Two Tests (Exponential Distribution) Flashcards
What is the parameter of interest in the Two Tests lecture?
The rate parameter λ of the Exponential distribution.
What are the hypotheses in the first test?
H0: λ = λ0 vs. HA: λ > λ0.
What statistic is used in the first test?
The sample mean X̄.
Why does a larger λ lead to a smaller X̄?
Because the mean of Exp(λ) is 1/λ, so increasing λ decreases expected values.
What is the rejection rule in the first test?
Reject H0 if X̄ < c.
How is X̄ related to the chi-squared distribution?
2nλX̄ ~ χ² with 2n degrees of freedom.
How do you compute the critical value c for the mean test?
c = χ²_{1−α,2n} / (2nλ0).
What R function is used to compute chi-square quantiles?
qchisq(probability, df).
What statistic is used in the second test?
The sample minimum Yₙ.
What is the rejection rule in the second test?
Reject H0 if Yₙ < c.
How is the sample minimum Yₙ distributed?
Yₙ ~ Exp(nλ).
How do you compute the critical value c for the minimum test?
c = -log(1−α) / (nλ0).
What is the power function?
The probability of rejecting H0 as a function of λ.
How does the power function behave for the mean-based test?
Increases quickly with λ and depends on n.
How does the power function behave for the minimum-based test?
Increases slowly and does not depend on n.
Which test is better and why?
The test based on X̄ is better because it has higher power and uses more information.
Why is the minimum not a great test statistic?
Because it always piles near zero and doesn’t improve with larger n.
What distribution does the minimum transformation become when scaled?
An Exp(1) distribution.
What is the formula for the power function of the minimum test?
γ(λ) = (1−α)^(λ/λ0).
What is the main advantage of the minimum test?
It has simple calculations and an exact rejection rule.
