Mathematical Statistics Lesson of the Day – Sufficient Statistics

The Chemical Statistician

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…

View original post 36 more words

About schapshow

Math & Statistics graduate who likes gymnastics, 90s alternative music, and statistical modeling. View all posts by schapshow

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