# Mathematical Statistics Lesson of the Day – Sufficient Statistics

Suppose that you collected data

\$latex mathbf{X} = X_1, X_2, …, X_n\$

in order to estimate a parameter \$latex theta\$.  Let \$latex f_theta(x)\$ be the probability density function (PDF)* for \$latex X_1, X_2, …, X_n\$.

Let

\$latex t = T(mathbf{X})\$

be a statistic based on \$latex mathbf{X}\$.  Let \$latex g_theta(t)\$ be the PDF for \$latex T(X)\$.

If the conditional PDF

\$latex h_theta(mathbf{X}) = f_theta(x) div g_theta[T(mathbf{X})]\$

is independent of \$latex theta\$, then \$latex T(mathbf{X})\$ is a sufficient statistic for \$latex theta\$.  In other words,

\$latex h_theta(mathbf{X}) = h(mathbf{X})\$,

and \$latex theta\$ does not appear in \$latex h(mathbf{X})\$.

Intuitively, this means that \$latex T(mathbf{X})\$ contains everything you need to estimate \$latex theta\$, so knowing \$latex T(mathbf{X})\$ (i.e. conditioning \$latex f_theta(x)\$ on \$latex T(mathbf{X})\$) is sufficient for estimating \$latex theta\$.

Often, \$latex T(mathbf{X})\$ is a summary statistic of \$latex X_1, X_2, …, X_n\$, such as their

• sample mean
• sample median
• sample minimum
• sample maximum

If such a summary…

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