Instructor: Colin Grudzien

## Instructions: We will work through the following series of activities as a group and hold small group work and discussions in Zoom Breakout Rooms. Follow the instructions in each sub-section when the instructor assigns you to a breakout room. ```{r} require("numDeriv") require("optimx") ``` ## Activity 1: Poisson maximum likelihood estimation Recall that the probability mass function for the Poisson distribution is given as, $$\begin{align} f_\xi(x,\lambda) = \frac{\lambda^x e^{-\lambda}}{x!} \end{align}$$ where 1. $x$ is a possible integer value for the realization of $\xi$; 2. $\lambda$ is equal to the mean and variance of the Poisson distribution. Following the same discussion as in the Gaussian maximum likelihood estimation, let's suppose that $\{x_i\}_{i=1}^n$ is an independent and identically distributed sample of observations of $\xi\sim \mathrm{Pois}(\lambda)$ for an unknown value of $\lambda$. The likelihood function for $\lambda$ is therefore given as, $$\begin{align} \mathcal{L}_{\mathbf{x}}(\lambda) &= \prod_{i=1}^n f_\xi(x_i,\lambda) \\ &=\prod_{i=1}^n \left(\frac{\lambda^{x_i}e^{-\lambda}}{x_i !}\right) \end{align}$$ Therefore, we define the log-likelihood function for $\lambda$ based on the sample $\mathbf{x} = \{x_i\}_{i=1}^n$ to be, $$\begin{align} L_\mathbf{x}(\lambda) = \log(\mathcal{L}_\mathbf{x}(\lambda)) \end{align}$$ ### Question 1: Derive the negative log-likelihood function for the Poisson distribution as a function of $\lambda$ for a known sample, to turn the maximum likelihood estimation into a minimization problem. Include the steps of your derivation to show your work. ### Question 2: We will make a maximum likelihood estimation of the parameter $\lambda$ for a known sample. Recall, $\lambda$ must be constrained to be positive, so we will again use the `optimx` package with a constrained version of BFGS to solve the maximum likelihood estimation of $\lambda$ for a known sample ```{r} set.seed(0) n <- 100 x <- rpois(n, lambda=10) ``` You will also need the package `pracma` for the factorial function `factorial`. Define the sample call within the function as follows ```{r eval=FALSE} neg_pois_log_like <- function(lambda) { set.seed(0) n <- 100 x <- rpois(n, lambda=10) # define the negative log-likelihood below and return the value } ``` And use the `opmtix` function as follows with the lower bound specified. ```{r eval=FALSE} fBFGS = optimx(fn = neg_pois_log_like, # define the objective function par = 7, # define the initial first guess at the minimum method ="L-BFGS-B", # define the method of solution lower = 0 #specify a lower bound ) print(fBFGS) ``` Finally, make a plot of the negative log likelihood function over integer values 1 to 100. Describe how this plot relates to the optimization.