Maximum Likelihood – Exponential Distribution: Log-Likelihood and Maximum Likelihood Estimator

Question

For a random variable with its CDF given by $$F(x)=1-\exp(-\lambda x),$$ and its PDF given by $$f(x)=\lambda \exp(-\lambda x),$$ for $x>0$ and $\lambda >0$.

How would I write the log-likelihood function for a random sample $X_1,X_2,…,X_n$ i.i.d. Exp($\lambda$) and a maximum likelihood estimator for $\lambda$?

I know that the exponential distribution is . I'm not quite sure where to go from there. Any help would be appreciated.

Answer 1

The likelihood is given as

$$L(\lambda,x) = L(\lambda,x_1,...,x_N) = \prod_{i=1}^N f(x_i,\lambda)$$

where the second identity use the IID assumption and with $x = (x_1,...,x_N)$. The log-likelikelihood is given as

$$l(\lambda,x) := log L(\lambda,x) = \sum_{i=1}^N \log f(x_i, \lambda),$$

where $log f(x_i,\lambda) = log \lambda - \lambda x_i$. This implies that

$$l(\lambda,x) = \sum_{i=1}^N log \lambda - \lambda x_i = N \log \lambda - \lambda \sum_{i=1}^N x_i.$$ Since we are interested in maximum a positive monotone transformation such as dividing with $N$ is fine. This gets us to

$$\frac{1}{N} l(\lambda , x) = \log \lambda - \lambda \bar x$$

differentiate and set to zero to get first order condition

$$\frac{1}{\lambda} - \bar x = 0 \Leftrightarrow \lambda = \frac{1}{\bar x}$$

Small simulation in R

lambda_0 <- 0.5
N <- 10000
x <- rexp(N, rate = lambda_0)

loglik <- function(theta)
    {
        ll <- N * log(theta) - theta*sum(x)
        return(ll)
    }

# Calculate estimate
m_x <- 1/mean(x)

# Create vector for plot of loglikelihood
t <- seq(0.5*m_x,1.5*m_x,length.out=100)

plot(t,loglik(t),type="l")
abline(v=m_x,col="red")

This will genrate this plot of loglikelihood function to see maximum ...