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Chapter 3: Problem 10

Consider the loss function $$ \rho(t)=\left\\{\begin{aligned} -A t & \text { if } t<0 \\ B t & \text { if } t \geq 0 \quad(A, B \geq 0) . \end{aligned}\right. $$ If \(X\) is a random variable with density \(f\) and distribution function \(F\), show that \(E \rho(X-v)\) is minimized for any \(v\) satisfying \(F(v)=B /(A+B)\).

Short Answer

Expert verified
The expected loss \(E[\rho(X-v)]\) is minimized when \(F(v) = \frac{B}{A+B}\).

Step by step solution

01

Define the Expected Loss Function

The function we want to minimize is the expected value of the loss function, \(E[\rho(X-v)]\). This can be expressed as an integral: \[E[\rho(X-v)] = \int_{-\infty}^{\infty} \rho(x-v) f(x) \, dx.\] To evaluate this integral, we need to split it into two parts based on the definition of \(\rho(t)\).
02

Split the Integral According to the Loss Function

Split the integral at the point \(x = v\) since the function \(\rho(x-v)\) changes form at this point. The expected loss becomes: \[E[\rho(X-v)] = \int_{-\infty}^{v} (-A(x-v)) f(x) \, dx + \int_{v}^{\infty} (B(x-v)) f(x) \, dx.\]
03

Calculate Each Integral

Calculate the first integral: \[ -A \int_{-\infty}^{v} (x-v) f(x) \, dx = -A \left[ \int_{-\infty}^{v} x f(x) \, dx - v \int_{-\infty}^{v} f(x) \, dx \right].\] For the second integral: \[ B \int_{v}^{\infty} (x-v) f(x) \, dx = B \left[ \int_{v}^{\infty} x f(x) \, dx - v \int_{v}^{\infty} f(x) \, dx \right].\]
04

Formulate the Complete Expected Loss Expression

Combine the results from Step 3:\[E[\rho(X-v)] = -A \left[\int_{-\infty}^{v} x f(x)\, dx - v F(v) \right] + B \left[\int_{v}^{\infty} x f(x)\, dx - v (1-F(v)) \right],\] where \(F(v)\) is the cumulative distribution function at \(v\).
05

Differentiate with Respect to v and Set to Zero

Find the derivative of \(E[\rho(X-v)]\) with respect to \(v\) and set it to zero to find the minimum:\[ \frac{d}{dv}E[\rho(X-v)] = A F(v) - Av f(v) + B (1 - F(v)) - Bv f(v).\] Setting this equal to zero gives:\[ AF(v) - BF(v) = B.\]
06

Solve for F(v)

Combine like terms in the equation from Step 5:\[ (A+B)F(v) = B.\] Solve for \(F(v)\), giving:\[ F(v) = \frac{B}{A+B}.\] This is the condition that minimizes the expected loss function.

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Key Concepts

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

Expected Value
The expected value is a fundamental concept in probability and statistics, often represented as \(E[X]\), where \(X\) is a random variable. It is essentially the mean or average of a random variable, providing a single value that summarizes the entire distribution of possible outcomes. In terms of the loss function, the expected value evaluates the average loss we can expect, based on the probability distribution of \(X\).
This calculation is crucial when determining optimal decisions under uncertainty, as it helps to minimize potential losses or maximize benefits. By using the concept of expected value, one can aggregate all possible scenarios into a singular prediction, facilitating a more informed decision-making process.
The expected value of a function \(g(X)\) regarding a random variable \(X\) with density function \(f(x)\) is given by the integral: \[E[g(X)] = \int_{-\infty}^{\infty} g(x) f(x) \, dx. \]
This expression ensures that all possible outcomes are considered, weighted by their likelihoods.
Random Variable Distribution
Random variable distribution refers to how probabilities are assigned to different outcomes. It provides a complete description of the likelihood of these outcomes coming into play. For a continuous random variable, the distribution is often described using a probability density function (pdf), \(f(x)\).
The pdf indicates how the probability is distributed over different values. It allows us to calculate the probability of the random variable falling within a specific range by integrating the pdf over that range. A key feature of the pdf is that the area under the curve equals one.
Distribution functions provide a powerful tool to model and analyze stochastic behaviors. They serve as the foundation for various statistical measures, like mean and variance, and aid in understanding the characteristics of the data they represent. Different distributions, like normal, binomial, or Poisson, fit different types of data depending on their properties.
Minimization Problem
A minimization problem seeks to find the smallest possible value of a function, often called the objective function. In the context of the exercise, the goal is to minimize the expected value of the loss function \(E[\rho(X-v)]\).
Minimizing this expected loss means finding the optimal 'cut-off' point \(v\) that reduces the average loss we anticipate for various outcomes of the random variable \(X\). Such problems are common in economics, engineering, and various decision-making scenarios where one needs to minimize cost, error, or risk.
The solution typically involves setting the derivative of the objective function to zero, revealing the conditions under which the minimum occurs. In this problem, setting the derivative to zero and solving for \(v\) ensures that we find the point at which the expected loss function reaches its minimum value.
Cumulative Distribution Function
The cumulative distribution function (CDF) \(F(x)\) of a random variable \(X\) describes the probability that \(X\) will take a value less than or equal to \(x\). It provides a complete view of the probability structure by mapping each possible value of the random variable to its cumulative probability.
Mathematically, it is given by the integral of the probability density function \(f(x)\): \[F(x) = \int_{-\infty}^{x} f(t) \, dt.\]
The CDF is a non-decreasing function with values ranging from 0 to 1 as \(x\) moves from negative infinity to positive infinity.
In the context of the minimization problem, the CDF helps define the condition that minimizes the expected loss. Specifically, the problem's solution \(F(v) = \frac{B}{A+B}\) utilizes the CDF to balance the probabilities and values of losses such that overall expected loss is minimized. This elegant usage of the CDF captures both the likelihood and severity of outcomes, guiding to an optimal decision point.

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Most popular questions from this chapter

Lele (1993) uses invariance in the study of mophometrics, the quantitative analysis of biological forms. In the analysis of a biological object, one measures data \(\mathbf{X}\) on \(k\) specific points called landmarks, where each landmark is typically two- or three-dimensional. Here we will assume that the landmark is two-dimensional (as is a picture), so \(\mathbf{X}\) is a \(k \times 2\) matrix. A model for \(\mathbf{X}\) is $$ \mathbf{X}=(M+\mathbf{Y}) \Gamma+\mathbf{t} $$ where \(M_{k \times 2}\) is the mean form of the object, \(\mathbf{t}\) is a fixed translation vector, and \(\Gamma\) is a \(2 \times 2\) matrix that rotates the vector \(\mathbf{X}\). The random variable \(\mathbf{Y}_{k \times 2}\) is a matrix normal random variable, that is, each column of \(\mathbf{Y}\) is distributed as \(N\left(0, \Sigma_{k}\right)\), a \(k\)-variate normal random variable, and each row is distributed as \(N\left(0, \Sigma_{d}\right)\), a bivariate normal random variable. (a) Show that \(\mathbf{X}\) is a matrix normal random variable with columns distributed as \(N_{k}\left(M \Gamma_{j}, \Sigma_{k}\right)\) and rows distributed as \(N_{2}\left(M_{i} \Gamma, \Gamma^{\prime} \Sigma_{d} \Gamma\right)\), where \(\Gamma_{j}\) is the \(j\) th column of \(\Gamma\) and \(M_{i}\) is the \(i\) th row of \(M\). (b) For estimation of the shape of a biological form, the parameters of interest are \(M\), \(\Sigma_{k}\) and \(\Sigma_{d}\), with \(\mathbf{t}\) and \(\Gamma\) being nuisance parameters. Show that, even if there were no nuisance parameters, \(\Sigma_{k}\) or \(\Sigma_{d}\) is not identifiable. (c) It is usually assumed that the \((1,1)\) element of either \(\Sigma_{k}\) or \(\Sigma_{d}\) is equal to 1 . Show that this makes the model identifiable. (d) The form of a biological object is considered an inherent property of the form (a baby has the same form as an adult) and should not be affected by rotations, reflections, or translations. This is summarized by the transformation $$ \mathbf{X}^{\prime}=\mathbf{X} P+b $$ where \(P\) is a \(2 \times 2\) orthogonal matrix \(\left(P^{\prime} P=I\right)\) and \(b\) is a \(k \times 1\) vector. (See Note 9.3 for a similar group.) Suppose we observe \(n\) landmarks \(\mathbf{X}_{1}, \cdots, \mathbf{X}_{n}\). Define the Euclidean distance between two matrices \(A\) and \(B\) to be \(D(A, B)=\sum_{i j}\left(a_{i j}-b_{i j}\right)^{2}\), and let the \(n \times n\) matrix \(F\) have \((i, j)\) th element \(f_{i j}=D\left(\mathbf{X}_{i}, \mathbf{X}_{j}\right)\). Show that \(F\) is invariant under this group, that is \(F\left(\mathbf{X}^{\prime}\right)=F(\mathbf{X})\). (Lele (1993) notes that \(F\) is, in fact, maximal invariant.)

Let \(X_{i j}, j=1, \ldots, n_{i}, i=1, \ldots, s\), and \(W\) be distributed according to a density of the form $$ \left[\prod_{i=1}^{s} f_{i}\left(\mathbf{x}_{i}-\xi_{i}\right)\right] h(w) $$ where \(\mathbf{x}_{i}-\xi_{i}=\left(x_{i 1}-\xi_{i}, \ldots, x_{i n_{i}}-\xi_{i}\right)\), and consider the problem of estimating \(\theta=\Sigma c_{i} \xi_{i}\) with loss function \(L\left(\xi_{i}, \ldots, \xi_{s} ; d\right)=\rho(d-\theta)\). Show that: (a) This problem remains invariant under the transformations $$ \begin{gathered} X_{i j}^{\prime}=X_{i j}+a_{i}, \quad \xi_{i}^{\prime}=\xi_{i}+a_{i}, \quad \theta^{\prime}=\theta+\Sigma a_{i} c_{i} \\ d^{\prime}=d+\Sigma a_{i} c_{i} \end{gathered} $$ (b) An estimator \(\delta\) of \(\theta\) is equivariant under these transformations if $$ \delta\left(\mathbf{x}_{1}+a_{1}, \ldots, \mathbf{x}_{s}+a_{s}, w\right)=\delta\left(\mathbf{x}_{1}, \ldots, \mathbf{x}_{s}, w\right)+\Sigma a_{i} c_{i} $$

Suppose \(X_{1}, \ldots, X_{m}\) and \(Y_{1}, \ldots, Y_{n}\) have joint density \(f\left(x_{1}-\xi, \ldots, x_{m}-\xi ; y_{1}-\right.\) \(\left.\eta, \ldots, y_{n}-\eta\right)\) and consider the problem of estimating \(\Delta=\eta-\xi\). Explain why it is desirable for the loss function \(L(\xi, \eta ; d)\) to be of the form \(\rho(d-\Delta)\) and for an estimator \(\delta\) of \(\Delta\) to satisfy \(\delta(\mathbf{x}+a, \mathbf{y}+b)=\delta(\mathbf{x}, \mathbf{y})+(b-a)\)

For \(\mathbf{X}_{1}, \ldots, \mathbf{X}_{n}\) iid as \(N_{p}(\mu, \Sigma)\), the cross-products matrix \(S\) is defined by $$ S=\left\\{S_{i j}\right\\}=\sum_{k=1}^{n}\left(x_{i_{k}}-\bar{x}_{i}\right)\left(x_{j_{k}}-\bar{x}_{j}\right) $$ where \(\bar{x}_{i}=(1 / n) \sum_{k=1}^{n} x_{i_{k}}\). Show that, for \(\Sigma=I\), (a) \(E_{I}[\operatorname{tr} S]=E_{I} \sum_{i=1}^{p} \sum_{k=1}^{n}\left(X_{i_{k}}-\bar{X}_{i}\right)\left(X_{i_{k}}-\bar{X}_{i}\right)=p(n-1)\), (b) \(E_{I}\left[\operatorname{tr} S^{2}\right]=E_{I} \sum_{i=1}^{p} \sum_{j=1}^{p}\left\\{\sum_{k=1}^{n}\left(X_{i_{k}}-\bar{X}_{i}\right)\left(X_{j_{k}}-\bar{X}_{j}\right)\right\\}^{2}=(n-1)(n p-p-1)\) [These are straightforward, although somewhat tedious, calculations involving the chisquared distribution. Alternatively, one can use the fact that \(S\) has a Wishart distribution (see, for example, Anderson 1984\()\), and use the properties of that distribution.]

Let \(\left(X_{i}, Y_{i}\right), i=1, \ldots, n\), be distributed as independent bivariate normal random variables with mean \((\mu, 0)\) and covariance matrix $$ \left(\begin{array}{ll} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{array}\right) $$ (a) Show that the probability model is invariant under the transformations $$ \begin{aligned} \left(x^{\prime}, y^{\prime}\right) &=(a+b x, b y) \\ \left(\mu^{\prime}, \sigma_{11}^{\prime}, \sigma_{12}^{\prime}, \sigma_{22}^{\prime}\right) &=\left(a+b \mu, b^{2} \sigma_{11}, b^{2} \sigma_{12}, b^{2} \sigma_{22}\right) \end{aligned} $$ (b) Using the loss function \(L(\mu, d)=(\mu-d)^{2} / \sigma_{11}\), show that this is an invariant estimation problem, and equivariant estimators must be of the form \(\delta=\bar{x}+\psi\left(u_{1}, u_{2}, u_{3}\right) \bar{y}\), where \(u_{1}=\Sigma\left(x_{i}-\bar{x}\right)^{2} / \bar{y}^{2}, u_{2}=\Sigma\left(y_{i}-\bar{y}\right)^{2} / \bar{y}^{2}\), and \(u_{3}=\Sigma\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right) / \bar{y}^{2}\) (c) Show that if \(\delta\) has a finite second moment, then it is unbiased for estimating \(\mu\). Its risk function is a function of \(\sigma_{11} / \sigma_{22}\) and \(\sigma_{12} / \sigma_{22}\) (d) If the ratio \(\sigma_{12} / \sigma_{22}\) is known, show that \(\bar{X}-\left(\sigma_{12} / \sigma_{22}\right) \bar{Y}\) is the MRE estimator of \(\mu\).
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