Question

Obtain the sensitivity curves for the sample mean, the sample median and the Hodges-Lehmann estimator for the following data set. Evaluate the curves at the values \(-300\) to 300 in increments of 10 and graph the curves on the same plot. Compare the sensitivity curves. $$ \begin{array}{rrrrrrrr} -9 & 58 & 12 & -1 & -37 & 0 & 11 & 21 \\ 18 & -24 & -4 & -53 & -9 & 9 & 8 & \end{array} $$ Note that the \(\mathrm{R}\) command wilcox.test \((\mathrm{x}\), conf . int \(=\mathrm{T}\) ) \$est computes the Hodges Lehmann estimate for the \(\mathrm{R}\) vector \(\mathrm{x}\).

Short answer

The sensitivity curves for the sample mean, sample median, and Hodges-Lehmann estimator provide visual representations of how these three estimates respond to shifts in the data values from -300 to 300. Their comparative analysis helps understand their differing sensitivity to changes in the data. Particular behaviors of these curves depend on the actual values of the data points, hence will be different for each specific case.

Step-by-step solution

Load the data

First, load the data into a vector using the c() function in R. Here's an example for the given data: \[ \text{{x <- c(-9, 58, 12, -1, -37, 0, 11, 21, 18, -24, -4, -53, -9, 9, 8)}} \]

Define a function for each estimator

Next, create functions to calculate the sample mean, median, and Hodges-Lehmann estimator. The sample mean and median have built-in functions in R: mean() and median(). For the Hodges-Lehmann estimator, use the wilcox.test() function with conf.int = TRUE to get the estimate: \[ \text{{hl_estimate <- function(x) {wilcox.test(x, conf.int = TRUE)$estimate}}} \]

Generate curves

Now it's time to generate the sensitivity curves. Iteratively add values from -300 to 300 to each data point, calculate the estimators, and store the results. For example, for the sample mean: \[ \text{{mean_curve <- sapply(-300:300, function(i) mean(x + i))}} \]

Plot the curves

Plot the three sets of results on the same graph using the plot() and lines() functions in R, which allows to visually compare their sensitivities. Label each line for clarity. One possible example could be: \[ \text{{plot(-300:300, mean_curve, type = 'l', col = 'blue', ylim = range(c(mean_curve, median_curve, hl_curve)), main = 'Sensitivity Curves', xlab = 'Increments', ylab = 'Estimator')}} \] \[ \text{{lines(-300:300, median_curve, col = 'red')}} \] \[ \text{{lines(-300:300, hl_curve, col = 'green')}} \]

Compare Curves

Lastly, compare the sensitivity curves. Comment on how they behave with increments. Do they increase or decrease? How do they react to positive and negative shifts? Note any steep or shallow parts of the curves, as these indicate where the estimators are most and least sensitive to changes in the data values.