Determine the Maximum Likelihood Estimator for;
1. λ for the Poisson distribution.
2. θ for the Exponential distribution.
Caveat: These are examples of distributions for which the MLE
can be found analytically in terms of the data x1, . . . , xn and
so no advanced computational methods are required and also in each
assume a random sample of size n, x1, x2, . . . , xn
Can you explain me about what is Maximum Likelihood Estimator
(MLE) for bernouli, normal, uniform, and poisson? How to use
it(example), and what is its relation with likelihood function?
Given the likelihood of θ with respect to sample D p(D|θ) = Q j
p(xj |θ), where D = {x1, · · · , xn} is identically and
independently distributed (i.i.d) sample points. Briefly describe
how you would find the maximum likelihood estimation and the
Bayesian estimation of θ.
Let X1, . . . , Xn be a random sample from a uniform
distribution on the interval [a, b]
(i) Find the moments estimators of a and b.
(ii) Find the MLEs of a and b.
Let X1,X2, . . . , Xn be a random sample from the uniform
distribution with pdf f(x; θ1, θ2) =
1/(2θ2), θ1 − θ2 < x <
θ1 + θ2, where −∞ < θ1 < ∞
and θ2 > 0, and the pdf is equal to zero
elsewhere.
(a) Show that Y1 = min(Xi) and Yn = max(Xi), the joint
sufficient statistics for θ1 and θ2, are
complete.
(b) Find the MVUEs of θ1 and θ2.
Let Y1, Y2, . . ., Yn be a
random sample from a uniform distribution on the interval (θ - 2,
θ).
a) Show that Ȳ is a biased estimator of θ. Calculate the
bias.
b) Calculate MSE( Ȳ).
c) Find an unbiased estimator of θ.
d) What is the mean square error of your unbiased estimator?
e) Is your unbiased estimator a consistent estimator of θ?
Suppose that X1, ..., Xn form a random sample from a uniform
distribution for on the interval [0, θ]. Show that T = max(X1, ...,
Xn) is a sufficient statistic for θ.
2. Let X1, ..., Xn be a random sample from a uniform
distribution on the interval (0, θ) where θ > 0 is a parameter.
The prior distribution of the parameter has the pdf f(t) =
βαβ/t^(β−1) for α < t < ∞ and zero elsewhere, where α > 0,
β > 0. Find the Bayes estimator for θ. Describe the usefulness
and the importance of Bayesian estimation.
We are assuming that theta = t, but we are unsure if...