unbiased estimator
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?
Homework Answers
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...
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 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...
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)
X1, X2, ..., Xn constitute a random sample from a population with pdf 2 +0.03) |2|<1...
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.
If X, X2,..., Xn constitute a random sample from the population with pdf ffx) 0 elsewhere...
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,....
Let X1, X2, X3, and X4 be a random sample of observations from a population with...
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...
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...
a) Consider a random sample {X1, X2, ... Xn} of X from a uniform distribution over...
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...
To show an estimator can be consistent without being unbiased or even asymptotically unbiased, consider the...
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...
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...
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)
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.
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,....
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...
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...
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