7.2 A Sufficient Statistic for a Parameter

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Definition 7.2.1: Sufficient statistic

Let

• $X_1,\dots ,X_n$ : Random sample, with joint pdf/pmf $f(x;\theta )$, $\theta \in \Omega$
• $Y=u(\mathbf{X})$ : Statistic, with pdf/pmf $f_Y(y;\theta )$

Then $Y$ is a sufficient statistic for $\theta$

If and only if

• $\frac{{\prod }_{i=1}^{n}f(x_i;\theta )}{f_Y[u(\mathbf{x});\theta ]}=H(\mathbf{x})$
• $H(\mathbf{X})$ does not depend upon $\theta$

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Theorem 7.2.1: Neyman theorem

Let

• $X_1,\dots ,X_n=\mathbf{X}$ : Random sample, with distribution that has pdf/pmf $f(x;\theta )$, $\theta \in \Omega$
• $Y=u(\mathbf{X})$ : Statistic for $\theta$
• $L(\theta )$ : Likelihood function of $X$

Then $Y$ is a sufficient statistic for $\theta$

If and only if

• $\exists k,l\ni L(\theta )=k[u(\mathbf{x});\theta ]\cdot l(\mathbf{x})$
• $l(\mathbf{x})$ is independent of $\theta$

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