doing likelihood plot in R for binomial model

Question

Can anybody tell me how do i plot the maximum likelihood values L(ˆθM, M) versus M for a suitable range of M values for the count data provided in frogs and then estimate the total number of frogs living in the pond and the probability of appearance in R? These were the questions asked: questions and i have answered a and b

I have my pmf of my module and finded likelihood and log likelihood of my binomial model and you can see how much code i have written so far please help!

solutions to a,b and c so far

# importing the necessary modules
library(tidyverse)
library(ggplot2)

# loading the data
load("~/Statistical Modelling and Inference/aut2020.RData")

# Assigning a variable to the data
data <- frogs

# Assigning n to the length of the data
n <- length(frogs$counts)
n

theta_hat <- sum(frogs$counts)/M

loglik <- function(theta, y, M, data){
  # Computes the log_likelihood for binomial model
  sum_y <- sum(data$counts)
  M <- sum_y / n
  sum(log(dbinom(M,y))) + sum(y)*log(theta) + n*M - sum(y)*log(1-theta)}

Data looks like this: in r script when readed

Answer 1

Since you have already found the likelihood function in your answer (a), you can see that it is a function of M and theta - both unknown. After estimating theta you have the MLE estimator - let's call it theta_hat. In the dataframe frogs you have all the count observations y_i (known). So, using the known data and the ML estimate theta_hat that means the likelihood can be plotted for some (reasonable) range of values of M (you might need to try different ranges). So plot L(theta_hat, M) as a function of M. Bear in mind though that the estimate theta_hat will change as you change M so take that into account. The point where L(theta_hat, M) is maximized are you ML estimates for theta and M.