/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Let \(X_{i j}, j=1, \ldots, n_{i... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 23

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 $$ \delta\left(\mathbf{x}_{1}+a_{1}, \ldots, \mathbf{x}_{s}+a_{s}, w\right)=\delta\left(\mathbf{x}_{1}, \ldots, \mathbf{x}_{s}, w\right)+\Sigma a_{i} c_{i} $$

Short Answer

Expert verified
The estimation of \(\theta\) is invariant and the estimator \(\delta\) is equivariant under the given transformations.

Step by step solution

01

Define the Problem and Transformations

The problem involves a distribution density based on transformed variables: \( f_i(\mathbf{x}_i - \xi_i) \) for \( X_{ij} \) and \( W \). We aim to calculate a parameter \( \theta = \Sigma c_i \xi_i \) and show invariance under specific transformations. The transformations given are \( X_{ij}' = X_{ij} + a_i \), \( \xi_i' = \xi_i + a_i \), \( \theta' = \theta + \Sigma a_i c_i \), and \( d' = d + \Sigma a_i c_i \).
02

Analyze Distribution Invariance under Transformation

Consider the transformed variables: after applying \( X_{ij}' = X_{ij} + a_i \) and \( \xi_i' = \xi_i + a_i \), the shifts cancel out in \( X_{ij}' - \xi_i' = (X_{ij} + a_i) - (\xi_i + a_i) = X_{ij} - \xi_i \). This implies the forms \( f_i(\mathbf{x}_i' - \xi_i') \) remain the same as \( f_i(\mathbf{x}_i - \xi_i) \). Hence, the distribution remains invariant.
03

Analyze Loss Function Invariance

For the loss function \( L(\xi_1', ..., \xi_s'; d') = \rho(d' - \theta') \), substituting \( d' = d + \Sigma a_i c_i \) and \( \theta' = \theta + \Sigma a_i c_i \), we find: \( \rho((d + \Sigma a_i c_i) - (\theta + \Sigma a_i c_i)) = \rho(d - \theta) \). Thus, the loss function remains invariant, showing the problem setup is invariant under stated transformations.
04

Condition for Equivariant Estimator

An estimator \( \delta \) is equivariant if shifting the inputs by \( a_i \) leads to an appropriate increase in \( \delta \). That is, estimate with shifted inputs should equal the original estimate plus \( \Sigma a_i c_i \):\[ \delta(\mathbf{x}_1 + a_1, ..., \mathbf{x}_s + a_s, w) = \delta(\mathbf{x}_1, ..., \mathbf{x}_s, w) + \Sigma a_i c_i \].This reflects a consistent shift in \( \delta \) proportional to the change in \( \theta \), maintaining the estimator's relative evaluation.

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Key Concepts

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

Equivariant Estimator
An equivariant estimator ensures that changes made to input data reflect proportionally in the outcome of the estimation. In our context, consider the estimator \( \delta \) for the parameter \( \theta \). This estimator is said to be equivariant if for any shift \( a_i \) applied to the inputs \( \mathbf{x}_i \), the estimator鈥檚 output adjusts by the same amount as the shift applied to the estimated parameter. This is succinctly captured by the equation: \[ \delta(\mathbf{x}_1 + a_1, \, ..., \, \mathbf{x}_s + a_s, \, w) = \delta(\mathbf{x}_1, \, ..., \, \mathbf{x}_s, \, w) + \Sigma a_i c_i \] Essentially, this means that if you apply a transformation to the input data, the estimator will adapt by shifting its estimate accordingly. This property is crucial because it maintains the estimator鈥檚 consistency across transformed datasets and ensures predictions are stable as the data changes. It allows us to handle data transformations without compromising the accuracy or reliability of the estimations.
Loss Function Invariance
Invariance in a loss function indicates that the function鈥檚 outcome does not change despite transformations applied to input parameters. Within this exercise, the loss function \( L(\xi_1, \ldots, \xi_s; d) = \rho(d - \theta) \) features this invariance, even when transformations such as shifts are introduced.When inputs are shifted, the transformed loss function \( L(\xi_1', \ldots, \xi_s'; d') = \rho(d' - \theta') \) yields the same result as the original. This is evident by substituting: - \( d' = d + \Sigma a_i c_i \) - \( \theta' = \theta + \Sigma a_i c_i \)Subsequently, the expression simplifies back to the original form: \[ \rho((d + \Sigma a_i c_i) - (\theta + \Sigma a_i c_i)) = \rho(d - \theta) \] This invariance is essential for ensuring that the effectiveness and penalty of the estimator remain consistent, regardless of the transformation applied. By maintaining a stable evaluation criterion, practitioners can trust that any adjustments in data do not inadvertently skew outcomes, preserving the integrity of decision-making.
Transformation Invariance
Transformation invariance in a statistical context signifies that the property of interest remains unaffected by certain transformations of data. Specifically, the distribution of the variables in this exercise maintains its form under the given transformations. When each variable \( X_{ij} \) in a dataset is increased by a constant \( a_i \), the parameters \( \xi_i \) become \( \xi_i' = \xi_i + a_i \). Despite these changes, the difference \( X_{ij}' - \xi_i' \) simplifies back to \( X_{ij} - \xi_i \), showing that the actual form of the distribution remains untouched:\[ X_{ij}' - \xi_i' = (X_{ij} + a_i) - (\xi_i + a_i) = X_{ij} - \xi_i \] This aspect of transformation invariance ensures that no fundamental properties are altered when data undergoes linear transformations like shifts. Therefore, statistical inferences made are robust to these transformations, providing reliable estimates and decisions across different dataset configurations. Understanding transformation invariance aids in appreciating how data manipulations impact distributions and in making more rigorous conclusions.

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Most popular questions from this chapter

For any density \(f\) of \(\mathbf{X}=\left(X_{1}, \ldots, X_{n}\right)\), the probability of the set \(A=\\{\mathbf{x}: 0<\) \(\left.\int_{-\infty}^{\infty} f(\mathbf{x}-u) d u<\infty\right\\}\) is 1. [Hint: With probability 1, the integral in question is equal to the marginal density of \(\mathbf{Y}=\left(Y_{1}, \ldots, Y_{n-1}\right)\) where \(Y_{i}=X_{i}-X_{n}\), and \(P[0

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.)

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 \(\delta(\mathbf{x}+a, \mathbf{y}+b)=\delta(\mathbf{x}, \mathbf{y})+(b-a)\)

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 \(F(v)=B /(A+B)\).

Let \(A=\left(a_{i j}\right)\) be a nonsingular \(n \times n\) matrix with \(a_{i i}=a\) and \(a_{i j}=b\) for all \(i \neq j\). Determine the elements of \(A^{-1}\). [Hint: Assume that \(A^{-1}=\left(c_{i j}\right)\) with \(c_{i i}=c\) and \(c_{i j}=d\) for all \(i \neq j\), calculate \(c\) and \(d\) as the solutions of the two linear equations \(\Sigma a_{1 j} c_{j 1}=1\) and \(\Sigma a_{1 j} c_{j 2}=0\), and check the product \(A C\).]
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