Question

Let Y1,Y2.....,Yn be independent ,uniformly distributed random variables on the interval[0,θ].，Y(n)=max(Y1,Y2,....,Yn)，which is considered as an estimator of θ. Explain why Y is a good estimator for θ when sample size is large.

Answer #1

The random variable X is uniformly distributed in the interval
[0, α] for some α > 0. Parameter α is fixed but unknown. In
order to estimate α, a random sample X1, X2, . . . , Xn of
independent and identically distributed random variables with the
same distribution as X is collected, and the maximum value Y =
max{X1, X2, ..., Xn} is considered as an estimator of α.
(a) Derive the cumulative distribution function of Y .
(b)...

Included all steps. Thanks
The random variable X is uniformly distributed in the interval
[0, α] for some α > 0.
Parameter α is fixed but unknown. In order to estimate α, a
random sample X1, X2, . . . , Xn of independent and identically
distributed random variables with the same distribution as X is
collected, and the maximum value Y = max{X1, X2, ..., Xn} is
considered as an estimator of α.
(a) Derive the cumulative distribution function...

Let Y1, Y2, ... Yn be a random sample of an exponential
population with parameter θ. Find the density function of the
minimum of the sample Y(1) = min(Y1, Y2, ..., Yn).

Let Y1, Y2, . . ., Yn be a
random sample from a Laplace distribution with density function
f(y|θ) = (1/2θ)e-|y|/θ for -∞ < y < ∞
where θ > 0. The first two moments of the distribution are
E(Y) = 0 and E(Y2) = 2θ2.
a) Find the likelihood function of the sample.
b) What is a sufficient statistic for θ?
c) Find the maximum likelihood estimator of θ.
d) Find the maximum likelihood estimator of the standard
deviation...

Suppose that Y1, . . . , Yn are iid random variables from the
pdf
f(y | θ) = 6y^5/(θ^6) I(0 ≤ y ≤ θ). (a) Prove that Y(n) = max
(Y1, . . . , Yn) is sufficient for θ. (b) Find the MLE of θ

(a) Let Y1,Y2,··· ,Yn be i.i.d.
with geometric distribution P(Y = y) = p(1−p)y-1 y=1, 2,
........, 0<p<1. Find a suﬃcient statistic for p.
(b) Let Y1,··· ,yn be a random sample of size n from
a beta distribution with parameters α = θ and β = 2. Find the
suﬃcient statistic for θ.

Let Y1, Y2, . . ., Yn be a
random sample from a uniform distribution on the interval (θ - λ, θ
+ λ) where -∞ < θ < ∞ and λ > 0. Find the method of
moments estimators of θ and λ.

5. Let Y1, Y2, ...Yn (independent and identically distributed. ∼
f(y; α) = 1/6 α8y3 · e^(−α2y3 ), 0 ≤
y < ∞, 0 < α < ∞.
(a) (8 points) Find an expression for the Method of Moments
estimator of α, ˜α. Show all work.
(b) (8 points) Find an expression for the Maximum Likelihood
estimator for α, ˆα. Show all work.

Let Y1, Y2, . . . , Yn denote a random sample from a uniform
distribution on the interval (0, θ). (a) (5 points)Find the MOM for
θ. (b) (5 points)Find the MLE for θ.

Problem 3. Let Y1, Y2, and Y3 be independent, identically
distributed random variables from a population with mean µ = 12 and
variance σ 2 = 192. Let Y¯ = 1/3 (Y1 + Y2 +
Y3) denote the average of these three random
variables.
A. What is the expected value of Y¯, i.e., E(Y¯ ) =? Is Y¯ an
unbiased estimator of µ?
B. What is the variance of Y¯, i.e, V ar(Y¯ ) =?
C. Consider a different estimator...

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