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Chapter 8: Problem 17

Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) are i.i.d. random variables on the interval [0,1] with the density function $$f(x | \alpha)=\frac{\Gamma(2 \alpha)}{\Gamma(\alpha)^{2}}[x(1-x)]^{\alpha-1}$$ where \(\alpha>0\) is a parameter to be estimated from the sample. It can be shown that $$\begin{aligned} E(X) &=\frac{1}{2} \\ \operatorname{Var}(X) &=\frac{1}{4(2 \alpha+1)} \end{aligned}$$ a. How does the shape of the density depend on \(\alpha ?\) b. How can the method of moments be used to estimate \(\alpha ?\) c. What equation does the mle of \(\alpha\) satisfy? d. What is the asymptotic variance of the mle? e. Find a sufficient statistic for \(\alpha .\)

Short Answer

Expert verified
a. Larger \(\alpha\) sharpens the distribution. b. Use sample variance to estimate \(\alpha\). c. Solve digamma equation. d. Asymptotic variance is \(I(\alpha)^{-1}\). e. Sufficient statistic: \(\sum \log(x_i(1-x_i))\).

Step by step solution

01

Understanding the Shape of Density

The given density function is for a beta distribution, which is symmetric around 0.5 for all \(\alpha\). As \(\alpha\) increases, the distribution becomes more peaked and concentrated around 0.5. Conversely, a smaller \(\alpha\) leads to a flatter distribution, indicating more spread-out values across the interval \([0,1]\).
02

Using Method of Moments for Estimation

The method of moments involves equating the sample moments to the theoretical moments. Given that \(E(X) = 0.5\), the first moment condition doesn't help estimate \(\alpha\) directly. We use the variance to establish the equation \(\frac{1}{4(2\alpha + 1)} = \frac{s^2}{n}\), where \(s^2\) is the sample variance used to solve for \(\alpha\).
03

Maximum Likelihood Estimation (MLE) Equation

The log-likelihood of the data sample is \(\sum_{i=1}^{n} \log(f(x_i|\alpha))\). Differentiate this expression with respect to \(\alpha\), setting it equal to zero leads to the equation:\[ \sum_{i=1}^{n} [\psi(2\alpha) - 2\psi(\alpha) + \log(x_i) + \log(1-x_i)] = 0 \]where \(\psi\) is the digamma function.
04

Finding Asymptotic Variance of MLE

For the asymptotic variance, we use Fisher's information, \(I(\alpha)\), which can be obtained by differentiating \(-\log(f(x|\alpha))\) twice with respect to \(\alpha\). The asymptotic variance is \(I(\alpha)^{-1}\). Specifically, \[ I(\alpha) = n \left[ \psi'(2\alpha) - 2\psi'(\alpha) \right] \] and the asymptotic variance of the MLE is this inverse.
05

Finding a Sufficient Statistic

By the factorization theorem, the statistic \(T = \sum_{i=1}^{n} \log(x_i(1-x_i))\) is sufficient for \(\alpha\). It captures all the information about \(\alpha\) reflected in the sample, as it directly appears in the likelihood function and therefore summarizes the sample's relevant features concerning \(\alpha\).

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Key Concepts

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

Method of Moments
In statistics, the Method of Moments is an approach used to estimate parameters of a distribution. This technique involves comparing sample moments to theoretical moments derived from the probability distribution.

**What are Moments?**
  • Moments are quantitative measures related to the shape of a function's graph.
  • The nth moment can be thought of as the expected value of a variable raised to the nth power.

**How does the Method Work?**
  • The first step is to express the moments of the theoretical distribution in terms of its parameters.
  • These theoretical moments are set equal to the corresponding sample moments calculated from the observed data.
  • Solving these equations gives us estimates of the parameters.

In the exercise context, because the expected value of the beta distribution is 0.5, the variance is used to estimate the parameter \( \alpha \) by setting \(\frac{1}{4(2 \alpha+1)} = \frac{s^2}{n}\) and solving for \( \alpha \).
Maximum Likelihood Estimation (MLE)
Maximum Likelihood Estimation (MLE) is a method used for estimating the parameters of a statistical model. This approach identifies parameter values that maximize the likelihood of the observed data.

**Understanding Likelihood**
  • The likelihood function measures how probable the observed data is given certain parameter values.
  • It is constructed by taking the product of the probability density functions of the observed data points.

**The Process of MLE**
  • First, write down the likelihood function for the parameters.
  • Then, take its logarithm to simplify calculations鈥攖his is known as the log-likelihood function.
  • Differentiate the log-likelihood function with respect to the parameters and set the derivatives equal to zero to solve for the parameters.

In our example, the MLE for \( \alpha \) involves differentiating the log-likelihood of the beta distribution and setting it equal to zero to derive an equation involving the digamma function.
Asymptotic Variance
Asymptotic Variance refers to the variability of an estimator as the sample size tends towards infinity. It provides insight into the precision of an estimator in larger samples.

**Why is it Important?**
  • It helps determine how the estimated parameter would behave in large samples.
  • It is crucial in confidence interval construction and hypothesis testing.

**Relation to Fisher Information**
  • The asymptotic variance can be derived using Fisher's information, \( I(\alpha) \), which measures the amount of information a sample provides about a parameter.
  • The asymptotic variance is the inverse of Fisher's information.

In the problem we are considering, Fisher's information involves the derivative of the digamma function and allows the calculation of the asymptotic variance as \( I(\alpha)^{-1} \).
Sufficient Statistic
A Sufficient Statistic is a function of the data that provides as much information about a parameter as the entire dataset.

**Key Features**
  • It reduces the data to a simpler form with no loss of information about the parameter of interest.
  • Finding a sufficient statistic can simplify analyses and calculations.

**Factorization Theorem**
  • This theorem provides a way of finding a sufficient statistic by factoring the likelihood function into two parts: one that depends solely on the data through the sufficient statistic, and the other that does not involve the parameter.

In the exercise, the statistic \(T = \sum_{i=1}^{n} \log(x_i(1-x_i))\) is a sufficient statistic for \( \alpha \) because it appears directly in the likelihood function, capturing all required information about \( \alpha \).

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Most popular questions from this chapter

The exponential distribution is \(f(x ; \lambda)=\lambda e^{-\lambda x}\) and \(E(X)=\lambda^{-1} .\) The cumulative distribution function is \(F(x)=P(X \leq x)=1-e^{-\lambda x} .\) Three observations are made by an instrument that reports \(x_{1}=5\) and \(x_{2}=3,\) but \(x_{3}\) is too large for the instrument to measure and it reports only that \(x_{3}>10 .\) (The largest value the instrument can measure is \(10.0 .\) ) a. What is the likelihood function? b. What is the mle of \(\lambda ?\)

In Example A of Section \(8.4,\) we used knowledge of the exact form of the sampling distribution of \(\hat{\lambda}\) to estimate its standard error by $$s_{\hat{\lambda}}=\sqrt{\frac{\hat{\lambda}}{n}}$$ This was arrived at by realizing that \(\sum X_{i}\) follows a Poisson distribution with parameter \(n \lambda_{0} .\) Now suppose we hadn't realized this but had used the bootstrap, letting the computer do our work for us by generating \(B\) samples of size \(n=23\) of Poisson random variables with parameter \(\lambda=24.9,\) forming the mle of \(\lambda\) from each sample, and then finally computing the standard deviation of the resulting collection of estimates and taking this as an estimate of the standard error of \(\hat{\lambda}\) Argue that as \(B \rightarrow \infty,\) the standard error estimated in this way will tend to \(s_{\hat{\lambda}}\).

Let \(X_{1}, \ldots, X_{n}\) be i.i.d. uniform on \([0, \theta].\) a. Find the method of moments estimate of \(\theta\) and its mean and variance. b. Find the mle of \(\theta\) c. Find the probability density of the mle, and calculate its mean and variance. Compare the variance, the bias, and the mean squared error to those of the method of moments estimate. d. Find a modification of the mle that renders it unbiased.

Consider an i.i.d. sample of random variables with density function $$f(x | \sigma)=\frac{1}{2 \sigma} \exp \left(-\frac{|x|}{\sigma}\right)$$ a. Find the method of moments estimate of \(\sigma .\) b. Find the maximum likelihood estimate of \(\sigma\). c. Find the asymptotic variance of the mle. d. Find a sufficient statistic for \(\sigma\).

Suppose that in the population of twins, males \((M)\) and females \((F)\) are equally likely to occur and that the probability that twins are identical is \(\alpha .\) If twins are not identical, their genes are independent. a. Show that $$P(M M)=P(F F)=\frac{1+\alpha}{4} \quad P(M F)=\frac{1-\alpha}{2}$$ b. Suppose that \(n\) twins are sampled. It is found that \(n_{1}\) are \(M M, n_{2}\) are \(F F,\) and \(n_{3}\) are \(M F,\) but it is not known which twins are identical. Find the mle of \(\alpha\) and its variance.
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