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Chapter 6: Problem 10

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(N(0, \theta)\) distribution. We want to estimate the standard deviation \(\sqrt{\theta}\). Find the constant \(c\) so that \(Y=\) \(c \sum_{i=1}^{n}\left|X_{i}\right|\) is an unbiased estimator of \(\sqrt{\theta}\) and determine its efficiency.

Short Answer

Expert verified
The constant \(c=\sqrt{\frac{2\pi}{n}}\), makes the estimator \(Y=\sum_{i=1}^{n} |X_{i}|\) an unbiased estimator of \(\sqrt{\theta}\). The efficiency of the estimator cannot be evaluated because the Cram茅r鈥揜ao lower bound for the parameter \(\sqrt{\theta}\) does not exist.

Step by step solution

01

Definition of Unbiased Estimator

An estimator is termed as unbiased if its expected value is equal to the true parameter value that it is estimating. To find the constant \(c\), one needs to calculate the expected value of the estimator \(Y = c\sum_{i=1}^{n} |X_{i}|\) and set this equal to what it is estimating which is, \(\sqrt{\theta}\).
02

Derive the formula for the estimator

Multiply \(Y\) with the random variable \(X_i\), which gives \(YX_{i} = c|X_{i}|\). The expected value of \(YX_i\) will be \(E(YX_{i}) = cE(|X_{i}|)\). Given that the \(X_i\) are from a normal distribution with mean zero and standard deviation \(\sqrt{\theta}\), one can look up that \(E(|X_{i}|) = \sqrt{\theta} / \sqrt{2/\pi}\). Hence, \(E(Y) = c*n*\sqrt{\theta/2\pi}\). To make this estimator unbiased, we set \(E(Y)=\sqrt{\theta}\), which gives the constant \(c=\sqrt{\frac{2\pi}{n}}\).
03

Calculate the efficiency

Efficiency measures how 'precise' an estimator is, usually measured by the estimator's variance. Efficiency is calculated as the ratio of the variance of the theoretical best estimator and the variance of the given estimator. The theoretical best estimator for a parameter is the estimator that achieves the smallest possible variance, also known as the Cram茅r鈥揜ao lower bound. Since, we can't find such estimator for \(\sqrt{\theta}\), it isn't possible to calculate the efficiency for this case.
04

Conclusion

The constant \(c=\sqrt{\frac{2\pi}{n}}\) makes \(Y=\sum_{i=1}^{n} |X_{i}|\) an unbiased estimator for \(\sqrt{\theta}\). The efficiency of this estimator \(Y\) cannot be evaluated against the Cram茅r鈥揜ao lower bound as that bound cannot be derived for the parameter \(\sqrt{\theta}\).

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Most popular questions from this chapter

Recall that \(\widehat{\theta}=-n / \sum_{i=1}^{n} \log X_{i}\) is the mle of \(\theta\) for a \(\operatorname{Beta}(\theta, 1)\) distribution. Also, \(W=-\sum_{i=1}^{n} \log X_{i}\) has the gamma distribution \(\Gamma(n, 1 / \theta)\). (a) Show that \(2 \theta W\) has a \(\chi^{2}(2 n)\) distribution. (b) Using Part (a), find \(c_{1}\) and \(c_{2}\) so that $$P\left(c_{1}<\frac{2 \theta n}{\hat{\theta}}

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Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) be a random sample from a bivariate normal distribution with \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}, \rho=\frac{1}{2}\), where \(\mu_{1}, \mu_{2}\), and \(\sigma^{2}>0\) are unknown real numbers. Find the likelihood ratio \(\Lambda\) for testing \(H_{0}: \mu_{1}=\mu_{2}=0, \sigma^{2}\) unknown against all alternatives. The likelihood ratio \(\Lambda\) is a function of what statistic that has a well- known distribution?
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