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Chapter 7: Problem 3

Let \(Y_{1}

Short Answer

Expert verified
All three, \(4Y_{1}\), \(2Y_{2}\) and \(\frac{4}{3}Y_{3}\), are indeed unbiased estimators of \(\theta\). The variances are \(Var(4Y_{1}) = \frac{16}{n^2}\), \(Var(2Y_{2}) = \frac{4}{(n+1)^2}\), and \(Var(\frac{4}{3}Y_{3}) = \frac{16}{9(n+2)^2}\).

Step by step solution

01

Determine unbiased estimators

Given that an estimator is unbiased if the expected value equals the true parameter it is estimating, for \(4Y_{1}\), \(E[4Y_{1}] = \theta\). Similarly for \(2Y_{2}\), \(E[2Y_{2}] = \theta\) and for \(\frac{4}{3}Y_{3}\), \(E[\frac{4}{3}Y_{3}] = \theta\). As it stands, all three estimators are unbiased.
02

Find the variance of the estimators

Next we have to calculate the variances by using the formulas for the variances of order statistics in a uniform distribution. For \(4Y_{1}\), \(Var(4Y_{1}) = \frac{16}{n^2}\), representing four times the standard variance. The variance for \(2Y_{2}\) would be \(Var(2Y_{2}) = \frac{4}{(n+1)^2}\), representing two times. For \(\frac{4}{3}Y_{3}\), \(Var(\frac{4}{3}Y_{3}) = \frac{16}{9(n+2)^2}\), which gives a standard variance calculation but is multiplied by a factor of \(\frac{4}{3}\).

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Key Concepts

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

Uniform Distribution
The uniform distribution is a fundamental concept in probability and statistics that refers to a situation where all outcomes are equally likely to occur. In the context of continuous random variables, it is often described by its probability density function (pdf), which, for the uniform distribution on the interval from 0 to \theta, is defined as \(f(x; \theta)=\frac{1}{\theta}\), for \(0
Order statistics play a critical role when dealing with samples from a uniform distribution. In the original exercise, three statistics \(Y_{1}
Order Statistics
Order statistics are an essential aspect of statistical analysis which involves the sorting of sample data. When we take a simple random sample and order it from the smallest to the largest value, these are called the order statistics. The nth smallest value is called the nth order statistic. In the exercise given, \(Y_{1}, Y_{2}, Y_{3}\) are the first, second, and third order statistics of a sample size of 3 from a uniform distribution.

The importance of order statistics is seen in their potential to create unbiased estimators of population parameters. These extreme values are particularly insightful because they encapsulate the range of the data in a sample. Understanding how to manipulate these can facilitate efficient estimation of unknown parameters within a population. The problem at hand demonstrates this by deriving unbiased estimators based on each of the three order statistics.
Estimator Variance
In statistics, the concept of estimator variance is a measure of how much an estimator would vary if we were to repeatedly sample from the same population. The variance gives us an idea of the estimator's reliability; lower variance means more reliability or precision. When considering unbiased estimators, the challenge often is to find the one with the lowest variance, thereby achieving the most efficient estimation for the parameter of interest.

In the exercise, after establishing that \(4Y_{1}, 2Y_{2}\), and \(\frac{4}{3} Y_{3}\) are unbiased estimators of \(\theta\), the next step involves calculating their respective variances. The variances are obtained using formulas specific to the distribution of the order statistics stemming from a uniform distribution. The variance calculations reflect how each estimator's distribution spreads around the expected value (\(\theta\)) and thus their reliability in estimating the parameter. By comparing the variances of \(4Y_{1}, 2Y_{2}\), and \(\frac{4}{3} Y_{3}\), one can determine which estimator would on average be closest to the true value of \(\theta\) with repeated sampling.

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Most popular questions from this chapter

Let \(Y_{1}

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a distribution with pdf \(f(x ; \theta)=(1 / 2) \theta^{3} x^{2} e^{-\theta x}, 0

If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a distribution that has a pdf which is a regular case of the exponential class, show that the pdf of \(Y_{1}=\sum_{1}^{n} K\left(X_{i}\right)\) is of the form \(f_{Y_{1}}\left(y_{1} ; \theta\right)=R\left(y_{1}\right) \exp \left[p(\theta) y_{1}+n q(\theta)\right]\). Hint: Let \(Y_{2}=X_{2}, \ldots, Y_{n}=X_{n}\) be \(n-1\) auxiliary random variables. Find the joint pdf of \(Y_{1}, Y_{2}, \ldots, Y_{n}\) and then the marginal pdf of \(Y_{1}\).

Let \(Y\) denote the median and let \(\bar{X}\) denote the mean of a random sample of size \(n=2 k+1\) from a distribution that is \(N\left(\mu, \sigma^{2}\right)\). Compute \(E(Y \mid \bar{X}=\bar{x})\). Hint: See Exercise \(7.5 .4 .\)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a distribution that is \(N(\mu, \theta), 0<\theta<\infty\), where \(\mu\) is unknown. Let \(Y=\sum_{1}^{n}\left(X_{i}-\bar{X}\right)^{2} / n\) and let \(\mathcal{L}[\theta, \delta(y)]=[\theta-\delta(y)]^{2}\). If we consider decision functions of the form \(\delta(y)=b y\), where \(b\) does not depend upon \(y\), show that \(R(\theta, \delta)=\left(\theta^{2} / n^{2}\right)\left[\left(n^{2}-1\right) b^{2}-2 n(n-1) b+n^{2}\right]\). Show that \(b=n /(n+1)\) yields a minimum risk decision function of this form. Note that \(n Y /(n+1)\) is not an unbiased estimator of \(\theta\). With \(\delta(y)=n y /(n+1)\) and \(0<\theta<\infty\), determine \(\max _{\theta} R(\theta, \delta)\) if it exists.
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