Homework 6

Assignment is due 4/6/23

Published

March 31, 2023

Problem 1

Let \(Y_i\overset{iid}{\sim}Pois(\lambda_i)\) with \(\lambda_i=g^{-1}(\boldsymbol X_i^\mathrm T\boldsymbol \beta)\), \(\eta_i = -1 + 3 X_i\), and \(X_i\sim N(0, 0.25)\). Write the code that will provide the log-likelihood function of the model.

x <- rnorm(1000, sd = sqrt(0.25)) 

eta <- -1 + 2 * x

y <- sapply(eta, \(x) rpois(1, exp(x)))
glm(y~x, family = poisson())


Call:  glm(formula = y ~ x, family = poisson())

Coefficients:
(Intercept)            x  
    -0.9869       1.9444  

Degrees of Freedom: 999 Total (i.e. Null);  998 Residual
Null Deviance:      1443 
Residual Deviance: 842.5    AIC: 1671

Problem 2

Using the function above, minimize the likelihood function to identify the estimates of the parameters.

Problem 3

Repeat Problems 1 and 2 a thousand times to conduct a simulation study.

Problem 4

The lime data set contains 385 observations on small-leaved lime trees. The Foliage variable measures the foliage biomass. Fit a regression model between Foliage and the following covariates: DBH tree diameter and Age. Fit the model using 3 different link functions. Print out the results for the one’s that work. Comment why one’s may work and others do not.