Question

Let X be a random sample of size 1 from a population with cumulative distribution function \[ F(x) = \begin{cases} 0 & \text{if } x < 0, \\ 1 - (1 - x)^\theta & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1 \end{cases} \] where \( \theta > 0 \) is an unknown parameter. To test \( H_0 : \theta = 1 \) against \( H_1 : \theta = 2 \), consider using the critical region \( \{ x \in \mathbb{R} : x < 0.5 \} \). If \( \alpha \) and \( \beta \) denote the level and power of the test, respectively, then \( \alpha + \beta \) (rounded off to two decimal places) equals:

Hint:

- For convergence of sums of random variables to normal distributions, the Central Limit Theorem (CLT) is helpful.
- In this case, the variance is estimated using the properties of exponential distributions and sample means.

Answer 1

The problem involves a sequence of independent and identically distributed exponential random variables with mean 1. We are given that the sequence \( \left\{ \sqrt{n} \left( e^{-\bar{Y}_n} - e^{-1} \right) \right\} \) converges in distribution to a normal random variable with mean 0 and variance \( \sigma^2 \).
The variance of the limiting normal distribution can be computed using the asymptotic properties of the exponential distribution. The detailed steps of the calculation lead to an estimate of \( \sigma^2 \) to be approximately between 0.12 and 0.16.