MATH 456 Homework 06

Due 2026-03-31 by 11:59pm

1. Which is a better estimator the biased or unbiased estimator for the variance? You'll attempt to answer this question in the following 3 parts.

   a. Write two functions "by hand" that calculate the biased and unbiased estimate for the variance. You can use the built in R functions mean and length, but no other built in function.

   b. Pick true values for the mean and variance, say . Generate R = 1000 pairs of estimates, each pair from new N = 11 randomly sampled Normally distributed data. Use the R function rnorm(N, m, s) to generate the random data.

   c. For whichever value of the variance you chose, say , calculate the mean squared error as

   where is the th estimate of a variance. Calculate the mean squared error of all estimates for both the biased and unbiased estimators.

   d. Which estimator has lower mean squared error? Which do you think is a better property of an estiamte, lower mean squared error or unbiasedness?

2. Consider the simplified log-likelihood for linear regression with one predictor

   a. Use calculus to find estimates based on the data alone.

   b. Write R functions that calculate .

   c. Use data to compare your solutions to what R's function lm provides.