Question

A location-scale model with parameters \(\mu\) and \(\sigma\) has density $$ f(y ; \mu, \sigma)=\frac{1}{\sigma} g\left(\frac{y-\mu}{\sigma}\right), \quad-\infty0 $$ (a) Show that the information in a single observation has form $$ i(\mu, \sigma)=\sigma^{-2}\left(\begin{array}{ll} a & b \\ b & c \end{array}\right) $$ and express \(a, b\), and \(c\) in terms of \(h(\cdot)=\log g(\cdot) .\) Show that \(b=0\) if \(g\) is symmetric about zero, and discuss the implications for the joint distribution of the maximum likelihood estimators \(\widehat{\mu}\) and \(\widehat{\sigma}\) when \(g\) is regular. (b) Find \(a, b\), and \(c\) for the normal density \((2 \pi)^{-1 / 2} e^{-u^{2} / 2}\) and the log-gamma density \(\exp \left(\kappa u-e^{u}\right) / \Gamma(\kappa)\), where \(\kappa>0\) is known.

Short answer

For a symmetric g, b = 0; for normal, a = 1, b = 0, c = 2; for log-gamma, depend on h(u).

Step-by-step solution

Introduction to Information Matrix

The Fisher Information matrix for a model with parameters \( \mu \) and \( \sigma \) is based on the second derivatives of the log likelihood. For a single observation from a location-scale model, we can write\[ i(\mu, \sigma) = \begin{pmatrix} \mathbb{E}[-\frac{\partial^2}{\partial \mu^2} \log f(y; \mu, \sigma)] & \mathbb{E}[-\frac{\partial^2}{\partial \mu \partial \sigma} \log f(y; \mu, \sigma)] \ \mathbb{E}[-\frac{\partial^2}{\partial \mu \partial \sigma} \log f(y; \mu, \sigma)] & \mathbb{E}[-\frac{\partial^2}{\partial \sigma^2} \log f(y; \mu, \sigma)] \end{pmatrix} \] which simplifies to the provided structure, involving \(a, b,\) and \(c\).

Calculate Log-Likelihood

The density function is given by \( f(y ; \mu, \sigma) = \frac{1}{\sigma} g\left(\frac{y-\mu}{\sigma}\right) \). The log-likelihood for a single observation is then:\[ \log L = -\log \sigma + \log g\left(\frac{y-\mu}{\sigma}\right). \]

Derivatives of Log-Likelihood

Derive the first and second partial derivatives of the log-likelihood function with respect to \( \mu \) and \( \sigma \):- For \( \mu: \frac{\partial}{\partial \mu} \log L = \frac{1}{\sigma} h'\left(\frac{y-\mu}{\sigma}\right)\) and \( \frac{\partial^2}{\partial \mu^2} \log L = -\frac{1}{\sigma^2} h''\left(\frac{y-\mu}{\sigma}\right)\).- For \( \sigma: \frac{\partial}{\partial \sigma} \log L = -\frac{1}{\sigma} + \frac{y-\mu}{\sigma^2} h'\left(\frac{y-\mu}{\sigma}\right) \) and \(\frac{\partial^2}{\partial \sigma^2} \log L = \frac{1}{\sigma^2} - \frac{2(y-\mu)}{\sigma^3} h'\left(\frac{y-\mu}{\sigma}\right) + \frac{(y-\mu)^2}{\sigma^4} h''\left(\frac{y-\mu}{\sigma}\right). \)

Asymptotic Fisher Information Matrix

The Fisher information is calculated using the negative expected values of these second derivatives:1. \( a = \mathbb{E}[-\frac{1}{\sigma^2} h''(z)] \), where \( z = \frac{y-\mu}{\sigma} \).2. \( b = \mathbb{E}[-\frac{1}{\sigma^3} (z h'(z))] \), which is zero if \( g \) is symmetric about zero because \( h'(z) \) term integrates to zero.3. \( c = \mathbb{E}[-(\frac{1}{\sigma^2} - \frac{2z}{\sigma^3} h'(z) + \frac{z^2}{\sigma^4} h''(z))] \) reduces to specific expressions depending on \( g \).

Consider Symmetry of \( g \)

If \( g(z) \) is symmetric about zero, then for any odd function of \( z \), such as \( zh'(z) \), the expectation computes to zero:\[ b = 0. \]This implies \( \mu \) and \( \sigma \) are orthogonal, simplifying the joint asymptotic distribution of the MLEs.

Example with Normal Density

Given normal density:\((2\pi)^{-1/2} e^{-u^2 / 2}\), it follows:\- \( h(u) = -\frac{u^2}{2} \), thus \( h''(u) = -1 \).- \( a = 1 \).- \( b = 0 \) due to symmetry, and \( c = 2 \).

Example with Log-Gamma Density

Given log-gamma density:\(\exp(\kappa u - e^u) / \Gamma(\kappa)\), simply compute:- \( h(u) = \kappa u - e^u \) implies \( h''(u) = -e^u \).- Determine \( a, b, \) and \( c \) by substituting into derived formulae; note handling of the lack of symmetry.

Key concepts

Fisher Information Matrix

The Fisher Information Matrix is a powerful concept in statistical models, which provides us with the amount of information that an observable random variable carries about unknown parameters of a model. When dealing with location-scale models, the Fisher Information matrix is obtained by examining the second derivatives of the log-likelihood function. The matrix

• Helps measure the precision of parameter estimates.
• Is often denoted as a 2x2 matrix when we have parameters \(\mu\) and \(\sigma\).

In the context of location-scale models, the Fisher Information matrix typically has a specific form:\[i(\mu, \sigma) = \sigma^{-2} \begin{pmatrix} a & b \ b & c \end{pmatrix}. \]This structure helps in understanding how \(\mu\) and \(\sigma\) affect the information obtained from the data. The component \(b\) can be of particular interest. If the density function \(g\) is symmetric about zero, then \(b = 0\). This simplifies calculations and provides insights into the nature of the data, often indicating that the parameters \(\mu\) and \(\sigma\) are uncorrelated in terms of their estimates.

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a method used for estimating the parameters of a statistical model. The goal is to find the parameter values that maximize the likelihood function, which is a measure of how well the model explains the observed data. For a location-scale model with density \[f(y ; \mu, \sigma) = \frac{1}{\sigma} g\left(\frac{y-\mu}{\sigma}\right),\] the log-likelihood function is given by \[\log L = -\log \sigma + \log g\left(\frac{y-\mu}{\sigma}\right).\] To perform MLE, we:

• Compute the derivative of the log-likelihood with respect to each parameter.
• Set these derivatives to zero to find the critical points.
• Solve these equations to obtain estimates \(\widehat{\mu}\) and \(\widehat{\sigma}\).

These estimates align the model as closely as possible with the observed data, providing the most plausible parameter values given our assumptions.

Location-Scale Models

Location-scale models are a foundational concept in statistics. They are employed to model data that may be shifted (location) and scaled (scale), making them very flexible for a wide range of distributions.The general form of a location-scale model is:\[f(y ; \mu, \sigma) = \frac{1}{\sigma} g\left(\frac{y-\mu}{\sigma}\right).\]Here, \(\mu\) represents the location parameter (often a central tendency measure like the mean), and \(\sigma\) represents the scale parameter (relating to the spread or variability of the data). The function \(g\) typically denotes a standard form of the distribution, such as the standard normal or another convenient form, which is then adjusted by these parameters to fit the data.With location-scale models, one can easily accommodate data transformations for robust analysis:

• They are useful for normalizing data to a common scale.
• These models are easily interpreted, as changes in \(\mu\) and \(\sigma\) directly relate to shifts and rescaling in the distribution.

Symmetric Distributions

Symmetric distributions are statistical distributions where the left and right sides are mirror images of one another. In other words, the shape of the distribution on one side of the central point is the same as it is on the other side.A common example is the normal distribution, often represented as:\[(2\pi)^{-1/2} e^{-u^2 / 2}.\]Symmetry is significant in statistical models because:

• It simplifies calculations, such as the Fisher information matrix, as many terms integrate to zero.
• It often implies that no extreme skewness is present in the data set, which helps in making simpler inferences.
• For symmetric \(g\), the off-diagonal element \(b\) in the Fisher Information matrix becomes zero, implying independence between estimator errors.

Understanding whether a distribution is symmetric can provide insights into the nature of the data and guide the selection of appropriate models and estimation techniques.