Let (X1, Y1, ...,Yn,Xn) be independentandexponentiallydistrib-

uted randomvariableswhere Xj and Yj havedensitieswithrespecttostandard Lebesque measure:

fj(x; θ) = jθe−jθx, x> 0, gj(y; λ) = λe−λy, y> 0, j = 1, ...,n, and (θ,λ) ∈ R2

+ both unknown.Notethat Y1, ...,Yn are identicallydistributed whereas thisisnotthecasefor X1, ...,Xn.

a) Arguethatthisspecifiesaminimalandregularexponentialfamilyofdimension two,identifythebasemeasure,canonicalstatistic,andcumulantfunction.

b) Considerthesubfamilydeterminedbytherestriction (θ,λ) =(β, 1/β) and show that thisisacurvedexponentialfamilyofdimensiononeandordertwo.

c) Findthelog-likelihoodfunction,scorefunction,Fisherinformation,andquad-

ratic scoreinthissubfamily.