110年 - 110 國立政治大學_碩士班暨碩士在職專班招生考試_統計學系:數理統計學#103311

a. (5pt) Let X be one observation from a N(0,θ) distribution. Find the sufficient statistic for θ.

b. (Spl) Let X₁,X₂.... X_n be i..d. from the density . Find the sufficient statistic for

c. (5pt) Let X be one observation from the density

Find an unbiased estimate for θ.

d. (10pt) Let X₁, X₂, ..., X₈ be a random sample from Uniform, where u and o ＞ 0 are the unknown parameters. Find the MLE of I and a based on the 8 observations: X₁=2.88,X₂=-4.04,X₃=-3.26,X₄=-3.57,
 X₅=-0.09,X₆=0.25,X₇=-2.58,X₈=4.59.

e. (5pt) Let X be one observation from Uniform(θ, θ + 1). For testing H0:θ = 0 vs. H₁:θ＞ 0, find the power function for the decision rule that we reject H₀if X ＞ 0.9

f. (5pt) Let X₁,X₂,... , X_n be a random sample fromFind the MLE of θ.

b. (12pt) Are and X independent? Prove or disprove it with your work.

3.(15pt) Let X₁, x₂.,... X_n be a random sample from the density
 Test Ho: θ ≤θ₀ vs. H₁:θ＞ θ₀. Find a uniformly most powerftul size a test if such exists.

b. (4pt) Find the moment estimate of θ