PSTAT 105 UCSB Empirical Probability Functions Questions

“Donation”500
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PSTAT 105: Assignment 5
Due Friday, Feburary 19, 2021
Please complete these questions and submit the answers on GauchoSpace.
1. Suppose we have n independent observations Xi , and we want to use a triangular kernel
(
4x
2
−h/2 < x < 0 2 + h Kh (x) = h2 4x − 0 ≤ x < h/2, h h2 in order to generate a KDE fˆ(t). (a) What is the disadvantage if we let the bandwidth h get too large? (b) Calculate the variance of the kernel density estimator fˆ(0.5) when h < 1 and the true distribution of the Xi are uniform on [0, 1], i.e. f (x) = 1 on 0 < x < 1. Your answer should be a function of h and n. 2. Suppose that we have 3000 observations from a exponential distribution with parameter λ = 1, (ie. f (x) = e−x ) and we are going to apply our kernel density estimator to this data with a rectangular kernel. ( 1 −h/2 ≤ x ≤ h/2 Kh (x) = h 0 otherwise (a) Calculate the variance of this estimator at x = 1 when the bandwidth is h = 0.2. (b) Calculate the bias of this estimator at x = 2 when the h = 1.2. 3. Please submit your answers to this questions as a typed report including input and output from your R code. The Federal Election Commission makes available the amount of money donated to federal politicians. The data file Carbajal.txt contains the dollar amounts of 2,295 contributions made to Rep. Salud Carbajal from 2019 to 2020. (a) Create a histogram using a number of bins that you believe allows us to see the interesting details in the contribution amounts. (b) Plot a copy of this histogram but include a kernel density estimate using the density or geom_density function. Set the bandwidth to 50 and use a Gaussian kernel. (c) Repeat this graph but use the Rectangular and then the Epanechnikov kernels. Keep the bandwidth the same. (d) What difference does the kernel shape make? (e) Produce another plot of the data, but now use automated bandwidth options bw="ucv" and bw="nrd" to draw two density estimates on the histogram. Use the Gaussian kernel. (f) Find a bandwidth for the density estimate that you think works best and estimate the density at $50 and $1500. (g) Does the shape of the kernel or the bandwidth have a greater effect on the resulting density estimate?