Question

Definitions and UMVUE Problem 3. Let $X_1, \dots, X_n$ denote a random sample of size $n > 1$ from a distribution with pdf $$f_x(x) = \frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}}$$ (a) Determine the sufficient statistic for $\sigma^2$ using Neyman factorization. (b) Use the defintion to confirm your answer in A. (c) Find the MLE of $\sigma^2$ (d) Use MLE theory to determine the asymptotic distribution of a scaled version fo the MLE of $\sigma$(not squared). Does the CLT give the same result? (e) Use C. to identify the UMVUE for $\sigma^2$ (f) Identify two statistics that are independent from the sufficient statistic. (g) Determine the uniformly most powerful test for testing $H_0 : \sigma^2 = \sigma_0^2$ vs $H_1 \sigma^2 > \sigma_0^2$. (h) Derive an expression for the power of the test in G.

          Definitions and UMVUE
Problem 3. Let $X_1, \dots, X_n$ denote a random sample of size $n > 1$ from a distribution
with pdf
$$f_x(x) = \frac{x}{\sigma^2}e^{-\frac{x^2}{2\sigma^2}}$$
(a) Determine the sufficient statistic for $\sigma^2$ using Neyman factorization.
(b) Use the defintion to confirm your answer in A.
(c) Find the MLE of $\sigma^2$
(d) Use MLE theory to determine the asymptotic distribution of a scaled version fo the
MLE of $\sigma$(not squared). Does the CLT give the same result?
(e) Use C. to identify the UMVUE for $\sigma^2$
(f) Identify two statistics that are independent from the sufficient statistic.
(g) Determine the uniformly most powerful test for testing $H_0 : \sigma^2 = \sigma_0^2$ vs $H_1 \sigma^2 > \sigma_0^2$.
(h) Derive an expression for the power of the test in G.

Definitions and UMVUE
Problem 3. Let X1, …, Xn denote a random sample of size n > 1 from a distribution
with pdf

fx(x) = (x)/(σ^2)e^-(x^2)/(2σ^2)

(a) Determine the sufficient statistic for σ^2 using Neyman factorization.
(b) Use the defintion to confirm your answer in A.
(c) Find the MLE of σ^2
(d) Use MLE theory to determine the asymptotic distribution of a scaled version fo the
MLE of σ(not squared). Does the CLT give the same result?
(e) Use C. to identify the UMVUE for σ^2
(f) Identify two statistics that are independent from the sufficient statistic.
(g) Determine the uniformly most powerful test for testing H0 : σ^2 = σ0^2 vs H1 σ^2 > σ0^2.
(h) Derive an expression for the power of the test in G.

Added by Briana A.

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