The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ against $H_1:\lambda=500$. A statistician sets up an experiment with $50$ bulbs, with $5$ bulbs in each of $10$ different locations, to examine their lifetimes.
To get quick preliminary results,the statistician decides to stop the experiment as soon as one bulb fails at each location.Let $Y_i$ denote the lifetime of the first bulb to fail at location $i$.Obtain the most powerful test of $H_0$ against $H_1$ based on $Y_1,Y_2,...Y_{10}$ and compute its power.
Now, I have problems regarding how to formulate the condition that a particular bulb has failed. Otherwise, how can I write the likelihood function in either of the hypotheses?
Let $Y_i = \min(X_{i1}, \cdots, X_{i5})$, where each $X_{ij} \overset{\text{i.i.d}}{\sim} \text{Exp}(\lambda)$, denote the failure time of the first bulb. By properties of the exponential distribution, you can show $Y_i \sim \text{Exp}(5\lambda)$.
From this, the LRT would show that you reject for large values of $\overline{Y}$ (i.e. $\overline{Y} > c$ for some critical value $c$), where $\overline{Y} = \frac{1}{10}\sum_{i=1}^{10}Y_i \sim \text{Gamma}(10, 50\lambda)$, where I'm using the shape/rate parametrization. Under $H_0$, $\lambda = 1000$, and use that to find the appropriate critical value $c$.