The inversenormaldistribution has density with respecttostandardLebesguemeasureon R+. Youmaywithoutproofassume that R 0 f,(x) dx = 1 for all
Question:
The inversenormaldistribution has density
with respecttostandardLebesguemeasureon R+. Youmaywithoutproofassume that R∞
0 fμ,λ(x) dx = 1 for all (μ, λ) ∈ R2 +. Letnow
where Pμ,λ has densityfunction fμ,λ as above.
a) Representthefamily P as anexponentialfamilyofdimension2andidentifythe base measure,canonicalparameter,canonicalstatistic,andcumulantfunction.
b) Showthatthemeanandvarianceinthefamilyisgivenas
Nowconsiderthesubfamilyofinversenormaldistributionswherethemeanisequal to thevariance,i.e.where λ = μ2. Thisisknownasthe standard inversenormal distributions.
c) Arguethatthisfamilyisanaffinesubfamilyofthefullfamilyandthusformsa regularexponentialfamilyofdimension1;
d) Identifythebasemeasure,canonicalparameter,canonicalstatistic,andcumulant function inthissubfamily.
e) Determinethemeanandvarianceof Y = X−1.
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