Suppose that ( X_{1}, ldots, X_{n} ) form a random sample from a continuous distribution with the following pdf: [ f(x mid heta)=left{egin{array}{ll} heta x^{ heta-1} & 0<x<1 \ 0 & ext { otherwise } end{array} ight. ] It is assumed that the value of the parameter ( heta ) is unknown ( ( heta>0) ). Show that ( T=prod_{i=1}^{n} x_{i} ) is a sufficient statistic for ( heta )
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According to the Factorization Theorem, a statistic \( T(X) \) is sufficient for a parameter \( \theta \) if and only if the joint probability density function (pdf) of the sample \( X \) can be factored into two parts: one part that depends only on \( T(X) \) and Show more…
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