being the most efficient or unbiased.
Likelihood Function
The likelihood function measures how likely it is to observe the data we've collected, given certain parameters.
For a discrete random variable like \(X\), the likelihood function is derived from the probability of observing specific outcomes.
  • For our example, with observations \(x_1 = 1, x_2 = 2, x_3 = 2\), we set up the likelihood: \( L(\theta) = \theta \cdot (1-\theta)^2 \).
  • This function quantifies the likelihood of our data for different values of \(\theta\).
Understanding this function is crucial as it lays the groundwork for finding parameter estimates that fit our observed data best.
Maximum Likelihood Estimation
Maximum Likelihood Estimation (MLE) involves finding the parameter value that maximizes the likelihood function.
It provides a way to choose parameter estimates that make the observed data most probable.
  • We first take the natural log of the likelihood to simplify calculations, resulting in the log-likelihood equation: \( \log L(\theta) = \log(\theta) + 2\log(1-\theta) \).
  • To find the maximum, we differentiate this log-likelihood with respect to \(\theta\) and set the derivative to zero.
  • Solving the equation \( \frac{1}{\theta} - \frac{2}{1-\theta} = 0 \) gives us the MLE of \( \theta = \frac{1}{3} \).
The MLE provides a robust estimate but requires that the likelihood is well-behaved for this method to be effective.
Bayesian Inference
Bayesian Inference combines prior information with current data to estimate parameters.
This approach takes into account both previous beliefs and new evidence, offering a flexible framework for estimation.
  • In this exercise, we begin with a uniform prior distribution for \(\Theta\) over the interval \([0,1]\), implying no prior preference for any value of \(\theta\).
  • We then use Bayes' theorem, which allows us to update our prior beliefs based on the likelihood: \( f(\theta | x) \propto L(\theta)f(\theta) \).
Bayesian Inference is especially useful when prior knowledge is valuable and should be integrated into the estimation process.
Posterior Distribution
The Posterior Distribution is a core concept in Bayesian statistics, representing updated beliefs after considering new data.
It essentially provides a complete description of the uncertainty regarding model parameters after observing the data.
  • For our scenario, given the uniform prior and likelihood function \( \theta \cdot (1-\theta)^2 \), the posterior becomes a Beta distribution.
  • Specifically, the posterior is the Beta distribution with parameters 2 and 3: \( f(\theta | x) = \frac{\theta (1-\theta)^2}{B(2,3)} \).
  • Beta distributions are commonly used as prior probabilities for parameters bound between 0 and 1, making them suitable for \(\theta\).
Understanding the posterior is crucial for deriving inferences and making predictions based on observed data.

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Most popular questions from this chapter

This problem is concerned with the estimation of the variance of a normal distribution with unknown mean from a sample \(X_{1}, \ldots, X_{n}\) of i.i.d. normal random variables. In answering the following questions, use the fact that (from Theorem \(\mathrm{B}\) of Section 6.3) $$\frac{(n-1) s^{2}}{\sigma^{2}} \sim \chi_{n-1}^{2}$$ and that the mean and variance of a chi-square random variable with \(r\) df are \(r\) and \(2 r,\) respectively. a. Which of the following estimates is unbiased? $$s^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} \quad \hat{\sigma}^{2}=\frac{1}{n} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}$$ b. Which of the estimates given in part (a) has the smaller MSE? c. For what value of \(\rho\) does \(\rho \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}\) have the minimal MSE?

George spins a coin three times and observes no heads. He then gives the coin to Hilary. She spins it until the first head occurs, and ends up spinning it four times total. Let \(\theta\) denote the probability the coin comes up heads. a. What is the likelihood of \(\theta ?\) b. What is the MLE of \(\theta ?\)

Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) are i.i.d. random variables on the interval [0,1] with the density function $$f(x | \alpha)=\frac{\Gamma(3 \alpha)}{\Gamma(\alpha) \Gamma(2 \alpha)} x^{\alpha-1}(1-x)^{2 \alpha-1}$$ where \(\alpha>0\) is a parameter to be estimated from the sample. It can be shown that $$\begin{aligned} E(X) &=\frac{1}{3} \\ \operatorname{Var}(X) &=\frac{2}{9(3 \alpha+1)} \end{aligned}$$ a. How could the method of moments be used to estimate \(\alpha ?\) b. What equation does the mle of \(\alpha\) satisfy? c. What is the asymptotic variance of the mle? d. Find a sufficient statistic for \(\alpha .\)

If a thumbtack is tossed in the air, it can come to rest on the ground with either the point up or the point touching the ground. Find a thumbtack. Before doing any experiment, what do you think \(\pi\), the probability of it landing point up, is? Next, toss the thumbtack 20 times and graph the log likelihood of \(\pi .\) Then do another experiment: Toss the thumbtack until it lands point up 5 times, and graph the log likelihood of \(\pi\) based on this experiment. Find and graph the posterior distribution arising from a uniform prior on \(\pi\) Find the posterior mean and standard deviation and compare the posterior with a normal distribution with that mean and standard deviation. Finally, toss the thumbtack 20 more times and compare the posterior distribution based on all 40 tosses to that based on the first \(20 .\)

Use the factorization theorem to find a sufficient statistic for the exponential distribution.
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