Question

suppose we draw a random sample of size n from a Poisson
distribution with parameter λ. show that the maximum likelihood
estimator for λ is an efficient estimator

Answer #1

Suppose that X is Poisson, with unknown mean λ, and let X1, ...,
Xn be a random sample from X.
a. Find the CRLB for the variance of estimators based on X1,
..., Xn. ̂̅̂
b. Verify that ? = X is an unbiased estimator for λ, and then
show that ? is an efficient estimator for λ.

Let X1, ..., Xn be a sample from an exponential population with
parameter λ.
(a) Find the maximum likelihood estimator for λ. (NOT PI
FUNCTION)
(b) Is the estimator unbiased?
(c) Is the estimator consistent?

Let Xl, n be a random
sample from a gamma distribution with parameters a = 2 and p =
20.
a) Find an
estimator , using the method of maximum likelihood
b) Is the estimator obtained in part a) is unbiased and
consistent estimator for the parameter 0?
c) Using the
factorization theorem, show that the estimator found in part a) is
a sufficient estimator of 0.

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.

Suppose that X1,..., Xn form a random sample from the
uniform distribution on the interval [0,θ], where the value of the
parameter θ is unknown (θ>0).
(1)What is the maximum likelihood estimator of θ?
(2)Is this estimator unbiased? (Indeed, show that it underestimates
the parameter.)

5.2.12. Let the random variable Zn have a Poisson distribution
with parameter μ = n. Show that the limiting distribution of the
random variable Yn =(Zn−n)/√n is normal with mean zero and variance
1.
(Hint: by using the CLT, first show Zn is the sum
of a random sample of size n from a Poisson random variable with
mean 1.)

Suppose the random variable X follows the Poisson P(m) PDF, and
that you have a random sample X1, X2,...,Xn from it. (a)What is the
Cramer-Rao Lower Bound on the variance of any unbiased estimator of
the parameter m? (b) What is the maximum likelihood estimator
ofm?(c) Does the variance of the MLE achieve the CRLB for all
n?

1. Show that if X is a Poisson random variable with parameter
λ, then its variance is λ
2.Show that if X is a Binomial random variable with parameters
n and p, then the its variance is npq.

Suppose X1, . . . , Xn are a random sample from a N(0, σ^2)
distribution. Find the MLE of σ^2
and show that it is an unbiased efficient estimator.

Let X and Y be independent random variables following Poisson
distributions, each with parameter λ = 1. Show that the
distribution of Z = X + Y is Poisson with parameter λ = 2. using
convolution formula

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