Question

Let X1, X2, . . . , Xn be a random sample from the normal distribution N(µ, 36). (a) Show that a uniformly most powerful critical region for testing H0 : µ = 50 against H1 : µ < 50 is given by C2 = {x : x ≤ c}. Find the values of c for α = 0.10.

Answer #1

Let X1, X2, . . . , X12 denote a random sample of size 12 from
Poisson distribution with mean θ.
a) Use Neyman-Pearson Lemma to show that the critical region
defined by
(12∑i=1) Xi, ≤2
is a best critical region for testing H0 :θ=1/2 against H1
:θ=1/3.
b.) If K(θ) is the power function of this test, find K(1/2) and
K(1/3). What is the significance level, the probability of the 1st
type error, the probability of the 2nd type...

Let X1, X2, · · · , Xn be a random sample from an exponential
distribution f(x) = (1/θ)e^(−x/θ) for x ≥ 0. Show that likelihood
ratio test of H0 : θ = θ0 against H1 : θ ≠ θ0 is based on the
statistic n∑i=1 Xi.

Let X1, X2 · · · , Xn be a random sample from the distribution
with PDF, f(x) = (θ + 1)x^θ , 0 < x < 1, θ > −1.
Find an estimator for θ using the maximum likelihood

Let X1, X2, . . . Xn be iid
random variables from a gamma distribution with unknown α and
unknown β. Find the method of moments estimators for α and β

1. Let X1, X2, . . . , Xn be a random sample from a distribution
with pdf f(x, θ) = 1 3θ 4 x 3 e −x/θ , where 0 < x < ∞ and 0
< θ < ∞. Find the maximum likelihood estimator of ˆθ.

Let X1, X2, · · · , Xn be a random sample from the distribution,
f(x; θ) = (θ + 1)x^ −θ−2 , x > 1, θ > 0. Find the maximum
likelihood estimator of θ based on a random sample of size n
above

Exercise 4. Suppose that X = (X1, · · · , Xn) is a random sample
from a normal distribution with unknown mean µ and known variance
σ^2 . We wish to test the following hypotheses at the significance
level α. Suppose the observed values are x1, · · · , xn. For each
case, find the expression of the p-value, and state your decision
rule based on the p-values
a. H0 : µ = µ0 vs. Ha : µ...

Let X1,..., Xn be a random sample from the Rayleigh
distribution:
Use the likelihood ratio test to give a form of test (without
specifying the value of the critical value) for H0: θ= 1 versus
H1:≠1

Let θ > 1 and let X1, X2, ..., Xn be a random sample from the
distribution with probability density function f(x; θ) = 1/xlnθ , 1
< x < θ.
c) Let Zn = nlnY1. Find the limiting distribution of Zn.
d) Let Wn = nln( θ/Yn ). Find the limiting distribution of
Wn.

Let X1,..., Xn be a random sample from the Geometric
distribution:
Use the likelihood ratio test to give a form of test (without
specifying the value of the critical value) for H0: p= 1 versus
H1:p≠1

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