Question

Let ?? ~ ??? (?,θ ) independent random variable. for ? = 1,2,… ?. Find an estimator for ? by the maximum likelihood method.(MLE)

Answer #1

Let ?? ~ ???? (2?, 4?) independet random variables. for ? =
1,2,… ?.
a)Find an estimator for ? by the method of moments.
b) Find an estimator for ? by the maximum likelihood estimator
(MLE)

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...

Let X1,X2,...,Xn be i.i.d. Geometric(θ), θ = 1,2,3,... random
variables.
a) Find the maximum likelihood estimator of θ.
b) In a certain hard video game, a player is confronted with a
series of AI opponents and has an θ probability of defeating each
one. Success with any opponent is independent of previous
encounters. Until ﬁrst win, the player continues to AI contest
opponents. Let X denote the number of opponents contested until the
player’s ﬁrst win. Suppose that data of...

6. 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.
a) Obtain the maximum likelihood estimator of θ, ˆθ.
b) Is ˆθ a consistent estimator of θ? Justify your answer.

1. Find the maximum likelihood estimator (MLE) of θ based on a
random sample X1, . . . , Xn from each of the following
distributions
(a) f(x; θ) = θ(1 − θ) ^(x−1) , x = 1, 2, . . . ; 0 ≤ θ ≤ 1
(b) f(x; θ) = (θ + 1)x ^(−θ−2) , x > 1, θ > 0
(c) f(x; θ) = θ^2xe^(−θx) , x > 0, θ > 0

Suppose data collected suggest a Bernoulli distribution with
parameter θ (a special case of the binomial distribution).
a) Use the method of moments to obtain an estimator of θ.
b) Obtain the maximum likelihood estimator (MLE) of θ.

Let ?1, ?2,…. . , ?? (n random variables iid) as a
variable X whose pdf is given by ??-a-1
for ? ≥1.
(a) For ? ≥ 1 calculate ? (??? ≤ ?) = ? (?). Deduce the
function
density of probabilities of Y = lnX.
(b) Determine the maximum likelihood estimator (MLE) of ?
and show that he is without biais

Let X1,..., Xn be a random sample from a
distribution with pdf as follows:
fX(x) = e^-(x-θ) , x > θ
0 otherwise.
Find the sufficient statistic for θ.
Find the maximum likelihood estimator of θ.
Find the MVUE of θ,θˆ
Is θˆ a consistent estimator of θ?

Let we have a sample of 100 numbers from exponential
distribution with parameter θ
f(x, θ) = θ e- θx , 0
< x.
Find MLE of parameter θ. Is it unbiased estimator? Find unbiased
estimator of parameter θ.

Let X1,...,Xn be a random sample from the pdf f(x;θ) = θx^(θ−1)
, 0 ≤ x ≤ 1 , 0 < θ < ∞ Find the method of moments estimator
of θ.

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