Homework Help Question & Answers

Please show step by step solution. 7. Let X1, X2, ..., Xn be i.i.d. random variables...

7. Let X1, X2, ..., Xn be i.i.d. random variables drawn from a N(u,0%). Show that the Sample Variance (52) and the Maximum Li

Please show step by step solution.

7. Let X1, X2, ..., Xn be i.i.d. random variables drawn from a N(u,0%). Show that the Sample Variance (52) and the Maximum Likelihood Estimator (S) of o2 are both Consistent Estimators for o?. S2 27=2(X-X)2 and S 21-2(X;-) n-1 n (n-1)S Hint: has a Chi-Square Distr. with (n − 1) degrees of freedom. E(x{n-1)) = n-1,V(xin-1)) = 2(n − 1)
0 0
Add a comment
Answer #1

Sign Up to Unlock the answer FREE

Already have an account? Log in

estimator For T Conditions to be consistent of o are :- o as n iy E(T) iiy vlt) o no d Ž (xi-x)² I cxi-x)? n. n-1 n-1 n n-l Nconsistent . Hence, it g2 in estimator of s² si (x,-F)? n-1 n.(n-1) Ž (xi-x)² n izi n-1 af n. 5,² - (n-1).s² We know; Bom (n-زم؟ Var n-1).& Var (X2) = 2 (n-1) Now; n. 8,² = (nol).s² Var (a) = 2 (n-1) 2 n Var (13) = 2(n-1) => Var (8) 2 (n-1).04 n2 an

Add a comment
Know the answer?
Add Answer to:
Please show step by step solution. 7. Let X1, X2, ..., Xn be i.i.d. random variables...
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coin

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for? Ask your own homework help question. Our experts will answer your question WITHIN MINUTES for Free.
Similar Homework Help Questions
  • Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It...

    Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It is well-known that X is an unbiased estimator for l because I = E(X). 1. Show that X1+Xn is also an unbiased estimator for \. 2 2. Show that S2 (Xi-X) = is also an unbaised esimator for \. n-1 3. Find MSE(S2). (We will need two facts) E com/questions/2476527/variance-of-sample-variance) 2. Fact 2: For Poisson distribution, E[(X – u)4] 312 + 1. (See for...

  • Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p....

    Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known that p = ΣΧ; is an unbiased estimator for p. n 1. Find E(@2) and show that p2 is a biased estimator for p. (Hint: make use of the distribution of X, and the fact that Var(Y) = E(Y2) – E(Y)2) 2. Suggest an unbiased estimator for p2. (Hint: use the fact that the sample variance is unbiased for variance.) Xi+2...

  • Please answer question (a) X1 - X X2 – Å a. Let X1, ..., Xn i.i.d....

    Please answer question (a) X1 - X X2 – Å a. Let X1, ..., Xn i.i.d. random variables with X; ~ N(u, o). Express the vector in the | Xn – form AX and find its mean and variance covariance matrix. Show some typical elements of the vari- ance covariance matrix. b. Refer to question (a). The sample variance is given by S2 = n11 21–1(X; – X)2, which can be ex- pressed as S2 = n1X'(I – 111')X (why?)....

  • Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p....

    Q2 Suppose X1, X2, ..., Xn are i.i.d. Bernoulli random variables with probability of success p. It is known ΣΧ; is an unbiased estimator for p. that = n 2. Suggest an unbiased estimator for pa. (Hint: use the fact that the sample variance is unbiased for variance.) 3. Show that p= ΣΧ,+2 n+4 is a biased estimator for p. 4. For what values of p, MSE) is smaller than MSE)?

  • Multi-part question: Let X1, ..... , Xn be random variables that describe the height of students...

    Multi-part question: Let X1, ..... , Xn be random variables that describe the height of students from a class, in the logarithmic scale. A) Write the statistical model (there might be more than one suitable distribution). B) Assume that X1, ... ,Xn form a random sample from the normal distribution with known mean θ and unknown variance σ^2 . Find the maximum likelihood estimator of the variability of the height (in log scale) of the students, this is, find the...

  • Let X1. . . . Xn be i.i.d Uniform over the interval (θ, θ + 1].Show...

    Let X1. . . . Xn be i.i.d Uniform over the interval (θ, θ + 1].Show that X(1)+X(n) )/2- 1/2 is also an unbiased estimator of θ, whereX(1) is the minimum order statistic and X(n) is the maximum order statistic. If X - 1/2 is also an unbiased estimator of θ which of the two estimators would you prefer to use.

  • QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution,...

    QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...

  • 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...

  • Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a...

    Let X1, X2, ..., Xn be a random sample from the N(u, 02) distribution. Derive a 100(1-a)% confidence interval for o2 based on the sample variance S2. Leave your answer in terms of chi-squared critical values. (Hint: We will show in class that, for this normal sample, (n − 1)S2/02 ~ x?(n − 1).)

  • 1. (40) Suppose that X1, X2, .. , Xn, forms an normal distribution with mean /u and variance o2, both unknown: independ...

    1. (40) Suppose that X1, X2, .. , Xn, forms an normal distribution with mean /u and variance o2, both unknown: independent and identically distributed sample from 2. 1 f(ru,02) x < 00, -00 < u < 00, o20 - 00 27TO2 (a) Derive the sample variance, S2, for this random sample (b) Derive the maximum likelihood estimator (MLE) of u and o2, denoted fi and o2, respectively (c) Find the MLE of 2 (d) Derive the method of moment...

ADVERTISEMENT
Free Homework Help App
Download From Google Play
Scan Your Homework
to Get Instant Free Answers
Need Online Homework Help?
Ask a Question
Get Answers For Free
Most questions answered within 3 hours.
ADVERTISEMENT
Active Questions
ADVERTISEMENT