Question

Let X1, X2, . . . , Xn be a random sample from a population following a uniform(0,2θ)

(a) Show that if n is large, the distribution of Z =( X − θ) / ( θ/√ 3n) is approximately N(0, 1).

(b) We can estimate θ by X(sample mean) and define W = (X − θ ) / (X/√ 3n) which is also approximately N(0, 1) for large n. Derive a (1 − α)100% confidence interval for θ based on W.

(c) Find a (1 − α)100% confidence interval for θ directly from Z defined in (a).

(d) Using the result in (b), find a (1 − α)100% confidence interval for the variance of X which follows a uniform(0,2θ) distribution.

Answer #1

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

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

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 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, . . . , Xn be a random sample from a Bernoulli(θ)
distribution, θ ∈ [0, 1]. Find the MLE of the odds ratio, defined
by θ/(1 − θ) and derive its asymptotic distribution.

Let X1, X2, ..., Xn be a random sample from a distribution with
probability density function f(x; θ) = (θ 4/6)x 3 e −θx if 0 < x
< ∞ and 0 otherwise where θ > 0
. a. Justify the claim that Y = X1 + X2 + ... + Xn is a complete
sufficient statistic for θ. b. Compute E(1/Y ) and find the
function of Y which is the unique minimum variance unbiased
estimator of θ.
b. Compute...

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.

Suppose that (X1, · · · , Xn) is a random sample from uniform
distribution U(0, θ).
(a) Prove that T(X1, · · · , Xn) = X(n) is minimal sufficient
for θ. (X(n) is the largest order statistic, i.e., X(n) = max{X1, ·
· · , Xn}.)
(b) In addition, we assume θ ≥ 1. Find a minimal sufficient
statistic for θ and justify your answer.

Let X2, ... , Xn denote a random sample
from a discrete uniform distribution over the integers - θ, - θ +
1, ... , -1, 0, 1, ... , θ - 1, θ,
where θ is a positive integer. What is the maximum
likelihood estimator of θ?
A) min[X1, .. , Xn]
B) max[X1, .. , Xn]
C) -min[X1, .. , Xn]
D) (max[X1, .. , Xn] -
min[X1, .. , Xn]) / 2
E) max[|X1| , ... , |Xn|]

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

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