4. It is known that for any data, sample variance s^2 with divisor (n − 1) is an unbiased estimator of the population variance σ^2 . Then prove that E(SSE) = (n − ν)σ^2 in one way ANOVA.
4. It is known that for any data, sample variance s^2 with divisor (n − 1)...
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
2. 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 (o^)-( Σ (Xi-X )2 ) / N. Calculate the bias of(o^)2 .
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,.
Find a consistent estimator of µ 2 , where E(Y ) = µ is the population mean and Y¯ n is the sample mean. 2 If E(Y 2 ) = µ 0 2 then prove that 1 n Pn i=1 Y 2 i is an consistent estimator of µ 0 2 3 We define σ 2 = µ 0 2 − µ 2 . Show that S 2 n = 1 n Pn i=1 Y 2 i − Y¯ 2...
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...
The definition of the sample variance is S2- -Σ(X-X)2 Prove that is an unbiased estimator of σ
Let X, , x, be a random sample from some density which has mean μ and variance σ2. Show that Σ a, X, is an unbiased estimator of/e for any set of known constants a, , . . . , a, satisfying Σ a,-1. If Σ a.-1, show that var [ Σ a, xl] is minimized for ai = 1/n, i = 1, [HINT: Prove that Σ a-Σ (al-IMF + 1/n when Σ al = 1 .] (a) (b) ,...
Let X,, X,,...X be a random sample of size n from a normal distribution with parameters a. Derive the Cramer-Rao lower bound matrix for an unbiased estimator of the vector of parameters (μ, σ2). b. Using the Cramer-Rao lower bound prove that the sample mean X is the minimum variance unbiased estimator of u Is the maximum likelihood estimator of σ--σ-->|··( X,-X ) unbiased? c. Let X,, X,,...X be a random sample of size n from a normal distribution with...
Suppose population 1 has mean with variance σ2 and population 2 has mean μ2 with the same variance σ. Let sỈ and s denote the sample variances from two samples with size ni and n2 from the corresponding populations, respectively. Show that the pooled estimator pooled is an unbiased estimator of σ2
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...