Question

Let Y_1, … , Y_n be a random sample from a normal distribution with unknown mu and unknown variance sigma^2. We want to test H_0 : mu=0 versus H_a : mu !=0. Find the rejection region for the likelihood ratio test with level alpha.

Answer #1

TOPIC: Likelihood ratio test.

Suppose Y_1, Y_2,… Y_n denote a random sample of a geometric distribution with parameter p. Find the maximum likelihood estimator for p.

Let ?1 and ?2 be a random sample of size 2 from a normal
distribution ?(?, 1). Find the likelihood ratio critical region of
size 0.005 for testing the null hypothesis ??: ? = 0 against the
composite alternative ?1: ? ≠ 0?

suppose a random sample of 25 is taken from a population that
follows a normal distribution with unknown mean and a known
variance of 144. Provide the null and alternative hypothesis
necessary to determine if there is evidence that the mean of the
population is greater than 100. Using the sample mean ybar as the
test statistic and a rejection region of the form {ybar>k} find
the value of k so that alpha=.15 Using the sample mean ybar as the...

Let ?1, ?2, … , ?? denote a random sample from a normal
population distribution with a known value of ?.
(a) For testing the hypotheses ?0: ? = ?0 versus ??: ? > ?0
where ?0 is a fixed number, show that the test with test statistic
?̅ and rejection region ?̅≥ ?0 + 2.33(?⁄√?) has a significance
level 0.01.
(b) Suppose the procedure of part (a) is used to test ?0: ? ≤ ?0
versus ??: ? >...

Let X1,....,XN be a random sample from N(Mu, sigma squared).
Both parameters unknown.
a) Give two pivotal quantities based on the sufficient
statistics and determine their distribution functions. The best
solutions are those you can use to construct confidence intervals
for Mu and sigma squared.
b) If a random sample of size 5 is observed as follows: 9.67,
10.01, 9.31, 9.33, 9.28, find a 95% equal tailed confidence
interval for sigma squared.
c) Based on the above observations, find a...

Using a random sample of size 77 from a normal population with
variance of 1 but unknown mean, we wish to test the null hypothesis
that H0: μ ≥ 1 against the alternative that Ha: μ <1.
Choose a test statistic.
Identify a non-crappy rejection region such that size of the
test (α) is 5%. If π(-1,000) ≈ 0, then the rejection region is
crappy.
Find π(0).

Let X1,...,Xn be a random sample from a normal
distribution where the variance is known and the mean is
unknown.
Find the minimum variance unbiased estimator of the
mean. Justify all your steps.

Let X have a gamma distribution
with and which is unknown. Let be
a random sample from this distribution.
(1.1) Find a consistent estimator for using the
method-of-moments.
(1.2) Find the MLE of denoted by .
(1.3) Find the asymptotic variance of the MLE, i.e.
(1.4) Find a sufficient statistic for .
(1.5) Find MVUE for .

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 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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