Question

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.

Answer #1

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...

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.)

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 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

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...

Let B > 0 and let X1 , X2 , … , Xn be a random sample from
the distribution with probability density function.
f( x ; B ) = β/ (1 +x)^ (B+1), x > 0, zero otherwise.
(i) Obtain the maximum likelihood estimator for B, β ˆ .
(ii) Suppose n = 5, and x 1 = 0.3, x 2 = 0.4, x 3 = 1.0, x 4 =
2.0, x 5 = 4.0. Obtain the maximum likelihood...

#1 A sample of 4 observations (X1 = 0.4,
X2 = 0.6, X3 = 0.7, X4 = 0.9) is
collected from a continuous distribution with pdf
(a) Find the point estimate of θ by the Method of
Moments.
(b) Find the point estimate of θ by the Method of
Maximum Likelihood. Use two decimal places.

Suppose X1 ...... Xn is a random sample from the uniform
distribution on [a; b].
(a) Find the method of moments estimators of a and b.
(b) Find the maximum likelihood estimators of a and b.
please step by step

use R software
Suppose that X1, …, Xn
are a random sample from a lognormal distribution. Construct a 95%
confidence interval for the parameter μ. Use a Monte Carlo
method to obtain an empirical estimate of the confidence level when
data is generated from standard lognormal.

The shear strength of each of ten test spot welds is determined,
yielding the following data (psi). 367 415 391 409 362 389 396 382
375 358
(a) Assuming that shear strength is normally distributed,
estimate the true average shear strength and standard deviation of
shear strength using the method of maximum likelihood. (Round your
answers to two decimal places.)
average psi?
standard deviation psi?
(b) Again assuming a normal distribution, estimate the strength
value below which 95% of all...

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