Here is a question from old exam papers I am having trouble with:
Suppose the lifetime of a bulb is distributed exponentially with mean life $\theta$ (in hours). Let $X_i$ be the number of trials required to get a bulb surviving at least $t$ (known) hours for the first time in the $i$th lot, where lot sizes are large enough. Discuss a likelihood ratio test for testing $H_0:\theta\ge\theta_0$ against $H_1:\theta<\theta_0$.
I am not sure what exactly my observed sample is supposed to be. So I am stuck at setting up the problem correctly.
Say I have $k$ lots of bulbs, the $i$th lot having size $n_i\,, i=1,2\ldots,k$.
Then is the observed sample some $\mathbf X=(X_1,X_2,\ldots,X_k)$, where $X_i=(X_{ij})_{i=1,\ldots,k\,;\,j=1,\ldots,n_i}$ is again a vector for each $i$ ? I am using $X_{ij}$ to denote the number of trials needed to get the $j$th bulb surviving for at least $t$ hours for the first time in the $i$th lot.
If I assume $X_1,X_2,\ldots,X_k$ are all independent samples, then my likelihood function should be
$$L(\theta\mid \mathbf x)=\prod_{i=1}^k\prod_{j=1}^{n_i}p(\theta)(1-p(\theta))^{x_{ij}-1}\quad,\,\theta>0$$
, where $p(\theta)=\int_t^{\infty}\frac{1}{\theta}e^{-z/\theta}\,dz=e^{-t/\theta}\,,t>0$ is the success probability.
But this might be wrong since I am supposed to have a sample consisting of only $X_1,\ldots,X_k$ where each $X_i$ denotes number of trials (so not vectors by themselves). In the above setup, this does not seem to be the case.
If this formulation is wrong, I would like a hint on how to arrive at the correct one. And also, where exactly am I supposed to apply a large sample approximation with large $n_i$s? Is this referring to Wilks' theorem on the asymptotic distribution of the log-likelihood ratio statistic?
(I am not looking for a solution to the actual problem.)