Let $X_1,\ldots,X_n$ a random sample of lifetime of a type of components with exponential distribution with mean $\theta$. Suppose that observations have a precision of $2h$, i.e., a component have timelife $x_0$ if fails in interval $(x_0-h,x_0+h)$. Find the Maximum likelihood estimator for $\theta.$
My attempt
I think is possible to write the likelihood function as $L(\theta|x_1,\ldots,x_n)=\Pi_{i=1}^nP(X_i\in(x_i-h,x_i+h)|\theta)$
since each $X_i$ is exponential with mean $\theta$
$P(X_i\in(x_i-h,x_i+h)|\theta)=F(x_i+h)-F(x_i-h)=1-e^\frac{-x_i-h}{\theta}-1+e^\frac{-x_i+h}{\theta}=e^\frac{-x_i}{\theta}(e^\frac{h}{\theta}-e^\frac{-h}{\theta}).$
So
$L(\theta|x_1,\ldots,x_n)=\Pi_{i=1}^n e^\frac{-x_i}{\theta}(e^\frac{h}{\theta}-e^\frac{-h}{\theta})=e^-\frac{\sum x_i}{\theta}(e^\frac{h}{\theta}-e^\frac{-h}{\theta})^n.$
I'm trying to find the maximum solving the equation $\frac{d}{d\theta}L=0$ but I can't solve it. Any suggestion?