Question

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail:

f(x|x₀, θ) = θx₀^θ x^−θ−1, x ≥ x₀,θ > 1

Assume that x₀ > 0 is given and that X₁,...,X_n is an i.i.d. sample.

find a sufficient statistic for θ

Answer 1

$f(x|x_0, \theta) = \theta x_0^{\theta} x^{-\theta-1}$ for $x \ge x_0$, $\theta > 1$

The joint pdf for X = (X1, X2,..., Xn) is,

$f(x|\theta) = \prod_{i=1}^{n}\theta x_0^{\theta} x_i^{-\theta-1} = \theta^n x_0^{n\theta} \prod_{i=1}^{n}x_i^{-\theta-1} = \left ( \theta^n x_0^{n\theta} \right ) \left (\prod_{i=1}^{n}x_i \right )^{-\theta-1} = g(t, \theta) h(x)$

where,

$t = \prod_{i=1}^{n}x_i$

$g (t , \theta)= \left ( \theta^n x_0^{n\theta} \right ) t^{-\theta-1}$

$h(x) = 1$

Thus, by factorization theorem,

$T(X) = \prod_{i=1}^{n}X_i$

is a sufficient statistic for $\theta$