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Chapter 4: Problem 6

If \(Y_{1}, \ldots, Y_{n} \stackrel{\text { iid }}{\sim} N\left(\mu, c \mu^{2}\right)\), where \(c\) is a known constant, show that the minimal sufficient statistic for \(\mu\) is the same as for the \(N\left(\mu, \sigma^{2}\right)\) distribution. Find the maximum likelihood estimate of \(\mu\) and give its large-sample standard error. Show that the distribution of \(\bar{Y}^{2} / S^{2}\) does not depend on \(\mu\).

Short Answer

Expert verified
The minimal sufficient statistic for \(\mu\) is \(\bar{Y}\). MLE for \(\mu\) is \(\hat{\mu} = \bar{Y}\). The large-sample standard error is \(\sqrt{\frac{c\hat{\mu}^2}{n}}\).

Step by step solution

01

Identify the Likelihood Function

The likelihood function for the sample \(Y_1, \ldots, Y_n\) from \(N(\mu, c\mu^2)\) is given by:\[ L(\mu) = \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi c\mu^{2}}} \exp\left(-\frac{(Y_i - \mu)^2}{2c\mu^{2}}\right) \].
02

Simplify to Find Sufficient Statistic

Since \( -2 \) times the log-likelihood involves terms like \( \sum_{i=1}^n (Y_i - \mu)^2 \), observe that \( \sum (Y_i)^2, \bar{Y}, n \) appear. By the factorization theorem, the sufficient statistic for \( \mu \) is \(T = (\sum Y_i, \sum Y_i^2, n)\) which is minimized to \(\bar{Y}\) when the distribution is \(N(\mu, \sigma^2)\).
03

Maximum Likelihood Estimation of \(\mu\)

The likelihood is the same subject to the variance in terms of \(\mu\), so the MLE estimator for \(\mu\) following transformations is \(\hat{\mu} = \bar{Y}\), as derived by equating the derivative of the logarithm of the likelihood to zero.
04

Large-Sample Standard Error

The variance for population distribution is limited to \( \frac{c\mu^2}{n} \). Thus, the standard error of the sample mean \(\bar{Y}\) is on approximation \(\sqrt{\frac{c\mu^2}{n}}\), indicating the large-sample standard error gets obtained as \(\sqrt{\frac{c\hat{\mu}^2}{n}}\).
05

Verify Independence of Ratio \(\bar{Y}^2/S^2\)

Given \(\bar{Y}\) and \(S^2\) being statistics containing \(\mu\), substituting their expressions obtains terms cancelling out \(\mu\). Examining \(\bar{Y}=\frac{1}{n}\sum Y_i\) and \(S^2=\frac{1}{n-1}\sum (Y_i-\bar{Y})^2\), the expression \(\bar{Y}^2/S^2\) is independent of \(\mu\), as they rely on derived independent distributions of chi-square and normal.

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Key Concepts

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

Maximum Likelihood Estimation
Maximum Likelihood Estimation, or MLE, is a method used to estimate the parameters of a statistical model. The goal is to find the parameter values that make the observed data most probable. For instance, if our data are believed to follow a normal distribution with unknown mean \(8\), we want to find the \(\u00b5\) that maximizes the likelihood of observing the sample data we have.
  • We start by defining a likelihood function, \(L(\mu)\), which describes how likely each observed data point is, given various possible values of \(\u00b5\).
  • In this exercise, the likelihood function for data \((Y_1, \ldots, Y_n)\) from the normal distribution \(N(\u00b5, c\u00b5^2)\) involves the product of terms involving each \(Y_i\).
  • By taking the derivative of the log-likelihood, we solve for \(\u00b5\) by setting its derivative to zero, leading to the Maximum Likelihood Estimate: \(\hat{\mu} = \bar{Y}\).
Sufficient Statistic
A sufficient statistic is a valuable concept in statistical inference. It summarizes the data in a way that provides all the information needed to estimate a parameter.Essentially, if you know the value of the sufficient statistic, you don't gain any new information about the parameter from knowing the actual data itself.
  • In the given problem, the sufficient statistic for \(\mu\) in a normal distribution \(N(\u00b5, \sigma^2)\) is the sample mean \(\bar{Y}\).
  • By applying the factorization theorem, we identify that the data combination \(T = (\sum Y_i, \sum Y_i^2, n)\) serves as a base to form sufficient statistics.
  • It simplifies further to just the sample mean \(\bar{Y}\), which is a concise summary of the sample conceived as a sufficient statistic when minimizing redundancy.
Statistical Distribution
Statistical distributions describe how values are spread or distributed for a random variable. The normal distribution is fundamental in statistics due to its unique properties.When we say a sample \(Y_1, \ldots, Y_n\) is distributed as \(N(\u00b5, c\u00b5^2)\), we are referring to each data point being normally distributed with mean \(\u00b5\) and variance \(c\u00b5^2\).
  • These parameters (8 and variance) characterize the curve's center and spread, respectively.
  • In practice, many large datasets approximate normality due to the Central Limit Theorem, making this distribution essential for practical analysis.
  • Understanding how \(\mu\) and \(c\u00b5^2\) affect the appearance of the distribution helps in predicting the behavior of the sample data.
Large Sample Theory
Large sample theory, often known as asymptotic theory, deals with the behavior of estimators as the sample size grows larger. It is fundamental in understanding the reliability of statistical estimates for large samples.
  • An engaging example of this theory is the approxi mation of the sampling distribution of the estimator, \(\bar{Y}\), which, as per the Central Limit Theorem, is typically normal or nearly so for large \(n\).
  • As in the provided problem, knowing the population variance \(c\u00b5^2/n\), we derive the standard error that describes spread or variation around \(\hat{\mu}\) for large samples. This simplifies to \(\sqrt{c\u00b5^2/n}\).
  • Thus, large sample theory promises that as sample size increases, the estimates (like the MLE) converge to true parameter values, offering confidence in their use with growing data.

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Most popular questions from this chapter

Eggs are thought to be infected with a bacterium salmonella enteriditis so that the number of organisms, \(Y\), in each has a Poisson distribution with mean \(\mu .\) The value of \(Y\) cannot be observed directly, but after a period it becomes certain whether the egg is infected \((Y>0)\) or not \((Y=0)\). Out of \(m\) such eggs, \(r\) are found to be infected. Find the maximum likelihood estimator \(\widehat{\mu}\) of \(\mu\) and its asymptotic variance. Is the exact variance of \(\widehat{\mu}\) defined?

Let \(Y_{1}, \ldots, Y_{n}\) and \(Z_{1}, \ldots, Z_{m}\) be two independent random samples from the \(N\left(\mu_{1}, \sigma_{1}^{2}\right)\) and \(N\left(\mu_{2}, \sigma_{2}^{2}\right)\) distributions respectively. Consider comparison of the model in which \(\sigma_{1}^{2}=\sigma_{2}^{2}\) and the model in which no restriction is placed on the variances, with no restriction on the means in either case. Show that the likelihood ratio statistic \(W_{\mathrm{p}}\) to compare these models is large when the ratio \(T=\sum\left(Y_{j}-\bar{Y}\right)^{2} / \sum\left(Z_{j}-\bar{Z}\right)^{2}\) is large or small, and that \(T\) is proportional to a random variable with the \(F\) distribution.

Find the expected information for \(\theta\) based on a random sample \(Y_{1}, \ldots, Y_{n}\) from the geometric density $$ f(y ; \theta)=\theta(1-\theta)^{y-1}, \quad y=1,2,3, \ldots, 0<\theta<1 $$ A statistician has a choice between observing random samples from the Bernoulli or geometric densities with the same \(\theta\). Which will give the more precise inference on \(\theta ?\)

Suppose that \(\partial \eta^{\mathrm{T}} / \partial \theta\) is symbolically rank-deficient, that is, there exist \(\gamma_{r}(\theta)\), non-zero for all \(\theta\), such that $$ \sum_{r=1}^{p} \gamma_{r}(\theta) \frac{\partial \eta_{j}}{\partial \theta_{r}}=0, \quad j=1, \ldots, n $$ Show that the auxiliary equations $$ \frac{d \theta_{1}}{\gamma_{1}(\theta)}=\cdots=\frac{d \theta_{p}}{\gamma_{p}(\theta)} $$ have \(p-1\) solutions given implicitly by \(\beta_{t}(\theta)=c_{t}\) for constants \(c_{1}, \ldots, c_{p-1} .\) Deduce that the model is parameter redundant. (Catchpole and Morgan, 1997)

In a first-order autoregressive process, \(Y_{0}, \ldots, Y_{n}\), the conditional distribution of \(Y_{j}\) given the previous observations, \(Y_{1}, \ldots, Y_{j-1}\), is normal with mean \(\alpha y_{j-1}\) and variance one. The initial observation \(Y_{0}\) has the normal distribution with mean zero and variance one. Show that the log likelihood is proportional to \(y_{0}^{2}+\sum_{j=1}^{n}\left(y_{j}-\alpha y_{j-1}\right)^{2}\), and hence find the maximum likelihood estimate of \(\alpha\) and the observed information.
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