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Chapter 2: Problem 22

Let \(X\) take on the values 1 and 0 with probability \(p\) and \(q\), respectively, and assume that \(1 / 4

Short Answer

Expert verified
\(\delta^{*}\) is unbiased and generally has a lower risk than \(X\).

Step by step solution

01

Define the Randomized Estimator

The randomized estimator \( \delta^* \) is defined based on the value of \( X \). If \( X = 0 \), \( \delta^* \) uses a random variable \( Y_0 \) uniformly distributed over the interval \( (-1/2, 1/2) \). If \( X = 1 \), \( \delta^* \) uses a random variable \( Y_1 \) uniformly distributed over the interval \( (1/2, 3/2) \). These random variables provide randomized estimates based on the observed value of \( X \).
02

Check Unbiasedness of \(\delta^{*}\)

To show that \( \delta^{*} \) is unbiased, we calculate the expected value of \( \delta^{*} \). For \( X = 0 \), the expected value is calculated by integrating the uniform distribution over \((-1/2, 1/2)\). For \( X = 1 \), the expected value is calculated over \((1/2, 3/2)\). Since \( E[Y_0] = 0 \) and \( E[Y_1] = 1 \), the expected value across both conditions becomes \( q \cdot 0 + p \cdot 1 = p \). Thus, \( E[\delta^{*}] = p \), which shows that \( \delta^{*} \) is unbiased.
03

Calculate Risk of \(\delta^{*}\)

The risk function with loss \( L(p, d) \) is calculated as the probability that \( |\delta^{*} - p| \geq 1/4 \). Given the distribution of \( Y_0 \) and \( Y_1 \), for \( X = 0 \), \( \delta^* \) is more than \(1/4\) away from \(p\) if \(p \geq 1/4\), and for \( X = 1 \), \( \delta^* \) is less than \(1/4\) away from \(p\) if \(p \leq 3/4\). The probability of making an error is low because the distributions are constructed to cover the neighborhood of \(p\) effectively.
04

Risk Comparison with \(X\)

For the estimator \( X \), the risk is the probability that \( |X - p| \geq 1/4 \). If \( X = 0 \), the error is \( p \), and if \( X = 1 \), the error is \( 1-p \). Comparing with \( \delta^{*} \), which minimizes the distance to \( p \), \( \delta^{*} \) has a generally lower or equivalent risk because it adapts based on uniform distributions that cover the distance within \( 1/4 \) about \( p \). As a consequence, \( \delta^{*} \) potentially offers lower risk than the simple estimator \( X \).

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Key Concepts

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

Randomized Estimator
A randomized estimator is a technique used to estimate an unknown parameter in a way that involves some form of randomness. In the context of this exercise, the randomized estimator \( \delta^{*} \) is designed to provide estimates of the probability \( p \) by utilizing two random variables, \( Y_0 \) and \( Y_1 \), which are chosen based on the observed value of \( X \).

When \( X = 0 \), \( \delta^{*} \) selects \( Y_0 \), a random variable uniformly distributed over the interval \((-1/2, 1/2)\). When \( X = 1 \), it picks \( Y_1 \), which is uniformly distributed over the range \( (1/2, 3/2) \).

The randomness in these distributions helps cover the possible values around the true parameter \( p \). This method of estimation aims to minimize errors by distributing potential estimation outcomes within specific intervals tailored to the different possible states of \( X \).

To check for unbiased estimation, we consider the expected value of \( \delta^{*} \). Since \( E[Y_0] = 0 \) and \( E[Y_1] = 1 \), the overall expected value of the estimator \( \delta^{*} \) remains \( p \), which concludes that \( \delta^{*} \) is an unbiased estimator.
Risk Function
The risk function in the context of estimation measures the expected loss involved in using a particular estimator to approximate a parameter. It provides an assessment of an estimator's accuracy over many trials or samples.

For this exercise, the risk function is determined by how often \( \delta^{*} \) produces estimates significantly different from the true value \( p \). Specifically, it evaluates the probability of the difference \(|\delta^{*} - p| \) being greater than \( 1/4 \). If \( \delta^{*} \) stays within this range, the loss is zero, otherwise, the loss is one.

The risk is calculated by considering the probabilities associated with the distributions of \( Y_0 \) and \( Y_1 \). The carefully chosen intervals for these distributions ensure that the estimates generally hover around the true \( p \), reducing the likelihood of large errors.

When compared to using the estimator \( X \) itself, \( \delta^{*} \) provides a generally lower or equivalent risk due to its adaptation to the parameter \( p \). The tailored intervals effectively cover the neighborhood around \( p \), thereby maintaining the estimation errors within acceptable bounds.
Uniform Distribution
A uniform distribution is one where all outcomes in the specified interval are equally likely to occur. This type of distribution is straightforward because it assumes no additional biases toward any particular value within the range. In this exercise, random variables \( Y_0 \) and \( Y_1 \) used in the randomized estimator \( \delta^{*} \) are examples of uniform distributions.

\( Y_0 \) is uniformly distributed over \((-1/2, 1/2)\), which means that any number within this interval could be selected with equal probability when \( X = 0 \). Similarly, \( Y_1 \) is uniformly distributed over the interval \( (1/2, 3/2) \), applied when \( X = 1 \).

These uniform distributions are especially useful because they provide a simple and unbiased way of covering the potential values for \( p \), helping to stabilize the estimation process. The uniform distribution offers an equal chance of selecting any value within the range, thereby allowing for a well-balanced and dependable estimation process without a favor towards any specific outcome.

By employing these distributions, \( \delta^{*} \) is able to manage the uncertainty in such a way that the estimates largely remain close to the true parameter \( p \), minimizing larger errors.

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Most popular questions from this chapter

In this problem, we establish some facts about eigenvalues and eigenvectors of square matrices. (For a more general treatment, see, for example, Marshall and Olkin 1979, Chapter 20.) We use the facts that a scalar \(\lambda>0\) is an eigenvalue of the \(n \times n\) symmetric matrix \(A\) if there exists an \(n \times 1\) vector \(p\), the corresponding eigenvector, satisfying \(A p=\lambda p\). If \(\mathrm{A}\) is nonsingular, there are \(n\) eigenvalues with corresponding linearly independent eigenvectors. (a) Show that \(A=P^{\prime} D_{\lambda} P\), where \(D_{\lambda}\) is a diagonal matrix of eigenvalues of \(A\) and \(P\) is and \(n \times n\) matrix whose rows are the corresponding eigenvalues that satisfies \(P^{\prime} P=P P^{\prime}=I\), the identity matrix. (b) Show that \(\max _{x} \frac{x^{\prime} A x}{x^{\prime} x}=\) largest eigenvalue of \(A\). (c) If \(B\) is a nonsingular symmetric matrix with eigenvector-eigenvalue representation \(B=Q^{\prime} D_{\beta} Q\), then \(\max _{x} \frac{x^{\prime} A x}{x^{\prime} B x}=\) largest eigenvalue of \(A^{*}\), where \(A^{*}=\) \(D_{\beta}^{-1 / 2} Q A Q^{\prime} D_{\beta}^{-1 / 2}\) and \(D_{\beta}^{-1 / 2}\) is a diagonal matrix whose elements are the reciprocals of the square roots of the eigenvalues of \(B\). (d) For any square matrices \(C\) and \(D\), show that the eigenvalues of the matrix \(C D\) are the same as the eigenvalues of the matrix \(D C\), and hence that \(\max _{x} \frac{x^{\prime} A x}{x^{\prime} B x}=\) largest eigenvalue of \(A B^{-1}\). (e) If \(A=a a^{\prime}\), where \(a\) is a \(n \times 1\) vector ( \(A\) is thus a rank-one matrix), then \(\max _{x} \frac{x^{\prime} a a^{\prime} x}{x^{\prime} B x}=\) \(a^{\prime} B^{-1} a\).

(a) Let \(X\) have density (with respect to \(\mu\) ) \(p(x, \theta)\) which is \(>0\) for all \(x\), and let \(\Lambda_{1}\) and \(\Lambda_{2}\) be two distributions on the real line with finite first moments. Then, any unbiased estimator \(\delta\) of \(\theta\) satisfies $$ \operatorname{var}(\delta) \geq \frac{\left[\int \Delta d \Lambda_{1}(\Delta)-\int \Delta d \Lambda_{2}(\Delta)\right]^{2}}{\int \psi^{2}(x, \theta) p(x, \theta) d \mu(x)} $$ where $$ \psi(x, \theta)=\frac{\int_{\Omega_{\theta}} p(x, \theta+\Delta)\left[d \Lambda_{1}(\Delta)-d \Lambda_{2}(\Delta)\right]}{p(x, \theta)} $$ with \(\Omega_{\theta}=\left\\{\Delta: \theta+\Delta_{\varepsilon} \Omega\right\\}\). (b) If \(\Lambda_{1}\) and \(\Lambda_{2}\) assign probability 1 to \(\Delta=0\) and \(\Delta\), respectively, the inequality reduces to (5.6) with \(g(\theta)=\theta\). [Hint: Apply (5.1).] (Kiefer 1952.)

(a) Any two random variables \(X\) and \(Y\) with finite second moments satisfy the covariance inequality \([\operatorname{cov}(X, Y)]^{2} \leq \operatorname{var}(X) \cdot \operatorname{var}(Y)\). (b) The inequality in part (a) is an equality if and only if there exist constants \(a\) and \(b\) for which \(P(X=a Y+b)=1\).

Suppose \(X\) is distributed on \((0,1)\) with probability density \(p_{\theta}(x)=(1-\theta)+\theta / 2 \sqrt{x}\) for all \(0

Verify the following statements, asserted by Basu (1988, Chapter 1 ), which illustrate the relationship between information, sufficiency, and ancillarity. Suppose that we let \(I(\theta)=E_{\theta}\left[-\partial^{2} / \partial \theta^{2} \log f(x \mid(\theta)]\right.\) be the information in \(X\) about \(\theta\) and let \(J(\theta)=E_{\theta}\left[-\partial^{2} / \partial \theta^{2} \log g(T \mid \theta)\right]\) be the information about \(\theta\) contained in a statistic \(T\), where \(g(\cdot \mid \theta)\) is the density function of \(T\). Define \(\lambda(\theta)=I(\theta)-J(\theta)\), a measure of information lost by using \(T\) instead of \(X .\) Under suitable regularity conditions, show that (a) \(\lambda(\theta) \geq 0\) for all \(\theta\) (b) \(\lambda(\theta)=0\) if and only if \(T\) is sufficient for \(\theta\). (c) If \(Y\) is ancillary but \((T, Y)\) is sufficient, then \(I(\theta)=E_{\theta}[J(\theta \mid Y)]\), where $$ J(\theta \mid y)=E_{\theta}\left[-\frac{\partial^{2}}{\partial \theta^{2}} \log h(T \mid y, \theta) \mid Y=y\right] $$ and \(h(t \mid y, \theta)\) is the conditional density of \(T\) given \(Y=y\).
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