To see what might happen when the parameter space is not open, let f0(x) = x I{0 ≤ x ≤ 1} + (2 − x)I{1 < x ≤ 2} .
Consider the family of densities indexed by θ ∈ [0, 1) defined by pθ(x) = (1 − θ2 ) f0(x) + θ2 f0(x − 2) .
Show that the condition (14.5) holds when θ0 = 0, if it is only required that h tends to 0 through positive values. Investigate the behavior of the likelihood ratio (14.12)
for such a family. (For a more general treatment, consult Pollard (1997)).