Homework Answers
Add Answer to:
8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function...
Suppose X1, ?2, ... , ?? are i.i.d. exponential random variables with mean ?. a. Find...
Suppose X1, ?2, ... , ?? are i.i.d. exponential random variables with mean ?. a. Find the Fisher information ?(?) b. Find CRLB. c. Find sufficient statistic for ?. d. Show that ?̂ = ?1 is unbiased, and use Rao − Blackwellization to construct MVUE for ?.
2. Let X1,..., Xn be i.i.d. according to a normal distribution N(u,02). (a) Get a sufficient...
2. Let X1,..., Xn be i.i.d. according to a normal distribution N(u,02). (a) Get a sufficient statistic for u. Show your work. (b) Find the maximum likelihood estimator for u. (c) Show that the MLE in part (b) is an unbiased estimator for u. (d) Using Basu's theorem, prove that your MLE from before and sº, the sample variance, are independent. (Hint: use W; = X1-0 and (n-1)32)
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
Let X1, X2, ..., X, be a r.s. from P(X), f(x) = (a) Show that X1...
Let X1, X2, ..., X, be a r.s. from P(X), f(x) = (a) Show that X1 is the unbiased estimator for 1. (b) Find îmle for X. (c) Derive the Fisher information I(). (d) Show that Amle is the unbiased estimator (UE) for 1 and Var(AMLE) attains CRLB(= mas). i.e., İmle is minimum variance unbiased estimator (MVUE).
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 < x < 1, > 0 a) Find a sufficient statistic for . Is the statistic minimal sufficient? b) Find the MLE for and verify that it is a function of the statistic in a) c) Find IX() and hence give the CRLB for an unbiased estimator of . pdf means probability distribution function We were unable to transcribe this...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
Let X1, X2, ...... Xn be a random sample of size n from EXP() distribution , ,...
Let X1, X2, ......
Xn be a random sample of size n from
EXP()
distribution ,
, zero , elsewhere.
Given, mean of distribution
and variances
and mgf
a) Show that the mle
for
is
. Is
a consistent estimator for
?
b)Show that Fisher information
. Is mle of
an efficiency estimator for
? why or why not? Justify your answer.
c) what is the mle estimator of
? Is the mle of
a consistent estimator for
?
d) Is...
Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood...
Let X1,X2,...,Xn be iid exponential random variables with unknown mean β. (b) Find the maximum likelihood estimator of β. (c) Determine whether the maximum likelihood estimator is unbiased for β. (d) Find the mean squared error of the maximum likelihood estimator of β. (e) Find the Cramer-Rao lower bound for the variances of unbiased estimators of β. (f) What is the UMVUE (uniformly minimum variance unbiased estimator) of β? What is your reason? (g) Determine the asymptotic distribution of the...
Hint of HW2: 4. Let X = (X1, ..., Xn) be an i.i.d. sample from the...
Hint of HW2:
4. Let X = (X1, ..., Xn) be an i.i.d. sample from the shifted exponential distri- bution with density fa,1(x) = de-1(z–a) 1(x > a), 0 := (a, 1) E O = R (0,00). Use Neyman-Fisher's theorem to: (a) show that S = X(1) is an SS for the family {fa,1}QER; (b) find an SS for the family {f1,1}x>0; (c) find an SS for the family {fa,x}o=(0,1)€0. (d) In part (a), use the procedure from Rao- Blackwell's...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
2. Let X1,..., Xn be i.i.d. according to a normal distribution N(u,02). (a) Get a sufficient statistic for u. Show your work. (b) Find the maximum likelihood estimator for u. (c) Show that the MLE in part (b) is an unbiased estimator for u. (d) Using Basu's theorem, prove that your MLE from before and sº, the sample variance, are independent. (Hint: use W; = X1-0 and (n-1)32)
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
Let X1, X2, ..., X, be a r.s. from P(X), f(x) = (a) Show that X1 is the unbiased estimator for 1. (b) Find îmle for X. (c) Derive the Fisher information I(). (d) Show that Amle is the unbiased estimator (UE) for 1 and Var(AMLE) attains CRLB(= mas). i.e., İmle is minimum variance unbiased estimator (MVUE).
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
Let X1, X2, ......
Xn be a random sample of size n from
EXP()
distribution ,
, zero , elsewhere.
Given, mean of distribution
and variances
and mgf
a) Show that the mle
for
is
. Is
a consistent estimator for
?
b)Show that Fisher information
. Is mle of
an efficiency estimator for
? why or why not? Justify your answer.
c) what is the mle estimator of
? Is the mle of
a consistent estimator for
?
d) Is...
Hint of HW2:
4. Let X = (X1, ..., Xn) be an i.i.d. sample from the shifted exponential distri- bution with density fa,1(x) = de-1(z–a) 1(x > a), 0 := (a, 1) E O = R (0,00). Use Neyman-Fisher's theorem to: (a) show that S = X(1) is an SS for the family {fa,1}QER; (b) find an SS for the family {f1,1}x>0; (c) find an SS for the family {fa,x}o=(0,1)€0. (d) In part (a), use the procedure from Rao- Blackwell's...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
Most questions answered within 3 hours.
-
What is the wavelength, in cm, of a microwave having a frequency
of 3.44 x 10^9...
asked 14 minutes ago -
A new business venture must develop a comprehensive business
plan to borrow money to get started....
asked 15 minutes ago -
Madtack Company uses a job-order costing system and started the month
of March with only one...
asked 18 minutes ago -
Describe how to distinguish true motility from brownian motion
and fluid flow when viewing a wet...
asked 21 minutes ago -
A manufacturer of game controllers is concerned that its
controller may be difficult for left-handed users....
asked 23 minutes ago -
Water at gauge pressure of 3.8 atmospheres at street level
enters an office building at 0.60...
asked 37 minutes ago -
Quick question, I want to know how to convert decimal numbers to
binary number, when i...
asked 58 minutes ago -
You found out that now you are going to receive payments of
$8,500 for the next...
asked 1 hour ago -
17.
Technician A says that leaking components are usually repaired
rather than replaced. Technician B says...
asked 1 hour ago -
A) An Operon is:?
1- A gene that affect another gene
transcription.
2. A gene coding...
asked 1 hour ago -
When opening a file, use this method to determine whether a file
exists before you attempt...
asked 1 hour ago -
Sales total $300,000 when variable costs total $180,000 and
fixed costs are $60,000. Breakeven sales total:...
asked 1 hour ago