Question

Consider a location model (Example \(6.2 .2\) ) when the error pdf is the contaminated normal (3.4.17) with \(\epsilon\) as the proportion of contamination and with \(\sigma_{c}^{2}\) as the variance of the contaminated part. Show that the ARE of the sample median to the sample mean is given by $$ e\left(Q_{2}, \bar{X}\right)=\frac{2\left[1+\epsilon\left(\sigma_{c}^{2}-1\right)\right]\left[1-\epsilon+\left(\epsilon / \sigma_{c}\right)\right]^{2}}{\pi} $$ Use the hint in Exercise \(6.2 .5\) for the median. (a) If \(\sigma_{c}^{2}=9\), use \((6.2 .34)\) to fill in the following table: \begin{tabular}{|l|l|l|l|l|} \hline\(\epsilon\) & 0 & \(0.05\) & \(0.10\) & \(0.15\) \\ \hline\(e\left(Q_{2}, \bar{X}\right)\) & & & & \\ \hline \end{tabular} (b) Notice from the table that the sample median becomes the "better" estimator when \(\epsilon\) increases from \(0.10\) to \(0.15 .\) Determine the value for \(\epsilon\) where this occurs [this involves a third-degree polynomial in \(\epsilon\), so one way of obtaining the root is to use the Newton algorithm discussed around expression \((6.2 .32)]\).

Short answer

The efficiency of the sample median to sample mean can be computed for different values of \(\epsilon\) using the provided formula. The turning point where sample median becomes a better estimate can be found by equating the efficiency to 1 and solving the resulting third-degree polynomial for \(\epsilon\).

Step-by-step solution

Substitute in formula

Substitute given \(\epsilon\) values and \(\sigma_{c}^{2} = 9\) into the equation \(e(Q_{2}, \bar{X}) = \frac{2[1 + \epsilon(\sigma_{c}^{2} - 1)][1 - \epsilon + (\epsilon / \sigma_{c})^{2}]}{\pi}\) to get \(e(Q_{2}, \bar{X})\) for each \(\epsilon\) value.

Fill in the table

Use the computed \(e(Q_{2}, \bar{X})\) values to fill the table.

Formulate Polynomial

In order to find the value of \(\epsilon\) where sample median becomes a better estimator, equate the efficiency of the sample median to sample mean to 1. This will result in a third-degree polynomial in terms of \(\epsilon\).

Solve the Polynomial

Solve the third-degree polynomial using Newton's method or any other root finding algorithm to get the value of \(\epsilon\) where sample median is a better estimate than sample mean.

Key concepts

Contaminated Normal Distribution

Understanding the contaminated normal distribution is crucial when dealing with real-world data that deviates from the ideal normal distribution. Imagine you're dealing with height measurements of a population. Most data cluster around a mean value, reflecting the average height, yet occasionally, you might encounter data points that are significantly different due to measurement errors or outliers. These points can be described as a contamination in the otherwise normal dataset.

The contaminated normal distribution models this scenario by combining a standard normal distribution with a secondary normal distribution that has a higher variance. The higher variance reflects the greater spread in the measurements due to the contamination. In mathematical terms, the contaminated normal distribution is described by the mixing parameter \( \epsilon \) and the variance of the contaminated part \( \sigma_c^2 \). The mixing parameter determines the proportion of contamination, whereas \( \sigma_c^2 \) specifies the dispersion of that contaminated subset.

This hybrid model is more adept at describing datasets that include errors or outliers and is particularly useful in robust statistics, where the objective is to reduce the influence of these aberrant points on the analysis.

Sample Median

The sample median is a fundamental statistical measure that represents the middle value of a dataset when it is sorted in ascending or descending order. If the sample size is odd, the median is the value at the central position. However, if the sample size is even, the median is typically computed as the average of the two middle values.

The sample median is known for its robustness, which means it is less affected by outliers in a dataset compared to the sample mean. For instance, in the analysis of income levels where extreme values can skew the data, the median provides a more accurate reflection of the central tendency.

In the context of contaminated distributions, the sample median can serve as a resistant estimator of the location parameter, thereby giving a more reliable measure of central tendency. As contamination increases, the sample median often surpasses the mean in terms of efficiency, which is precisely why statisticians may prefer it in the presence of outliers or non-standard data distributions.

ARE (Asymptotic Relative Efficiency)

The Asymptotic Relative Efficiency (ARE) is a concept in statistics that compares the performance of two estimators as the sample size grows to infinity. In other words, it helps to determine which estimator would be more efficient in the long run under certain conditions. Efficiency, in this context, refers to the precision and reliability with which an estimator is able to pinpoint a population parameter.

The formula for ARE of two estimators involves the ratio of their respective variances, under the assumption of asymptotic normality. A higher ARE value indicates a more efficient estimator. In the exercise provided, we examine the ARE of the sample median compared to the sample mean in the setting of a contaminated normal distribution. The equation includes both the contamination proportion \( \epsilon \) and the contamination variance \( \sigma_c^2 \) to reflect their impact on the estimating efficiency.

As we can see from the exercise, when contamination increases, the ARE for the sample median also increases, suggesting that the median becomes a better estimator relative to the mean. This type of analysis gives us insights into how resistant an estimator is to deviations from the standard assumptions, like those of a pure normal distribution, and guides statisticians when choosing the most suitable method for analyzing their data.