Question

2. \( (\mathbf{2 4 \%}) \) Let \( X_{1}, \cdots, X_{n} \) be iid with pdf \[ f(x ; \theta)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-\theta)^{2}}, \quad x \in(-\infty, \infty) . \] (a) Show that \( Y=\sum_{i=1}^{n} X_{i} \) is sufficient and complete for \( \theta \). (b) Find the UMVUE of \( \theta^{2} \).

          2. \( (\mathbf{2 4 \%}) \) Let \( X_{1}, \cdots, X_{n} \) be iid with pdf
\[
f(x ; \theta)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-\theta)^{2}}, \quad x \in(-\infty, \infty) .
\]
(a) Show that \( Y=\sum_{i=1}^{n} X_{i} \) is sufficient and complete for \( \theta \).
(b) Find the UMVUE of \( \theta^{2} \).

2. (2 4 %) Let X1, ⋯, Xn be iid with pdf

f(x ; θ)=(1)/(√(2 π)) e^-(1)/(2)(x-θ)^2, x ∈(-∞, ∞) .

(a) Show that Y=∑i=1^n Xi is sufficient and complete for θ.
(b) Find the UMVUE of θ^2.

Added by William S.

Introduction to Mathematical Statistics

Robert V. Hogg, Joseph W. McKean, Allen… 8th Edition

Chapter 6

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2. \( (\mathbf{2 4 \%}) \) Let \( X_{1}, \cdots, X_{n} \) be iid with pdf \[ f(x ; \theta)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-\theta)^{2}}, \quad x \in(-\infty, \infty) . \] (a) Show that \( Y=\sum_{i=1}^{n} X_{i} \) is sufficient and complete for \( \theta \). (b) Find the UMVUE of \( \theta^{2} \).