For SRSWOR (N, n), find the ratio estimator of the population mean. Show that the estimator is biased.
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Suppose X , X ,..., X n 1 2 be a random sample drawn from a population with mean θand varianceσ2 . Show that Ti=1/i∑Xj is unbiased estimator of θ (i = 1,2,…,n). Also show var(Ti ) is monotonically decreasing. Hence find the best unbiased estimator for θ.
7-27. Let X1, X2,..., X, be a random sample of size n from a population with mean u and variance o?. (a) Show that X² is a biased estimator for u?. (b) Find the amount of bias in this estimator. c) What happens to the bias as the sample size n increases?
Suppose that Y1 , Y2 ,..., Yn denote a random sample of size n from a normal population with mean μ and variance 2 . Problem # 2: Suppose that Y , Y,,...,Y, denote a random sample of size n from a normal population with mean u and variance o . Then it can be shown that (n-1)S2 p_has a chi-square distribution with (n-1) degrees of freedom. o2 a. Show that S2 is an unbiased estimator of o. b....
Please give detailed steps. Thank you. 5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
1. Select all true statements about sample mean and sample median. A) When the population distribution is skewed, sample mean is biased but sample median is an unbiased estimator of population mean. B) When the population distribution is symmetric, both mean and sample median are unbiased estimators of population mean. C) Sampling distribution of sample mean has a smaller standard error than sample median when population distribution is normal. D) Both mean and median are unbiased estimators of population mean...
7.Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ:⊝1 = 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4. Is this a biased estimator for the mean? What is the variance of the estimator? Can you find a more efficient estimator?
, X, be a random sample from a population with mean μ and variance Show let XI. . . . , 5.4.8. that ¡2 -X* is a biased estimator of that-T 2, and compute the bias.
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...
(a) Show that Σ-1(n-X)2 İs a biased estimator of σ2 (b) Find the amount of bias in the estimator. (c) What happens to the bias as the sample size n increases?
5. For each n = 1, 2, . . . , the random variable Xn is such that P(Xn-01 = 1-1 and PlXn-θ + n) n. Show that Xn is biased estimator of θ and find its MSE. Show that Xn is consistent estimator of θ.