Sufficient Statistic

Definition

Let $X_1,X_2,\dots ,X_n$ be a Random Sample with PDF $f(x|\theta )$, where $\theta \in \Omega$, and $Y_1=u_1(X_1,X_2,\dots ,X_n)$ be a Statistic with PDF $f_{Y_1}(y_1|\theta )$. The $Y_1$ is a sufficient statistic for $\theta$ if and only of $\frac{{\prod }_{i=1}^{n}f(x_i|\theta )}{f_{Y_i}(y_1|\theta )}=H(x_1,x_2,\dots ,x_n)$

No other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter.

Facts

Any one-to-one transformation of a sufficient statistic is also sufficient statistic

MLE is a function of sufficient statistic

Let $X_1,X_2,\dots ,X_n$ be a Random Sample with PDF $f(x|\theta )$, where $\theta \in \Omega$, $Y_1=u_1(X_1,X_2,\dots ,X_n)$ be a Statistic with PDF $f_{Y_1}(y_1|\theta )$, and $\hat{\theta }$ be a unique MLE of $\theta$, then $\hat{\theta }$ is a function of $Y_1$