Question

Consider the straight-line regression model \(y_{j}=\alpha+\beta x_{j}+\sigma \varepsilon_{j}, j=1, \ldots, n\). Suppose that \(\sum x_{j}=0\) and that the \(\varepsilon_{j}\) are independent with means zero, variances \(\varepsilon\), and common density \(f(\cdot)\) (a) Write down the variance of the least squares estimate of \(\beta\). (b) Show that if \(\sigma\) is known, the log likelihood for the data is $$ \ell(\alpha, \beta)=-n \log \sigma+\sum_{j=1}^{n} \log f\left(\frac{y_{j}-\alpha-\beta x_{j}}{\sigma}\right) $$ derive the expected information matrix for \(\alpha\) and \(\beta\), and show that the asymptotic variance of the maximum likelihood estimate of \(\beta\) can be written as \(\sigma^{2} /\left(i \sum x_{j}^{2}\right)\), where $$ i=\mathrm{E}\left\\{-\frac{d^{2} \log f(\varepsilon)}{d \varepsilon^{2}}\right\\} $$ Hence show that the the least squares estimate of \(\beta\) has asymptotic relative efficiency \(i / v \times 100 \%\) (c) Show that the cumulant-generating function of the Gumbel distribution, \(f(u)=\) \(\exp \\{-u-\exp (-u)\\},-\infty

Short answer

The least squares estimate of \(\beta\) has a variance of \(\sigma^2/\sum x_j^2\), and its asymptotic relative efficiency compared to MLE in a Gumbel distribution is approximately 61\%.

Step-by-step solution

Determine Variance of Least Squares Estimate for \(\beta\)

The variance of the least squares estimate of \(\beta\) is given by the formula:\[ \text{Var}(\hat{\beta}) = \frac{\sigma^2}{\sum x_j^2} \] This comes from the property of the least squares estimate, specifically under the assumption that \(\sum x_j = 0\). This aligns with the orthogonality principle in regression where the errors are uncorrelated with the independent variable.

Write Log Likelihood for Known \(\sigma\)

The log likelihood function \(\ell(\alpha, \beta)\) under the condition that \(\sigma\) is known is given by:\[ \ell(\alpha, \beta) = -n \log \sigma + \sum_{j=1}^{n} \log f\left(\frac{y_{j} - \alpha - \beta x_{j}}{\sigma}\right) \] This expression results from substituting the error term \((y_j - \alpha - \beta x_j)/\sigma\) into the probability density function \(f\) and applying logarithmic transformation.

Derive Expected Information Matrix

The expected information matrix \(I(\theta)\) for parameters \(\alpha\) and \(\beta\) is constructed from the second derivatives of the log likelihood with respect to \(\alpha\) and \(\beta\). Specifically for \(\beta\):\[ I_{\beta, \beta}(\theta) = n \frac{i \sum x_{j}^2}{\sigma^2} \] where \(i = \mathrm{E}\left\{-\frac{d^2 \log f(\varepsilon)}{d \varepsilon^2}\right\}\). This matrix allows us to derive the asymptotic variance of the maximum likelihood estimates.

Show Asymptotic Variance of MLE of \(\beta\)

The asymptotic variance of the maximum likelihood estimate of \(\beta\) is given by:\[ \text{Var}(\hat{\beta}_{\text{MLE}}) = \frac{1}{I_{\beta, \beta}(\theta)} = \frac{\sigma^2}{i \sum x_{j}^2} \] Thus, showing how \(i\) and \(\sigma^2\) contribute to the variance of the MLE.

Calculate Asymptotic Relative Efficiency

The asymptotic relative efficiency (ARE) of the least squares estimate compared to the MLE is calculated as:\[ \text{ARE} = \frac{i \sum x_{j}^2 / \sigma^2}{1 / \sum x_{j}^2} \times 100\% = \frac{i}{1} \times 100\% \] which simplifies to \(i \times 100\%\) due to the variance relationship between the estimates.

Find Cumulant-Generating Function for Gumbel Distribution

The cumulant-generating function \(K(t)\) for the Gumbel distribution with density \(f(u) = \exp{-u - \exp(-u)}\) is \(\log \Gamma(1-t)\). This result is obtained by recognizing the standard Gumbel distribution properties and its connection to extreme value theory.

Calculate Variance and Information \(i\) for Gumbel Distribution

For the Gumbel distribution, the variance is calculated as \(\approx 1.65\). The information \(i\) is derived from the second derivative of the log density function with respect to the standard Gumbel variable, which gives an insight into its curvature and strength of signal in estimation.

Calculate Asymptotic Relative Efficiency for Gumbel

With \(i \approx 0.61\), the asymptotic relative efficiency of the least squares estimate for\(\beta\) is calculated as \(61\%\). This implies that with a Gumbel error distribution, the least squares method has an efficiency of 61\% relative to the MLE approach.

Key concepts

Least Squares Estimate

The Least Squares Estimate is a method commonly used in regression analysis to determine the best-fitting line through data points. The idea is to minimize the sum of the squares of the residuals, which are the differences between observed and estimated values. In the context of a straight-line model, we're often focused on estimating the slope. If we have data points where the sum of the x-values is zero, the variance of the estimate for the slope coefficient \(\beta\) is simplified to \( \frac{\sigma^2}{\sum x_j^2} \), where \( \sigma \) is the standard deviation of the error term, and \( \sum x_j^2 \) is the sum of the squared values of the predictor variable. This reduction is particularly useful as it ties the variance directly to the distribution of x-values.

Asymptotic Variance

Asymptotic Variance is a concept used to describe the variance of an estimator as the sample size tends toward infinity. It's a measure of the spread of an estimator around the true parameter value in large samples. For maximum likelihood estimates (MLE), the asymptotic variance can be derived from the expected information matrix, which involves second derivatives of the log-likelihood function. In our scenario, this is computed as \( \frac{\sigma^2}{i \sum x_j^2} \), where \(i\) is the expected value of the negative second derivative of the log density function of the error term. This variance tells us how the accuracy of an MLE improves as more data is used, and it’s crucial for understanding the efficiency of our estimates.

Gumbel Distribution

The Gumbel Distribution is a probability distribution used in scenarios involving extreme values, such as predicting the maximum or minimum values in datasets. It's particularly useful in fields like meteorology and hydrology. This distribution has a probability density function given by \( f(u) = \exp \{-u - \exp(-u)\} \), applicable for all real numbers \(u\). A fascinating feature of the Gumbel distribution is its cumulant-generating function, \( \log \Gamma(1-t) \), which provides a compact way to understand its behavior and moments. For the Gumbel distribution, the variance is approximately 1.65, and significant is \(i\), which reflects the curvature of the log density function's second derivative, crucial for efficiency calculations.

Maximum Likelihood Estimation

The Maximum Likelihood Estimation (MLE) is a method of estimating the parameters of a statistical model. It does so by maximizing the likelihood function, thus finding the parameter values that make the observed data most probable. For regression, this technique helps determine the best values for intercept and slope in a model. With a known \( \sigma \), the log-likelihood function is formulated to incorporate the errors, yielding \(-n \log \sigma + \sum \log f\left(\frac{y_j - \alpha - \beta x_j}{\sigma}\right)\). The MLE’s strength comes from its asymptotic properties, as the estimation process assumes an infinite data sample, leading to the desirable outcome of minimizing the asymptotic variance. This approach is instrumental in advanced statistical modeling due to its precision and theoretical foundation.