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Chapter 8: Problem 3

Suppose that \(\hat{\theta}\) is an estimator for a parameter \(\theta\) and \(E(\hat{\theta})=a \theta+b\) for some nonzero constants \(a\) and \(b\) a. In terms of \(a, b,\) and \(\theta\), what is \(B(\hat{\theta}) ?\) b. Find a function of \(\hat{\theta}-\operatorname{say}, \hat{\theta}^{*}-\) that is an unbiased estimator for \(\theta\)

Short Answer

Expert verified
a. \( B(\hat{\theta}) = (a-1)\theta + b \); b. \( \hat{\theta}^{*} = \frac{\hat{\theta} - b}{a} \) is an unbiased estimator.

Step by step solution

01

Understanding Bias

An estimator is biased if the expected value of the estimator does not equal the true parameter. The bias \( B(\hat{\theta}) \) of an estimator \( \hat{\theta} \) is given by:\[ B(\hat{\theta}) = E(\hat{\theta}) - \theta \]Substituting \( E(\hat{\theta}) = a\theta + b \), the bias becomes:\[ B(\hat{\theta}) = (a\theta + b) - \theta \]
02

Calculating the Bias

Simplify the expression to find the bias:\[ B(\hat{\theta}) = a\theta + b - \theta \]\[ B(\hat{\theta}) = (a-1)\theta + b \]This expression gives the bias in terms of \( a, b, \) and \( \theta \).
03

Finding an Unbiased Estimator

We need a function of \( \hat{\theta} \), say \( \hat{\theta}^{*} \), such that it is unbiased, meaning:\[ E(\hat{\theta}^{*}) = \theta \]Start by recalling \( E(\hat{\theta}) = a\theta + b \). To correct for bias, define:\[ \hat{\theta}^{*} = \frac{\hat{\theta} - b}{a} \]
04

Verifying Unbiased Estimator Function

Calculate the expected value of \( \hat{\theta}^{*} \):\[ E(\hat{\theta}^{*}) = E\left( \frac{\hat{\theta} - b}{a} \right) = \frac{1}{a}E(\hat{\theta} - b) = \frac{1}{a}(E(\hat{\theta})-b) \]Substitute \( E(\hat{\theta}) = a\theta + b \):\[ E(\hat{\theta}^{*}) = \frac{1}{a}((a\theta + b) - b) = \frac{1}{a}(a\theta) = \theta \]Thus, \( \hat{\theta}^{*} \) is an unbiased estimator for \( \theta \).

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Key Concepts

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

Unbiased Estimator
In statistics, an unbiased estimator is a valuable tool because it provides an accurate estimate of a population parameter. The key characteristic of an unbiased estimator is that its expected value equals the true parameter. This means there is no systematic error in the estimate.
For example, if we have an estimator \( \hat{\theta} \) for a parameter \( \theta \), and its expected value \( E(\hat{\theta}) \) is exactly \( \theta \), then \( \hat{\theta} \) is unbiased. When an estimator is unbiased, it implies that over many samples, its average estimate converges to the true parameter value.
To shift a biased estimator to an unbiased one, adjustments are made based on its bias. The expression for bias is given as \( B(\hat{\theta}) = E(\hat{\theta}) - \theta \). By altering the estimator in a way that this bias becomes zero, we achieve an unbiased estimator.
Expected Value
The expected value is a foundational concept in probability and statistics, often referred to as the mean or average. It provides a measure of the central tendency of a random variable. In the context of estimators, it serves as the average or "expected" output of the estimator if applied an infinite number of times to different samples.
Mathematically, the expected value of an estimator \( \hat{\theta} \) is denoted as \( E(\hat{\theta}) \). When deriving an unbiased estimator, the goal is for this expected value to equal the parameter \( \theta \).
When the expected value of an estimator is different from the parameter, bias is present. Therefore, by making the expected value of an estimator equal to the parameter, we effectively remove any bias. This principle plays an important role in refining estimators for accurate predictions.
Parameter Estimation
Parameter estimation involves using sample data to infer or estimate the unknown parameters of a statistical model. It is a crucial part of statistical inference, allowing us to draw conclusions about the population from which the data is drawn.
The parameter is a number describing some characteristic of the population. Estimators are utilized to estimate these parameters. For example, with a parameter \( \theta \), our estimator \( \hat{\theta} \) will aim to approximate this value based on sample data.
It is important to select or develop appropriate estimators to ensure accurate parameter estimation. Unbiased estimators are preferred as they ensure the expected value equals the parameter, thereby providing reliable estimates over time. Estimation methods that adjust for potential bias result in more precise and trustworthy conclusions.

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Most popular questions from this chapter

The number of persons coming through a blood bank until the first person with type \(A\) blood is found is a random variable \(Y\) with a geometric distribution. If \(p\) denotes the probability that any one randomly selected person will possess type \(A\) blood, then \(E(Y)=1 / p\) and \(V(Y)=(1-p) / p^{2}\) a. Find a function of \(Y\) that is an unbiased estimator of \(V(Y)\). b. Suggest how to form a 2 -standard-error bound on the error of estimation when \(Y\) is used to estimate \(1 / p\).

Using the identity $$(\hat{\theta}-\theta)=[\hat{\theta}-E(\hat{\theta})]+[E(\hat{\theta})-\theta]=[\hat{\theta}-E(\hat{\theta})]+B(\hat{\theta})$$ show that $$ \operatorname{MSE}(\hat{\theta})=E\left[(\hat{\theta}-\theta)^{2}\right]=V(\hat{\theta})+(B(\hat{\theta}))^{2} $$

A factory operates with two machines of type \(A\) and one machine of type \(B\). The weekly repair costs \(X\) for type \(A\) machines are normally distributed with mean \(\mu_{1}\) and variance \(\sigma^{2}\). The weekly repair costs \(Y\) for machines of type \(B\) are also normally distributed but with mean \(\mu_{2}\) and variance \(3 \sigma^{2} .\) The expected repair cost per week for the factory is thus \(2 \mu_{1}+\mu_{2} .\) If you are given a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) on costs of type \(A\) machines and an independent random sample \(Y_{1}, Y_{2}, \ldots, Y_{m}\) on costs for type \(\mathrm{B}\) machines, show how you would construct a \(95 \%\) confidence interval for \(2 \mu_{1}+\mu_{2}\) a. if \(\sigma^{2}\) is known. b. if \(\sigma^{2}\) is not known.

Suppose that \(S^{2}\) is the sample variance based on a sample of size \(n\) from a normal population with unknown mean and variance. Derive a \(100(1-\alpha) \%\) a. upper confidence bound for \(\sigma^{2}\). b. lower confidence bound for \(\sigma^{2}\).

Organic chemists often purify organic compounds by a method known as fractional crystallization. An experimenter wanted to prepare and purify \(4.85 \mathrm{g}\) of aniline. Ten 4.85 -gram specimens of aniline were prepared and purified to produce acetanilide. The following dry yields were obtained: $$\begin{array}{llllllll} 3.85, & 3.88, & 3.90, & 3.62, & 3.72, & 3.80, & 3.85, & 3.36, & 4.01, & 3.82 \end{array}$$ Construct a \(95 \%\) confidence interval for the mean number of grams of acetanilide that can be recovered from 4.85 grams of aniline.
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