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Given the pdf $$f(x ; \theta)=\frac{1}{\pi\left[1+(x-\theta)^{2}\right]}, \quad-\infty

Short Answer

Expert verified
The Rao-Cramér lower bound is \(2 / n\). Asymptotically as \(n\) goes to infinity, the distribution of \(\sqrt{n}(\widehat{\theta}-\theta)\) becomes standard normal \(N(0,1)\).

Step by step solution

01

Calculation of Fisher Information

From the given p.d.f., we first compute the first derivative of the log-likelihood \(\log f(x ; \theta)\) w.r.t. \(\theta\). Then calculate the Fisher Information by taking expected value of its square.
02

Find the Rao-Cramér Lower Bound

Use the formula for Rao-Cramér Lower Bound which is the reciprocal of the Fisher Information. We compute this by substituting the calculated Fisher Information into this formula.
03

Find the MLE

To compute the MLE, we first write down the likelihood function, which is the product of the density functions. Then take a log and differentiate w.r.t. \(\theta\). Equate this to zero and solve for \(\theta\). The result is \(\widehat{\theta}\).
04

Determine the Asymptotic Distribution

We can use standard results regarding the asymptotic properties of the MLE, which state that as \(n\) approaches infinity, \(\sqrt{n}(\widehat{\theta}-\theta)\) converges in distribution to a normal distribution with mean 0 and variance equal to the reciprocal of the Fisher Information.

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Most popular questions from this chapter

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a \(N\left(\mu_{0}, \sigma^{2}=\theta\right)\) distribution, where \(0<\theta<\infty\) and \(\mu_{0}\) is known. Show that the likelihood ratio test of \(H_{0}: \theta=\theta_{0}\) versus \(H_{1}: \theta \neq \theta_{0}\) can be based upon the statistic \(W=\sum_{i=1}^{n}\left(X_{i}-\mu_{0}\right)^{2} / \theta_{0}\) Determine the null distribution of \(W\) and give, explicitly, the rejection rule for a level \(\alpha\) test.

Consider a location model $$X_{i}=\theta+e_{i}, \quad i=1, \ldots, n$$ where \(e_{1}, e_{2}, \ldots, e_{n}\) are iid with pdf \(f(z)\). There is a nice geometric interpretation for estimating \(\theta .\) Let \(\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)^{\prime}\) and \(\mathbf{e}=\left(e_{1}, \ldots, e_{n}\right)^{\prime}\) be the vectors of observations and random error, respectively, and let \(\mu=\theta 1\) where 1 is a vector with all components equal to one. Let \(V\) be the subspace of vectors of the form \(\mu_{i}\) i.e, \(V=\\{\mathbf{v}: \mathbf{v}=a \mathbf{1}\), for some \(a \in R\\} .\) Then in vector notation we can write the model as $$\mathbf{X}=\boldsymbol{\mu}+\mathbf{e}, \quad \boldsymbol{\mu} \in V$$

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are iid with pdf \(f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0

The Pareto distribution is frequently used a model in study of incomes and has the distribution function $$F\left(x ; \theta_{1}, \theta_{2}\right)=\left\\{\begin{array}{ll} 1-\left(\theta_{1} / x\right)^{\theta_{2}} & \theta_{1} \leq x \\ 0 & \text { elsewhere }\end{array}\right.$$ where \(\theta_{1}>0\) and \(\theta_{2}>0 .\) If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from this distribution, find the maximum likelihood estimators of \(\theta_{1}\) and \(\theta_{2}\).

A machine shop that manufactures toggle levers has both a day and a night shift. A toggle lever is defective if a standard nut cannot be screwed onto the threads. Let \(p_{1}\) and \(p_{2}\) be the proportion of defective levers among those manufactured by the day and night shifts, respectively. We shall test the null hypothesis, \(H_{0}: p_{1}=p_{2}\), against a two-sided alternative hypothesis based on two random samples, each of 1000 levers taken from the production of the respective shifts. Use the test statistic \(Z^{*}\) given in Example \(6.5 .3\). (a) Sketch a standard normal pdf illustrating the critical region having \(\alpha=0.05\). (b) If \(y_{1}=37\) and \(y_{2}=53\) defectives were observed for the day and night shifts, respectively, calculate the value of the test statistic and the approximate \(p-\) value (note that this is a two-sided test). Locate the calculated test statistic on your figure in Part (a) and state your conclusion. Obtain the approximate \(p\) -value of the test.

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