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Q23Section I (Objective (Q1–Q80))23 of 80

Statistical Inference — sufficient statistic for a gamma shape parameter

Let x1,x2,x3,…,xnx_1, x_2, x_3, …, x_n be a random sample from the gamma distribution having pdf f(x;θ,μ)=1Γ(θ)μθe−xμxθ−1f(x; \theta, \mu) = \frac{1}{\Gamma(\theta)\mu^\theta}e^{-\frac{x}{\mu}}x^{\theta - 1}; x>0x > 0, θ>0\theta > 0 and μ>0\mu > 0 is known constant. What is the sufficient statistic for θ\theta?

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