1. In this question you'll explore some pieces of density estimators. (a) First consider the kernel functions: calculate K2 for the uniform, Epanechnikov, biweight and Gaussian kernels. Can you explain why, even though K2u, K2e, K2b and K2g are very different, they are unlikely to play a large role in determining the quality of the resulting density estimate? (b) Next, suppose that we are estimating a standard normal density (in practice, we wouldn't know that). For a given bandwidth h, calculate the bias of estimates f using the different kernels from the previous part, at the median of the distribution, one standard deviation above the median, and two standard deviations above it. What is common to all the biases? What does the pattern tell us about the estimates? 1. In this question you'll explore some pieces of density estimators. (a) First consider the kernel functions: calculate K2 for the uniform, Epanechnikov, biweight and Gaussian kernels. Can you explain why, even though K2u, K2e, K2b and K2g are very different, they are unlikely to play a large role in determining the quality of the resulting density estimate? (b) Next, suppose that we are estimating a standard normal density (in practice, we wouldn't know that). For a given bandwidth h, calculate the bias of estimates f using the different kernels from the previous part, at the median of the distribution, one standard deviation above the median, and two standard deviations above it. What is common to all the biases? What does the pattern tell us about the estimates