7.4 Completeness and Uniqueness

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Definition 7.4.1: Complete family of pdf/pmf

Definition

Let:

• $Z$ : Random variable
• pdf/pmf of $Z$ is one member of the family $\{h(z;\theta ):\theta \in \Omega \}$

If $\forall \theta \in \Omega :E[u(Z)]=0⟹u(z)=0$ almost surely with respect to $h(z;\theta ),\forall \theta \in \Omega$

Then the family $h(z;\theta ):\theta \in \Omega$ is called a complete family of pdf/pmfs.

Definition: Complete sufficient statistic

Definition

Let $X_1,\dots ,X_n$ : Random sample, with pdf/pmf $f(x;\theta )$, $\theta \in \Omega$

If

• $Y_1=u_1(X_1,\dots ,X_n)$ : Sufficient statistic for $\theta$
• Family $\{f_{Y_1}(y_1;\theta ):\theta \in \Omega \}$ be complete

Then $Y_1$ is a complete sufficient statistic for $\theta$

Definition 7.4.1: Unique unbiased minimum variance estimator (UMVUE)

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample, with
  • pdf/pmf $f(x;\theta )$, $\theta \in \Omega$
• $Y_1=u_1(X_1,\dots ,X_n)$ : Complete sufficient statistic for $\theta$

If $\varphi (Y_1)$ is an unbiased estimator of $\theta$

Then $\varphi (Y_1)$ is the unique MVUE (UMVUE) of $\theta$