STAT 311: Mathematical Statistics Homework 02 Chapter 5: Point Estimation 1- Let X, ,...,X, be a random sample of size n from the geometric distribution for which p is the probability of success. (a) Use the method of moments to find a point estimator for p. (b) Use the following data (simulated from geometric distribution) to find the moment estimator for p: 5 43 18 19 16 11 22 4 34 19 21 23 6. 21 12 2- The probability density of a one-parameter Weibull distribution is given by S2ax e¬ax² 0, x > 0 otherwise f(x) = - (a) Using a random sample of size n, obtain a moment estimator for a. (b) Assuming that the following data are from a one-parameter Weibull population, 1.87 1.60 2.36 1.12 0.15 1.83 0.64 1.53 0.73 2.26 3- Suppose X, ,...,X, are a random sample from an exponential distribution with parameter 0. Find the MLE of Ô. Also using the invariance property, obtain an MLE for the variance. 4- Let X1 .,X, be a random sample from a two-parameter Weibull distribution with pdf .... xa-1e(x/ß)“, x >0 f(x) 0, otherwise Find the MLES of a and ß. 5- Consider the problem of estimating p in a binomial distribution. Let X be number of successes in a sample of size n. (a) Let the prior distribution of p be given by Beta(3,1), that is T(p) = {3p², 0 0 f(x) = f(x) = 0, %3D otherwise Is the method of moments estimator for a consistent? 10- Let X,,..., X,,n > 4, be a random sample from a population with a mean µ and variance o². Consider the following three estimators of µ: ô =7(X, + 2X2 + 5X3 + X4) Ôz = X1 +X2 + -(X3 + ……·+ Xn-1) + n - 3) 5(n Ôz = X (a) Show that each of the three estimators is unbiased. (b) Find e(@2, ô1), e(@3, ô1) and e(Ô3, Ô2). 11- Let X, ,...,X, be a random sample from the Weibull density (2x -x²/a x > 0 f(x) = f(x) = 0, otherwise Find an UMVUE for a.
STAT 311: Mathematical Statistics Homework 02 Chapter 5: Point Estimation 1- Let X, ,...,X, be a random sample of size n from the geometric distribution for which p is the probability of success. (a) Use the method of moments to find a point estimator for p. (b) Use the following data (simulated from geometric distribution) to find the moment estimator for p: 5 43 18 19 16 11 22 4 34 19 21 23 6. 21 12 2- The probability density of a one-parameter Weibull distribution is given by S2ax e¬ax² 0, x > 0 otherwise f(x) = - (a) Using a random sample of size n, obtain a moment estimator for a. (b) Assuming that the following data are from a one-parameter Weibull population, 1.87 1.60 2.36 1.12 0.15 1.83 0.64 1.53 0.73 2.26 3- Suppose X, ,...,X, are a random sample from an exponential distribution with parameter 0. Find the MLE of Ô. Also using the invariance property, obtain an MLE for the variance. 4- Let X1 .,X, be a random sample from a two-parameter Weibull distribution with pdf .... xa-1e(x/ß)“, x >0 f(x) 0, otherwise Find the MLES of a and ß. 5- Consider the problem of estimating p in a binomial distribution. Let X be number of successes in a sample of size n. (a) Let the prior distribution of p be given by Beta(3,1), that is T(p) = {3p², 0 0 f(x) = f(x) = 0, %3D otherwise Is the method of moments estimator for a consistent? 10- Let X,,..., X,,n > 4, be a random sample from a population with a mean µ and variance o². Consider the following three estimators of µ: ô =7(X, + 2X2 + 5X3 + X4) Ôz = X1 +X2 + -(X3 + ……·+ Xn-1) + n - 3) 5(n Ôz = X (a) Show that each of the three estimators is unbiased. (b) Find e(@2, ô1), e(@3, ô1) and e(Ô3, Ô2). 11- Let X, ,...,X, be a random sample from the Weibull density (2x -x²/a x > 0 f(x) = f(x) = 0, otherwise Find an UMVUE for a.
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Transcribed Image Text:STAT 311: Mathematical Statistics
Homework 02
Chapter 5: Point Estimation
1- Let X, ,...,X, be a random sample of size n from the geometric distribution for which p is the
probability of success.
(a) Use the method of moments to find a point estimator for p.
(b) Use the following data (simulated from geometric distribution) to find the moment estimator
for p:
5
43
18
19
16
11
22
4
34
19
21
23
6.
21
12
2- The probability density of a one-parameter Weibull distribution is given by
S2ax e¬ax²
0,
x > 0
otherwise
f(x) = -
(a) Using a random sample of size n, obtain a moment estimator for a.
(b) Assuming that the following data are from a one-parameter Weibull population,
1.87
1.60 2.36
1.12
0.15 1.83
0.64
1.53
0.73
2.26
3- Suppose X, ,...,X, are a random sample from an exponential distribution with parameter 0. Find
the MLE of Ô. Also using the invariance property, obtain an MLE for the variance.
4- Let X1
.,X, be a random sample from a two-parameter Weibull distribution with pdf
....
xa-1e(x/ß)“,
x >0
f(x)
0,
otherwise
Find the MLES of a and ß.
5- Consider the problem of estimating p in a binomial distribution. Let X be number of successes
in a sample of size n.
(a) Let the prior distribution of p be given by Beta(3,1), that is
T(p) = {3p², 0 <p<1
0,
otherwise
Find the posterior distribution of p.
(b) Let the prior distribution of p be given by Beta(a,b) (that is, 7(p) x pa-1(1 – p)b-1. Find
the posterior distribution of p.
6- Let X,,X, ,...,X, be exponential random variables with parameter 2. Let the prior a(2) be
exponentially distributed with parameter u, which is a fixed and known constant.
(a) Show that the posterior distribution of 1 is Gamma (1+E-1xi ,n+ 1)
(b) Obtain the Bayes estimate of A.
STAT 311: Mathematical Statistics, By Dr. Abdelfattah Mustafa,
Transcribed Image Text:7- Let X,, X, ,...,X, be Poisson random variables with parameter 2. Assume that 2 has a Gamma
(a, ß) prior.
(a) Compute the posterior distribution of 2.
(b) Obtain the Bayes estimate of A.
(c) Compare the MLE of 1 with the Bayes estimate of l.
(d) Which of the two estimates is better? Why?
8- Let X1 .
,X, be a random sample from a Poisson distribution with parameter 2. Show that the
sample mean X is sufficient for 2.
9- Let X, ,...,X„ be a random sample from a population with pdf
0 < x < 1, a > 0
f(x) = f(x) =
0,
%3D
otherwise
Is the method of moments estimator for a consistent?
10- Let X,,..., X,,n > 4, be a random sample from a population with a mean µ and variance o².
Consider the following three estimators of µ:
ô =7(X, + 2X2 + 5X3 + X4)
Ôz = X1 +X2 +
-(X3 + ……·+ Xn-1) + n
- 3)
5(n
Ôz = X
(a) Show that each of the three estimators is unbiased.
(b) Find e(@2, ô1), e(@3, ô1) and e(Ô3, Ô2).
11- Let X, ,...,X, be a random sample from the Weibull density
(2x
-x²/a
x > 0
f(x) = f(x) =
0,
otherwise
Find an UMVUE for a.
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