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Chapter 5: Problem 9

A family of functions \(\mathcal{F}\) is equicontinuous at the point \(x_{0}\) if, given \(\varepsilon>0\), there exists \(\delta\) such that \(\left|f(x)-f\left(x_{0}\right)\right|<\varepsilon\) for all \(\left|x-x_{0}\right|<\delta\) and all \(f \in \mathcal{F}\). (The same \(\delta\) works for all \(f\).) The family is equicontinuous if it is equicontinuous at each \(x_{0}\). Theorem 8.6 (Communicated by L. Gajek) Consider estimation of \(\theta\) with loss \(L(\theta, \delta)\), where \(X \sim f(x \mid \theta)\) is continuous in \(\theta\) for each \(x\). If (i) The family \(L(\theta, \delta(x))\) is equicontinuous in \(\theta\) for each \(\delta\). (ii) For all \(\theta, \theta^{\prime} \in \Omega\), $$ \sup _{x} \frac{f\left(x \mid \theta^{\prime}\right)}{f(x \mid \theta)}<\infty $$ Then, any finite-valued risk function \(R(\theta, \delta)=E_{\theta} L(\theta, \delta)\) is continuous in \(\theta\) and, hence, the estimators with finite, contimous risks form a complete class. (a) Prove Theorem 8.6. (b) Give an example of an equicontinuous family of loss functions. [Hint: Consider squared error loss with a bounded sample space.]

Short Answer

Expert verified
Theorem 8.6 holds due to equicontinuity and bounded likelihood ratios ensuring continuity of risk functions. Squared error loss in a bounded space is an equicontinuous family.

Step by step solution

01

Understanding Equicontinuity and Its Application

The concept of equicontinuity refers to a family of functions \( \mathcal{F} \) being equicontinuous at a point \( x_0 \) if, for any \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that \( |f(x) - f(x_0)| < \varepsilon \) holds uniformly for all \( f \in \mathcal{F} \). This implies that the behavior of all functions in the family is constrained similarly around \( x_0 \), ensuring a form of uniform continuity. In Theorem 8.6, this property is applied to loss functions across a parameter space.
02

Analyzing Condition (i): Equicontinuity of Loss Functions

First, ensure that for the loss function family \( L(\theta, \delta(x)) \), the condition of equicontinuity in \( \theta \) is satisfied for each decision function \( \delta \). This implies that for any \( \varepsilon > 0 \), there is a \( \delta \theta > 0 \) such that for all \( \theta \) and \( \theta' \) with \( |\theta - \theta'| < \delta \theta \), the inequality \( |L(\theta, \delta(x)) - L(\theta', \delta(x))| < \varepsilon \) holds uniformly for all functions in the family.
03

Evaluating Condition (ii): Bounded Likelihood Ratios

Condition (ii) ensures the family of likelihood functions is uniformly bounded. The supremum of the likelihood ratio \( \sup _{x} \frac{f\left(x \mid \theta^{\prime}\right)}{f(x \mid \theta)} \) is bounded, ensuring that for any pair \( \theta, \theta' \) in the parameter space, the ratio of likelihoods cannot grow unbounded. This plays a critical role in ensuring the continuity of the risk function.
04

Proving Continuity of Risk Function

Combine Conditions (i) and (ii) to show \( R(\theta, \delta) = E_{\theta} L(\theta, \delta) \) is continuous in \( \theta \). Because the loss functions are equicontinuous in \( \theta \), any change in \( \theta \) has a uniformly small effect across the sample space, and the bounding of likelihood ratios controls the contribution of individual events. Thus, the overall expectation changes smoothly with \( \theta \). Therefore, \( R(\theta, \delta) \) is continuous.
05

Example: Squared Error Loss and Uniform Boundedness

For an example of an equicontinuous family of loss functions, consider squared error loss \( L(\theta, \delta(x)) = (\theta - \delta(x))^2 \) with a uniformly bounded sample space. If the decision \( \delta(x) \) varies within bounded limits, \( L(\theta, \delta(x)) \) remains bounded and equicontinuous over \( \theta \) because within any small interval, changes in \( \theta \) will cause only small uniform changes in \( L(\theta, \delta(x)) \).

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Key Concepts

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

Likelihood Ratio
The likelihood ratio is a concept used to compare two statistical models' fit to observed data. It is defined as the ratio of the likelihoods of two different hypotheses given the same observed data. In mathematical terms, the likelihood ratio for comparing two parameter values \(\theta\) and \(\theta'\) is given by:\[\text{Likelihood Ratio} = \frac{f(x \mid \theta')}{f(x \mid \theta)}\]where \(f(x \mid \theta)\) is the probability density function for the observed data when the parameter \(\theta\) is true. The use of likelihood ratios is crucial because:
  • It allows for the evaluation of how well an assumed model explains observed data compared to an alternative model.
  • In hypothesis testing, the likelihood ratio helps determine which of the two hypotheses is more consistent with the observed data.
  • For the given condition in Theorem 8.6, ensuring that the likelihood ratio is bounded is necessary for risk function continuity. This boundedness prevents the likelihood ratio from diverging, which can lead to smoother estimates of parameters.
Understanding this concept is particularly useful in situations where continuous probability models describe the data, and comparisons need to be made between different possible parameter values.
Risk Function
The risk function is a mathematical function used in statistics to measure the expected loss when using a statistical procedure, such as an estimator. It helps quantify how "risky" or reliable an estimator might be with respect to unknown parameters. The risk function \( R(\theta, \delta) \) is expressed as:\[R(\theta, \delta) = E_{\theta} L(\theta, \delta)\]where \(L(\theta, \delta)\) is the loss function, \(\delta\) represents a decision or estimation rule, and \(E_{\theta}\) denotes the expected value taken over the distribution defined by \(\theta\).Key points to understand about risk functions include:
  • The function provides a way to evaluate the performance of different decision rules under various parameter settings.
  • In the context of Theorem 8.6, if the conditions are met, the risk function is continuous. This continuity is important because it indicates that small changes in parameter values result in small changes in the expected loss.
  • The process of minimizing the risk function over a class of decision rules is a central theme in decision theory and statistical estimation.
Squared Error Loss
Squared error loss is a specific type of loss function often used in statistical estimation and regression analysis. It measures the squared difference between the actual parameter value and the estimator (or decision rule). Mathematically, it is defined as:\[L(\theta, \delta(x)) = (\theta - \delta(x))^2\]This form of loss function is simple yet very powerful and is frequently used because:
  • It penalizes larger errors more than smaller ones, which makes it sensitive to large discrepancies between the estimated and true values.
  • It has useful analytical properties, like differentiability, which facilitates mathematical manipulation and optimization.
  • In terms of equicontinuity, the squared error loss with a bounded sample space ensures that small changes in the parameter \(\theta\) lead to small, uniform changes in the loss value across the sample space.
The properties of the squared error loss make it a practical choice in many real-world applications where balancing between estimation accuracy and computational simplicity is desired.

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Most popular questions from this chapter

For \(X \mid \theta \sim N_{r}(\theta, I)\), George (1986a, 1986b) looked at multiple shrinkage estimators, those that can shrink to a number of different targets. Suppose that \(\theta \sim \pi(\theta)=\) \(\sum_{j=1}^{k} \omega_{i} \pi_{i}(\theta)\), where the \(\omega_{i}\) are known positive weights, \(\sum \omega_{i}=1\). (a) Show that the Bayes estimator against \(\pi(\theta)\), under squared error loss, is given by \(\delta^{*}(\mathbf{x})=\mathbf{x}+\nabla \log m^{*}(\mathbf{x})\) where \(m^{*}(\mathbf{x})=\sum_{j=1}^{k} \omega_{j} m_{j}(\mathbf{x})\) and $$ m_{i}(\mathbf{x})=\int_{\Omega} \frac{1}{(2 \pi)^{p / 2}} e^{-(1 / 2)|\mathbf{x}-\boldsymbol{\theta}|^{2}} \pi_{i}(\theta) d \theta $$ (b) Clearly, \(\delta^{*}\) is minimax if \(m^{*}(\mathbf{x})\) is superharmonic. Show that \(\delta^{*}(\mathbf{x})\) is minimax if either (i) \(m_{i}(\mathbf{x})\) is superharmonic, \(i=1, \ldots, k\), or (ii) \(\pi_{i}(\theta)\) is superharmonic, \(i=1, \ldots, k_{.}[\)Hint: Problem 1.7.16] (c) The real advantage of \(\delta^{*}\) occurs when the components specify different targets. For \(\rho_{j}=\omega_{j} m_{j}(x) / m^{*}(x)\), let \(\delta^{*}(\mathbf{x})=\sum_{j=1}^{k} \rho_{j} \delta_{j}^{+}(\mathbf{x})\) where $$ \delta_{j}^{+}(\mathbf{x})=\mu_{j}+\left(1-\frac{r-2}{\left|\mathbf{x}-\mu_{j}\right|^{2}}\right)^{+}\left(\mathbf{x}-\mu_{j}\right) $$ and the \(\mu_{j}\) 's are target vectors. Show that \(\delta^{*}(\mathbf{x})\) is minimax. [Hint: Problem 5.19] [George (1986a, 1986b) investigated many types of multiple targets, including multiple points, subspaces, and clusters and subvectors. The subvector problem was also considered by Berger and Dey (1983a, 1983b). Multiple shrinkage estimators were also investigated by \(\mathrm{Ki}\) and Tsui (1990) and Withers (1991).]

For the most part, the risk function of a Stein estimator increases as \(|\theta|\) moves away from zero (if zero is the shrinkage target). To guarantee that the risk function is monotone increasing in \(|\theta|\) (that is, that there are no "dips" in the risk as in Berger's 1976 a tail minimax estimators) requires a somewhat stronger assumption on the estimator (Casella 1990). Let \(X \sim N_{r}(\theta, I)\) and \(L(\theta, \delta)=|\theta-\delta|^{2}\), and consider the Stein estimator $$ \delta(\mathbf{x})=\left(1-c\left(|\mathbf{x}|^{2}\right) \frac{(r-2)}{|\mathbf{x}|^{2}}\right) \mathbf{x} $$ (a) Show that if \(0 \leq c(\cdot) \leq 2\) and \(c(-)\) is concave and twice differentiable, then \(\delta(\mathbf{x})\) is minimax. [Hint: Problem 1.7.7.] (b) Under the conditions in part (a), the risk function of \(\delta(\mathbf{x})\) is nondecreasing in \(|\theta|\). [Hint: The conditions on \(c(\cdot)\), together with the identity $$ (d / d \lambda) E_{\dot{\lambda}}\left[h\left(\chi_{p}^{2}(\lambda)\right)\right]=E_{\lambda}\left\\{\left[\partial / \partial \chi_{p+2}^{2}(\lambda)\right] h\left(\chi_{p+2}^{2}(\lambda)\right)\right\\} $$ where \(\chi_{\rho}^{2}(\lambda)\) is a noncentral \(\chi^{2}\) random variable with \(p\) degrees of freedom and noncentrality parameter \(\lambda\), can be used to show that \(\left(\partial / \partial|\theta|^{2}\right) R(\theta, \delta)>0\).]

The estimation problem of \((5.18)\), $$ \begin{aligned} \mathbf{X} & \sim N(\theta, \Sigma) \\ L(\theta, \delta) &=(\theta-\delta)^{\prime} Q(\theta-\delta) \end{aligned} $$ where both \(\Sigma\) and \(Q\) are positive definite matrices, can always be reduced, without loss of generality, to the simpler case, $$ \begin{aligned} \mathbf{Y} & \sim N(\boldsymbol{\eta}, I) \\ L\left(\boldsymbol{\eta}, \delta^{*}\right) &=\left(\eta-\delta^{*}\right)^{\prime} D_{q^{*}}\left(\eta-\delta^{*}\right) \end{aligned} $$ where \(D_{q^{\circ}}\) is a diagonal matrix with elements \(\left(q_{1}^{*}, \ldots, q_{r}^{*}\right)\), using the following argument. Define \(R=\Sigma^{1 / 2} B\), where \(\Sigma^{1 / 2}\) is a symmetric square root of \(\Sigma\) (that is, \(\left.\Sigma^{1 / 2} \Sigma^{1 / 2}=\Sigma\right)\), and \(B\) is the matrix of eigenvectors of \(\Sigma^{1 / 2} Q \Sigma^{1 / 2}\) (that is, \(B^{\prime} \Sigma^{1 / 2} Q \Sigma^{1 / 2} B=D_{4^{*}}\) ). (a) Show that \(R\) satisfies $$ R^{\prime} \Sigma^{-1} R=I, \quad R^{\prime} Q R=D_{q^{*}} $$ (b) Define \(\mathbf{Y}=R^{-1} \mathbf{X}\). Show that \(\mathbf{Y} \sim N(\eta, I)\), where \(\eta=R^{-1} \boldsymbol{\theta}\). (c) Show that estimations problems are equivalent if we define \(\delta^{*}(\mathbf{Y})=\) \(R^{-1} \delta(R \mathbf{Y})\) [Note: If \(\Sigma\) has the eigenvalue-eigenvector decomposition \(P^{\prime} \Sigma P=D=\) diagonal \(\left(d_{1}, \cdots, d_{r}\right)\), then we can define \(\Sigma^{1 / 2}-P D^{1 / 2} P^{\prime}\), where \(D^{1 / 2}\) is a diagonal matrix with elements \(\sqrt{d_{i}}\). Since \(\Sigma\) is positive definite, the \(d_{i}\) 's are positive.]

Show that an estimator \([1 /(1+\lambda)+\varepsilon] X\) of \(E_{\theta}(X)\) is inadmissible (with squared error loss) under each of the following conditions: (a) if \(\operatorname{var}_{\theta}(X) / E_{\theta}^{2}(X)>\lambda>0\) and \(\varepsilon>0\) (b) if \(\operatorname{var}_{\theta}(X) / E_{\theta}^{2}(X)<\lambda\) and \(\varepsilon<0\) [ Hint: (a) Differentiate the risk function of the estimator with respect to \(\varepsilon\) to show that it decreases as \(\varepsilon\) decreases (Karlin 1958).]

Consider the problem of estimating the mean based on \(X \sim N_{r}(\theta, I)\), where it is thought that \(\theta_{l}=\sum_{j=1}^{s} \beta_{j} t_{i}^{j}\) where \(\left(t_{i}, \ldots, t_{r}\right)\) are known, \(\left(\beta_{1}, \ldots, \beta_{s}\right)\) are unknown, and \(r-s>2\). (a) Find the MLE of \(\theta\), say \(\bar{\theta}_{R}\), if \(\theta\) is assumed to be in the linear subspace $$ (b) Show that \(\mathcal{L}\) can be written in the form \((6.7)\), and find \(K\). (c) Construct a Stein estimator that shrinks toward the MLE of part (a) and prove that it is minimax. \mathcal{L}=\left\\{\theta: \sum_{j=1}^{s} \beta_{j} t_{i}^{\prime}=\theta_{i}, \quad i=1, \ldots, r\right\\} $$
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