Let $X_1, . . . , X_n ∼ Exp(λ)$, for some unknown parameter $λ > 0$ and let $λ_0$ be a (known) fixed positive number.
- Consider the following hypotheses: $H_0 : ”λ = λ_0” vs. H_1 : ”λ ≠ λ_0”$. Give a test with asymptotic level α, for any α ∈ (0, 1).
- Consider the following hypotheses: $H_0 : ”λ ≤ λ_0” vs. H_1 : ”λ > λ_0”.$ Give a test with asymptotic level (at most) $α$, for any $α ∈ (0, 1)$
I wanted to proceed with Wald's test. Is it a good choice or I should do this with likelihood ratio test?
$$λ^{MLE}=\frac{n}{\sum_{i=1}^{n}x_i}\\ I(λ_0)=-E\left[\left(n\ln λ-λ\sum_{i=1}^{n}x_i\right)''\right] =-E\left[-\frac{n}{λ_0^2}\right] =\frac{n}{λ_0^2};$$ so
$$ \lim_{n \to \infty} \sqrt{n} \left(\frac{n}{\sum_{i=1}^{n}x_i}-λ_0\right) \to \mathcal{N}\left(0,\frac{λ_0^2}{n}\right),$$ in distribution, so
$$T_n=\left[\sqrt{n} \left(\frac{n}{\sum_{i=1}^{n}x_i}-λ_0\right)I(λ_0)\right]^2 =n\left(\frac{n}{\sum_{i=1}^{n}x_i}-λ_0\right)^2\frac{n^2}{λ_0^4}$$
and in a) we must compute $P(T_n > q_{α/2})$ for $q_{α/2}$ from chi-squared with $n$ degrees of freedom, while in b) $P(T_n > q_{α})$, because in this case its one-sided test.
Is it right?