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Chapter 5: Problem 15

Let \(X=1\) or 0 with probabilities \(p\) and \(q\), respectively, and consider the estimation of \(p\) with loss \(=1\) when \(|d-p| \geq 1 / 4\), and 0 otherwise. The most general randomized estimator is \(\delta=U\) when \(X=0\), and \(\delta=V\) when \(X=1\) where \(U\) and \(V\) are two random variables with known distributions. (a) Evaluate the risk function and the maximum risk of \(\delta\) when \(U\) and \(V\) are uniform on \((0,1 / 2)\) and \((1 / 2,1)\), respectively. (b) Show that the estimator \(\delta\) of (a) is minimax by considering the three values \(p=0\), \(1 / 2,1\) [Hint: (b) The risk at \(p=0,1 / 2,1\) is, respectively, \(P(U>1 / 4), 1 / 2[P(U<1 / 4)+\) \(P(V>3 / 4)]\), and \(P(V<3 / 4)]\).

Short Answer

Expert verified
The maximum risk is \(\frac{1}{2}\) and the estimator \(\delta\) is minimax.

Step by step solution

01

Calculate Risk for p = 0

For the case where \( p = 0 \), \( X = 0 \) with probability 1. Therefore, \( \delta = U \) always. We check the condition \( |U - p| \geq \frac{1}{4} \), which simplifies to \( U \geq \frac{1}{4} \) because \( p = 0 \). Since \( U \) is uniformly distributed on \([0, \frac{1}{2}]\), the risk is \( P(U > \frac{1}{4}) = 1 - P(U \leq \frac{1}{4}) = 1 - \frac{1}{2} \times \frac{1}{4} = \frac{1}{2} \).
02

Calculate Risk for p = \(\frac{1}{2}\)

For \( p = \frac{1}{2} \), \( X = 1 \) with probability \( \frac{1}{2} \) and \( X = 0 \) with probability \( \frac{1}{2} \). Hence, the risk is a combination: \[R(\delta) = \frac{1}{2} P(U \leq \frac{1}{4}) + \frac{1}{2} P(V > \frac{3}{4})\]On \([0, \frac{1}{2}]\), \( P(U \leq \frac{1}{4}) = \frac{1}{2} \) and for \([\frac{1}{2}, 1]\), \( P(V > \frac{3}{4}) = 1 - P(V \leq \frac{3}{4}) = \frac{1}{4} \). Therefore, the risk comes out to be \( \frac{1}{2}(\frac{1}{2} + \frac{1}{4}) = \frac{3}{8} \).
03

Calculate Risk for p = 1

When \( p = 1 \), \( X = 1 \) with probability 1, meaning \( \delta = V \). We check \( |V - p| \geq \frac{1}{4} \), giving us \( V \leq \frac{3}{4} \). Since \( V \) is uniform on \([\frac{1}{2}, 1]\), \( P(V \leq \frac{3}{4}) = \frac{1}{2} \).
04

Determine Maximum Risk

We identified the risks as \( \frac{1}{2} \), \( \frac{3}{8} \), and \( \frac{1}{2} \) for \( p = 0 \), \( p = \frac{1}{2} \), and \( p = 1 \) respectively. The maximum risk among these values is \( \frac{1}{2} \).
05

Verify Minimax Condition

To be minimax, the maximum risk of the estimator must be equal to the minimum of the maximum risks across all possible estimators. Since the maximum risk is cleary \( \frac{1}{2} \) and applies fairly for the chosen \( p \'s \), no estimator can achieve a lower maximum risk compared to \( \delta \), hence the estimator \( \delta \) is minimax.

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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 type of estimator used in estimation theory. It involves utilizing additional randomness when making estimates about a parameter. In this context, the estimator
  • Facilitates flexibility by incorporating random variables into the estimation process.
  • Handles situations where traditional estimators may not be optimal.
With randomized estimators, decisions are made according to some random process. This can offer advantages over deterministic estimators, especially in pure strategies where accuracy might be compromised by restrictions not present in a probabilistic approach.
In the exercise, the estimator
  • When \(X=0\), assigns a random variable \(U\) defined uniformly on \((0, 1/2)\),
  • When \(X=1\), assigns another random variable \(V\) defined uniformly on \((1/2, 1)\).
This addition of random variables helps to achieve a strategic estimate of the parameter \(p\), aiming to minimize biases in contexts that would be suboptimal without including additional randomization.
Uniform Distribution
Uniform distribution is a fundamental concept in probability and statistics. It refers to a situation where all outcomes in a specific range are equally likely.
  • The distribution is characterized by two parameters, the lower limit \(a\) and the upper limit \(b\), and the probability density function (PDF) is constant between them.
  • For a continuous uniform distribution on an interval \([a, b]\), the probability is given by \[ f(x) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \ 0 & \text{otherwise} \end{cases} \]
In our problem, both \(U\) and \(V\) are distributed uniformly. The random variables have their respective ranges: \(U\) is in the interval \((0,1/2)\) and \(V\) in the interval \((1/2,1)\). This choice reflects different behavior for estimating \(p\) under the constraints, helping ensure decisions made are consistent and reliable despite the randomness introduced.
Risk Function
The risk function quantifies the accuracy of an estimator. It is defined as the expected value of the loss function for a given estimator.
  • It takes into account possible errors in estimation by measuring how far the estimator deviates from the true value.
  • The risk function helps identify how optimal an estimator is, based on minimizing expected losses.
In the given estimation problem, the risk function is calculated under different circumstances - for \(p=0\), \(p=\frac{1}{2}\), and \(p=1\).
For example, when \(p=1\), \(X=1\) occurs with certainty and the risk is derived from \(V\)'s distribution, specifically from the probability \(P(V \leq \frac{3}{4})\). Likewise, when \(p=\frac{1}{2}\), the risk is a combination of \(U\) and \(V\)'s probability values reflecting the chance of significant deviation.
The careful calculation of risk for each situation is pivotal, as it points out where the estimator performs well or may falter under different true value scenarios.
Minimax Estimator
A minimax estimator is a type of estimator that minimizes the maximum possible risk. The idea is to safeguard against the worst-case loss in estimation terms.
  • This approach is vital in decision-making scenarios where avoiding the worst possible outcome is crucial.
  • The essence is to find an estimator that performs uniformly well across all possible parameter values.
In the exercise, the estimator's minimax nature is validated by examining the calculated risks for \(p=0\), \(p=\frac{1}{2}\), and \(p=1\). Each scenario yielded a risk within the expected conditions, showing that the highest risk was limited by an upper threshold of \( \frac{1}{2} \). This indicates that no alternative estimator could lessen the greatest risk, verifying the estimator \(\delta\) as minimax for these particular settings.
The minimax property is crucial for robustness, ensuring estimates are not unreasonable even in the least favorable cases, which ultimately enhances estimator reliability under uncertain conditions.

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Most popular questions from this chapter

A natural extension of risk domination under a particular loss is to risk domination under a class of losses. Hwang (1985) defines universal domination of \(\delta\) by \(\delta^{\prime}\) if the inequality $$ E_{\theta} L\left(\left|\theta-\delta^{\prime}(\mathbf{X})\right|\right) \leq E_{\theta} L(|\theta-\delta(\mathbf{X})|) \text { for all } \theta $$ holds for all loss functions \(L(\cdot)\) that are nondecreasing, with at least one loss function producing nonidentical risks. (a) Show that \(\delta^{\prime}\) universally dominates \(\delta\) if and only if it stochastically dominates \(\delta\), that is, if and only if $$ P_{\theta}\left(\left|\boldsymbol{\theta}-\delta^{\prime}(\mathbf{X})\right|>k\right) \leq P_{\theta}(|\theta-\delta(\mathbf{X})|>k) $$ for all \(k\) and \(\theta\) with strict inequality for some \(\theta\). [Hint: For a positive random variable \(Y\), recall that \(E Y=\int_{0}^{\infty} P(Y>t) d t .\) Alternatively, use the fact that stochastic ordering on random variables induces an ordering on expectations. See Lemma 1, Section \(3.3\) of TSH2.] (b) For \(X \sim N_{r}(\boldsymbol{\theta}, I)\), show that the James-Stein estimator \(\delta^{c}(\mathbf{x})=\left(1-c /\left[\left.\mathbf{x}\right|^{2}\right) \mathbf{x}\right.\) does not universally dominate \(\mathbf{x}\). [From (a), it only need be shown that \(P_{\theta}\left(\left|\theta-\delta^{c}(\mathbf{X})\right|>\right.\) \(k)>P_{\theta}(|\theta-\mathbf{X}|>k)\) for some \(\theta\) and \(k .\) Take \(\theta=0\) and find such a \(k\).] Hwang (1985) and Brown and Hwang (1989) explore many facets of universal domination. Hwang (1985) shows that even \(\delta^{+}\)does not universally dominate \(X\) unless the class of loss functions is restricted. We also note that although the inequality in part (a) may seem reminiscent of the "Pitman closeness" criterion, there is really no relation. The criterion of Pitman closeness suffers from a number of defects not shared by stochastic domination (see Robert et al. 1993).

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 \(\left(\partial / \partial|\theta|^{2}\right) R(\theta, \delta)>0\).]

Let \(X\) and \(Y\) be independently distributed according to Poisson distributions with \(E(X)=\xi\) and \(E(Y)=\eta\), respectively. Show that \(a X+b Y+c\) is admissible for estimating \(\xi\) with squared error loss if and only if either \(0 \leq a<1, b \geq 0, c \geq 0\) or \(a=1, b=c=0\) (Makani 1972).

Let \(X_{i}(i=1, \ldots, n)\) be iid with unknown distribution \(F\). Show that $$ \delta=\frac{\text { No. of } X_{i} \leq 0}{\sqrt{n}} \cdot \frac{1}{1+\sqrt{n}}+\frac{1}{2(1+\sqrt{n})} $$ is minimax for estimating \(F(0)=P\left(X_{i} \leq 0\right)\) with squared error loss. [Hint: Consider the risk function of \(\delta\).]

Efron (1990), in a discussion of Brown's (1990a) ancillarity paradox, proposed an alternate version. Suppose \(\mathbf{X} \sim N_{r}(\mu, I), r>2\), and with probability \(1 / r\), independent of \(\mathbf{X}\), the value of the random variable \(J=j\) is observed, \(j=1,2, \ldots, r\). The problem is to estimate \(\theta_{j}\) using the loss function \(L\left(\theta_{j}, d\right)=\left(\theta_{j}-d\right)^{2}\). Show that, conditional on \(J=j, X_{j}\) is a minimax and admissible estimator of \(\theta_{j}\). However, unconditionally, \(X_{f}\) is dominated by the \(j\) th coordinate of the James-Stein estimator. This version of the paradox may be somewhat more transparent. It more clearly shows how the presence of the ancillary random variable forces the problem to be considered as a multivariate problem, opening the door for the Stein effect.
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