2. The sample variance s2 is known to be an unbiased estimator of the variance σ2....
The sample variance s2 is known to be an unbiased estimator of the variance σ2. Consider the estimator (σ^)2 of the variance σ2, where (σ^)2 = ( ∑ (Xi − )2 ) / N. Calculate the bias of (σ^)2.
The definition of the sample variance is S2- -Σ(X-X)2 Prove that is an unbiased estimator of σ
4. Xi ,i = 1, , n are iid N(μ, σ2). (a) Find the MLE of μ, σ2. Are these unbiased estimators of μ and of σ2 respectively? Aside: You can use your result in (b) to justify your answer for the bias part of the MLE estimator of σ2 (b) In this part you will show, despite that the sample variance is an unbiased estimator of σ2, that the sample standard deviation is is a biased estimator of σ....
4. It is known that for any data sample variance s2 with divisor (n - 1) is an unbiased estimator of the population variance σ2. Then prove that E(SSE) = (n-v)o2 in one way ANOVA
Denoting the variance of by ơ, prove that n' ) σ ơy _ (N-1) n State (without proof) the expected value of the sample variance s2. Derive an unbiased estimator, so, for σ,.
Denoting the variance of by ơ, prove that n' ) σ ơy _ (N-1) n State (without proof) the expected value of the sample variance s2. Derive an unbiased estimator, so, for σ,.
x, and S1 are the sample mean and sample variance from a population with mean μ| and variance ơf. Similarly, X2 and S1 are the sample mean and sample variance from a second population with mean μ and variance σ2. Assume that these two populations are independent, and the sample sizes from each population are n,and n2, respectively. (a) Show that X1-X2 is an unbiased estimator of μ1-μ2. (b) Find the standard error of X, -X. How could you estimate...
is taken from N(μ, σ2), where the mean 2. A randorn sample X1, X2, , xn of size μ is a known real num ber. Show that the m axim urn likelihood estimator for σ2 is ớmle n Σ.i(Xi μ)2 and that this estimator is an unbiased estinator of σ2. (I lint: Σ.JX _ μ)-g. Σ.i My L and Σ. (Xcpl, follows X2(n))
4. Let X1,X2, ,Xn be a randonn sample from N(μ, σ2) distribution, and let s* Ση! (Xi-X)2 and S2-n-T Ση#1 (Xi-X)2 be the estimators of σ2 (i) Show that the MSE of s is smaller than the MSE of S2 (ii) Find E [VS2] and suggest an unbiased estimator of σ.
DISTRIBUTION OF SAMPLE VARIANCE:
Xn ~ N(μ, σ2), where both μ and σ are Problem 4 (25 points). Assume that Xi unknowin 1. Using the exact distribution of the sample variance (Topic 1), find the form of a (1-0) confidence interval for σ2 in terms of quantiles of a chi-square distribution. Note that this interval should not be symmetric about a point estimate of σ2. [10 points] 2. Use the above result to derive a rejection region for a level-o...
Let P be a distribution on R with variance σ2. Let X1, and let S2 be the associated unbiased estimator of σ2. 1, ,Xn be a random sample form P n-1 i-1 Show that 4 2ơ 2 Wa Feel free to "Cheat" and use the fact that (n - 1)s2 2 n-1 (Can you do it without "Cheating"?)