Question

Consider a location model (Example \(6.2 .2\) ) when the error pdf is the contaminated normal (3.4.14) with \(\epsilon\) 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: $$\begin{array}{|l|l|l|l|l|}\hline \epsilon & 0 & 0.05 & 0.10 & 0.15 \\\\\hline e\left(Q_{2}, \bar{X}\right) & & & & \\ \hline\end{array}$$ (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 solution involves understanding the Asymptotic Relative Efficiency (ARE) formula, calculating the ARE for different contamination proportions, and finding the value of \(\epsilon\) where the sample median becomes the 'better' estimator using the Newton's method. The values for ARE for the contamination proportions \(0, 0.05, 0.10, 0.15\) can be computed by directly substituting into the ARE formula. The crossover \(\epsilon\) value can be found by initializing \(\epsilon\) and repeatedly updating it using the Newton's method until the difference in the updates is below a set accuracy level.

Step-by-step solution

Understand the ARE formula

In order to successfully solve the exercise, you need to understand the Asymptotic Relative Efficiency (ARE) formula given in the problem. The formula presents the ARE of the sample median \(Q_{2}\) to the sample mean \(\overline{X}\), where \(\epsilon\) is the proportion of contamination and \(\sigma_c^2\) is the variance of the contaminated part. The ARE is found using the formula \[e\left(Q_{2},\bar{X}\right)=\frac{2\left[1+\epsilon\left(\sigma_{c}^{2}-1\right)\right]\left[1-\epsilon+\left(\epsilon/ \sigma_{c}\right)\right]^{2}}{\pi}\]

Calculate the ARE for different contamination proportions

The question asks to fill the table for \(\sigma_c^2 = 9\) and four different values of \(\epsilon\): \(0, 0.05, 0.10, 0.15\). Using the ARE equation, we find:For \(\epsilon = 0\), \[e\left(Q_{2},\bar{X}\right)=\frac{2\left[1+0\left(9-1\right)\right]\left[1-0+\left(0/ 3\right)\right]^{2}}{\pi} = \frac{2}{\pi} \],For \(\epsilon = 0.05\), \[e\left(Q_{2},\bar{X}\right)=\frac{2\left[1+0.05\left(9-1\right)\right]\left[1-0.05+\left(0.05/ 3\right)\right]^{2}}{\pi}\],For \(\epsilon = 0.10\),\[e\left(Q_{2},\bar{X}\right)=\frac{2\left[1+0.10\left(9-1\right)\right]\left[1-0.10+\left(0.10 / 3\right)\right]^{2}}{\pi}\],For \(\epsilon = 0.15\),\[e\left(Q_{2},\bar{X}\right)=\frac{2\left[1+0.15\left(9-1\right)\right]\left[1-0.15+\left(0.15/ 3\right)\right]^{2}}{\pi}\]

Determine the value of \(\epsilon\) where the sample median becomes the better estimator

It is asked to determine the value for \(\epsilon\) where the sample median becomes the 'better' estimator when \(\epsilon\) increases from \(0.10\) to \(0.15\). This means that we are looking for a value of \(\epsilon\) where \(e(Q_{2},\bar{X}) < 1\). To perform this, the Newton's method can be used to find the root for the ARE equation. The steps are: 1. Initialize \(\epsilon\) and set a desired accuracy for the root; 2. Compute the value of the equation and its derivative at that point; 3. Update \(\epsilon\) using the Newton's method formula (subtracts the value/derivative ratio); 4. Repeat steps 2-3 until the update to \(\epsilon\) is less than the desired accuracy (the root is found).

Key concepts

Location Model

A location model is a statistical framework used to describe the distribution of data points centered around a "location" parameter, typically the population mean or median. The primary purpose of this model is to explore how data is spread about this central value. Unlike the variance, which measures the spread or dispersion of data, the location model focuses on the data's position around the location parameter.
In practical terms, imagine plotting a series of values that naturally gravitate around a particular point on a graph — that's essentially the concept of a location model. Whether using the mean or median as the location, this model helps in capturing the "central tendency" of data.

• Useful for regression analysis, where the goal is to determine how other variables relate to this central tendency.
• The assumptions of normality in data can be examined within this framework.

Understanding the location model is crucial for analyzing how additional factors, such as contamination, affect standard statistical estimators. For students, it's essential to grasp how these models simplify data interpretation and forecast outcomes.

Contaminated Normal Distribution

The contaminated normal distribution describes a complex situation where data is not purely representative of one normal distribution but is instead mixed, or "contaminated," with another distribution. This contamination typically involves a small proportion of the data having a much larger variance.
This can arise in real-world scenarios where there's noise or error in measured data, often following a normal distribution but having a subset of unusual values — for example, measurement errors or outliers.

Properties of Contaminated Normal Distribution

• Two-Part Composition: A combination of a primary normal distribution and a secondary one with greater variance.
• Contamination Proportion ( "): This parameter illustrates the mix between the primary and contaminated distributions.

For students, visualizing this distribution helps understand why usual statistical measures, like means, can become unreliable. This understanding is crucial for implementing appropriate adjustments or choosing alternative statistical measures, such as the median, when analyzing data with potential contamination.

Sample Median vs Sample Mean

When comparing the sample median to the sample mean, it's fundamental to understand their respective roles in statistical analysis and how they respond to data, especially in the presence of outliers or skewed distributions.
The sample mean (average) is the sum of all data points divided by their number, providing a quick summary of data values. However, it can be overly sensitive to extremes or outliers. In contrast, the sample median, which is the middle value when data is ordered, is more robust to such anomalies.

Comparing Characteristics

• Resilience to Outliers: The sample median is less affected by extreme values, making it a better choice when analyzing contaminated data.
• Simplicity of Calculation: The mean is often quicker to compute but can be misleading if the data contains outliers.

Understanding the distinction between sample median and sample mean is vital, especially in exercises and scenarios involving distributions like the contaminated normal distribution. Recognizing when one is preferred over the other supports informed statistical analysis, ensuring the results are realistic and representative of the actual data characteristics.