The singly truncated normal (STN) distribution, indexed by parameters and has support the positive real

The singly truncated normal (STN) distribution, indexed by parameters ν and λ has support the positive real line with density p(x; ν, λ) = C(ν, λ) exp(−νx − λx 2

) , where C(ν, λ) is a normalizing constant. Based on an i.i.d. sample, show there exists a UMPU test of the null hypothesis that the observations are exponential against the STN alternative, and describe the form of rejection region as explicitly as possible.

[See Castillo and Puig (1999).]

Section 4.5