Let LN(γ, μ, σ2) denote the shifted (three-parameter) lognormal distribution, which has de...

Let LN(γ, μ, σ²) denote the shifted (three-parameter) lognormal distribution, which has density

for σ > 0 and any real numbers γ and μ. [Thus, LN(0, μ, σ²) is the original LN(μ, σ²) distribution.]

(a) Verify that X ~ LN(γ, μ, σ²) if and only if X – γ ~ LN(μ, σ²).

(b) Show that for a fixed, known value of γ, the MLEs of μ and σ in the LN(γ, μ, σ²) distribution are

i.e., we simply shift the data by an amount –γ and then treat them as being (un-shifted) lognormal data.