Conflict Among Criteria for Testing Hypotheses: Examples from Non-Normal Distributions. This is based on Baltagi (2000). Berndt and Savin (1977) showed that W ≥ LR ≥ LM for the case of a multivariate regression model with normal distrubances. Ullah and Zinde-Walsh (1984) showed that this inequality is not robust to non-normality of the disturbances. In the spirit of the latter article, this problem considers simple examples from non-normal distributions and illustrates how this conflict among criteria is affected.

(a) Consider a random sample x1, x2, . . . , xn from a Poisson distribution with parameter λ. Show that for testing λ = 3 versus λ = 3 yields W ≥ LM for ¯x ≤ 3 and W ≤ LM for ¯x ≥ 3.

(b) Consider a random sample x1, x2, . . . , xn from an Exponential distribution with parameter

θ. Show that for testing θ = 3 versus θ = 3 yields W ≥ LM for 0 < ¯x ≤ 3 and W ≤ LM for

¯x ≥ 3.

(c) Consider a random sample x1, x2, . . . , xn from a Bernoulli distribution with parameter θ.

Show that for testing θ = 0.5 versus θ = 0.5, we will always get W ≥ LM. Show also, that for testing θ = (2/3) versus θ = (2/3) we get W ≤ LM for (1/3) ≤ ¯x ≤ (2/3) and W ≥ LM for (2/3) ≤ ¯x ≤ 1 or 0 < ¯x ≤ (1/3).