![]() ![]() If λ|k,θ ~ Gamma(shape = k, scale = θ) and X|λ ~ Exponential(rate = λ) then the marginal distribution of X|k,θ is Lomax(shape = k, scale = 1/θ). The Lomax distribution arises as a mixture of exponential distributions where the mixing distribution of the rate is a gamma distribution. Gamma-exponential (scale-) mixture connection This implies that a Lomax(shape = 1.0, scale = λ)-distribution equals a log-logistic distribution with shape β = 1.0 and scale α = log(λ). ![]() The logarithm of a Lomax(shape = 1.0, scale = λ)-distributed variable follows a logistic distribution with location log(λ) and scale 1.0. Relation to the (log-) logistic distribution
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