# Likelihood ratio test for the exponential distribution

Let $$Y$$ be a random exponentially distributed variable, with mean $$\lambda$$. That is, its probability density function is $$f(y) = \left\{ \begin{array}{ c c } \frac{1}{\lambda}e^{\frac{-y}{\lambda}} & y > 0 \\ 0 & y \leq 0 \end{array} \right.$$

I want to perform a likelihood ratio test to test $$\left\{\begin{array}{ c c c } H_0 & : & \lambda \leq \lambda_0 \\ H_1 & : & \lambda > \lambda_0 \end{array} \right.$$

I derive the likelihood $$L(\lambda \mid y) = \left(\frac{1}{\lambda}\right)^n \exp\left(\frac{n\overline{y}}{\lambda}\right)$$ as well as the log-likelihood $$l(\lambda \mid y) = n \log\left(\frac{1}{\lambda}\right) - \frac{n\overline{y}}{\lambda}$$

I also know the MLE for $$\lambda$$, $$\hat{\lambda} = \overline{y}$$

I have read this question as well as this one, but just do not understand how to proceed now.

• If you know the procedure, then study the likelihood ratio criterion. For that you need the (restricted) MLE of $\lambda$ when $\lambda\le \lambda_0$ alongwith the unrestricted MLE $\bar y$. Commented May 15, 2019 at 6:02
• It is not clear where you are stuck or what you don't understand. So if you add details, maybe there is a better chance of getting a response. Commented May 17, 2019 at 6:05