Assignment 1: Estimating CDF and PDF

Assignment Objectives

• Develop a clear technical understanding of nonparametric cumulative distribution function (CDF) estimation and various kernel density estimators.

• Translate mathematical formulas into R functions and apply them to solve related problems.

• Create effective visualizations to demonstrate your understanding of key concepts in the following questions.

Question 1: Cumulative Distribution Function (CDF) Estimation

The following failure times (in hours) were observed for 8 electronic components:

23, 45, 67, 89, 112, 156, 189, 245

1. Write an R function implementing the ECDF \(\hat{F}_n(t)\) according to its mathematical definition. Validate your implementation using R’s ecdf() function on the given data, with comparison based on their step functions.

2. A colleague claims that the probability of failure before 100 hours is 0.5 based on these data. Do you agree? Explain your reasoning using the empirical cumulative distribution function (ECDF).

Question 2: Density Function Estimation

Consider the following failure times from a mechanical system:

12.3, 14.7, 15.2, 16.8, 18.1, 19.4, 20.6, 22.3, 23.9, 25.4

1. Create a histogram of the data using 3 equally spaced bins. What is the estimated density in each bin? Describe the shape of the histogram’s distribution.

2. Write an R function that computes kernel density estimates using a Gaussian kernel with \(h=2\). Validate your implementation against R’s built-in density() function.

\[ \hat{f}_h(t) = \frac{1}{nh}\sum_{i=1}^n K\left( \frac{t-t_i}{h}\right), \ \ \text{ where } \ \ K(u) = \frac{1}{\sqrt{2\pi}} e^{-u^2/2}. \]

3. Write a custom R function that computes kernel density estimates using the Epanechnikov kernel with \(h=2\). Validate your implementation by comparing results with R’s built-in density() function for Gaussian kernel estimation.

\[ \hat{f}_h(t) = \frac{1}{nh}\sum_{i=1}^n K\left( \frac{t-t_i}{h}\right), \ \ \text{ where } \ \ K(u) = \frac{3}{4}(1 - u^2) \ \ \text{ for } \ \ |u| \le 1. \]

4. How does the choice of kernel (Gaussian vs. Epanechnikov) affect the density estimate? For both kernel estimators applied to this dataset, what happens when we select \(h=1.5\) versus \(h=2.5\)?