If X1,X2, . . . ,Xn constitute a random sample from a
population with the mean μ, what condition must be
imposed on the constants a1, a2, . . . , an so that
a1X1 +a2X2 + · · · + anXn
is an unbiased estimator of μ?
Let is a random sample from a population with mean and
Here, we need to identify a condition that imposes on the constants so that is an unbiased estimator of.
To show that a statistic is unbiased estimator of the parameter, we need to show that.
Now, let us consider
From (a), if we want is an unbiased estimator of, we should have the condition that
By substituting the value in (a) we get
Hence, is an unbiased estimator of if sum of all constant value is equal to 1.
5. (Chihara and Hestelberg : Exercise 6.4.25) Let X1, X2, . . . , Xn be random variables with E(X) = μ, for all i-1, 2, . . . , n. Under what condition on the constants ai, a2, . . . , an İs an unbiased estimator of μ?
Let {x1, x2, ..., xn} be a sample from Bernoulli(p). Find an unbiased estimator for p^2 . Let {x1,x2,..., Xn} be a ..., Xn} be a sample from Bernoulli(p). Find an unbiased estimator for p?.
2. Show that: When X is a binomial rv, the sample proportion is the unbiased estimator of the population proportion. IfX1.Хг, estimator of the population mean a) xn is a random sample with mean ,, then the sample mean is the unbiased b)
If X, X2,..., Xn constitute a random sample from the population with pdf ffx) 0 elsewhere a) ind the E(X) and hence show that X is a biased estimator of 0. What is the bias? b)What estimator based on X would be an unbiased estimator of 0? Why? nen( y1-0) y, > c Given g(y,)- show that Yı= min ( X1, X2, Х. ) is a consistent 0 otherwise estimator of the parameter 0 d) Obtain the mean of Y,....
X1, X2, ..., Xn constitute a random sample from a population with pdf 2 +0.03) |2|<1 f(0) = 0 {ila. 0.W. where 101 < 1. Determine if X is an unbiased estimator of 8. If not, modify it to make it unbiased, and determine if it is consistent. Justify.
Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. The observations are independent because they were randomly drawn. Consider the following two point estimators of the population mean μ: 1 = 0.10 X1 + 0.40 X2 + 0.40 X3 + 0.10 X4 and 2 = 0.20 X1 + 0.30 X2 + 0.30 X3 + 0.20 X4 Which of the following statements is true? HINT: Use the definition of...
If a null hypothesis is rejected at a significance level of 1%, then we should say that it was rejected at 1%. Reporting that the null was also rejected at the 5% level of significance is unnecessary and unwise. True False The p-value equals alpha, the level of significance of the hypothesis test. True False THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION: Let X1, X2, X3, and X4 be a random sample of observations from a population with...
To show an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the following estimation procedure: To estimate the mean of a population with the nite variance 2, we rst take a random sample of size n. Then, we randomly draw one of n slips of paper numbered from 1 through n, and if the number we draw is 2, 3, , or n, we use as our estimator the mean of the random sample; otherwise, we...
To show an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the following estimation procedure: To estimate the mean of a population with the finite variance σ 2 , we first take a random sample of size n . Then, we randomly draw one of n slips of paper numbered from 1 through n , and • if the number we draw is 2, 3, ··· , or n , we use as our estimator the...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over [0,0], where 0 <0 < co and e is unknown. Is п Х1 п an unbiased estimator for 0? Please justify your answer. b) Consider a random sample {X1,X2, ...Xn] of X from N(u, o2), where u and o2 are unknown. Show that X2 + S2 is an unbiased estimator for 2 a2, where п п Xi and S (X4 - X)2. =- п...