Homework Answers
Given that,
X1, X2, ....., Xn are the observed sample items of size n.
Then we know that a random sample itself is an unbiased estimator for distribution mean and the sample mean is an unbiased estimator of the population mean.
So, a) X1, and b)
is an unbiased estimator for distribution mean.
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If X Xny are the observed values of n sample...
3. Suppose we observe 5 values from an unknown distribution: (1,7,5, 16,4). (a) Find the sample...
3. Suppose we observe 5 values from an unknown distribution: (1,7,5, 16,4). (a) Find the sample mean (which is often used as an estimator of the population mean) (b) Find the sample variance (which is often used as an estimator of the population variance). (c) In general, the estimators above are both unbiased and consistent for the population mean and variance, respectively Bias and consistency are both measures of the central tendency of an estimators. One is more relevant in...
14. Multiple Choice Question Let X1,..., X10 be a random sample from a population with mean...
14. Multiple Choice Question Let X1,..., X10 be a random sample from a population with mean y and variance 02. Consider the following estimators for ju: X1 + ... + X 10 3X1 - 2X3 + 3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to w)? A. Both estimators are unbiased. B. Both estimators are biased. C. Only the second is unbiased. D. Only the first is unbiased. E. Insufficient information.
2. Suppose XX2,X is a random sample from an exponential distribution with . Let X(1) minX1,X2,...
2. Suppose XX2,X is a random sample from an exponential distribution with . Let X(1) minX1,X2, Xn), the minimum of the sample mean (a) Show that the estimator 6nx is an unbiased estimator of 8. (hint: you were asked to derive the distribution of X for a random sample from an exponential distribution on assignment 2 -you may use the result) (b) X, the sample mean, is also an unbiased estimator of . Which of the unbiased estimators, or X,...
7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X...
7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X and mx X, be estimators for 0. It is given that the mean and variance of oz are (a) Give an expression for the bias of cach of the two estimators. Are they unbiased? (b) Give an expression for the MSE of cach of the two estimators. (c) Com pute the MSE of each of the two ctrnators for n...
10. Multiple Choice Question It is known that a particular company produces products of which 30%...
10. Multiple Choice Question It is known that a particular company produces products of which 30% are defective. We select items at random and identify it as being defective or not. Calculate the probability that the sixth selected item will be the 3rd defective. A. 0.38282 B. 0.09261 C. 0.24518 D. 0.75494 E. none of the above 11. Multiple Choice Question Let X1,..., X10 be a random sample from a population with mean y and variance o?. Consider the following...
, X,.), the minimum of the sample. mean θ. Let X(1)-min( X1.x2, (a) show that the...
, X,.), the minimum of the sample. mean θ. Let X(1)-min( X1.x2, (a) show that the estimator θ nx(1) is an unbiased estimator of θ. (hint: you were asked to derive the distribution of Xa for a random sample from an exponential distribution on assignment 2 -you may use the result). (b) X, the sample mean, is also an unbiased estimator of o. Which of the unbiased Suppose you observe the following random sample from the population: Calculate point estimates...
Multiple Choice Question Let X1, ..., X10 be a random sample from a population with mean...
Multiple Choice Question Let X1, ..., X10 be a random sample from a population with mean y and variance o?. Consider the following estimators for je: X1 +...+ X10 Ô, 3X1 - 2X3 +3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to j)?|| A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
(a) Are they unbiased estimators for µ? (b) Compute the MSE for all the 4 estimators. (c) Which one is the best estimat...
(a) Are they unbiased estimators for µ?
(b) Compute the MSE for all the 4 estimators.
(c) Which one is the best estimator for µ? Why.
PLEASE answer all parts, thanks
Let X1, X2, ..., X, be and i.i.d. sample from some distribution with mean y and variance o? Let us construct several estimators for . Let îi = X, iz = X1, A3 = (X1 + X2)/2, W = X1 + X2 (a) Are they unbiased estimators for ?...
Which of the following is not one of the least squares assumptions used in Stock and...
Which of the following is not one of the least squares assumptions used in Stock and Watson to show that the OLS estimators are unbiased and consistent and have approximately a normal distribution in large samples? 1) large outliers are unlikely 2) the error term is homoskedastic, i.e., Var(ui ∣ X=x) does not depend on x 3) the sample (Xi,Yi),i=1,…,n constitutes an i.i.d. random sample from the population joint distribution of X and Y 4) the conditional mean of the...
3. Suppose we observe 5 values from an unknown distribution: (1,7,5, 16,4). (a) Find the sample mean (which is often used as an estimator of the population mean) (b) Find the sample variance (which is often used as an estimator of the population variance). (c) In general, the estimators above are both unbiased and consistent for the population mean and variance, respectively Bias and consistency are both measures of the central tendency of an estimators. One is more relevant in...
14. Multiple Choice Question Let X1,..., X10 be a random sample from a population with mean y and variance 02. Consider the following estimators for ju: X1 + ... + X 10 3X1 - 2X3 + 3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to w)? A. Both estimators are unbiased. B. Both estimators are biased. C. Only the second is unbiased. D. Only the first is unbiased. E. Insufficient information.
2. Suppose XX2,X is a random sample from an exponential distribution with . Let X(1) minX1,X2, Xn), the minimum of the sample mean (a) Show that the estimator 6nx is an unbiased estimator of 8. (hint: you were asked to derive the distribution of X for a random sample from an exponential distribution on assignment 2 -you may use the result) (b) X, the sample mean, is also an unbiased estimator of . Which of the unbiased estimators, or X,...
7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X and mx X, be estimators for 0. It is given that the mean and variance of oz are (a) Give an expression for the bias of cach of the two estimators. Are they unbiased? (b) Give an expression for the MSE of cach of the two estimators. (c) Com pute the MSE of each of the two ctrnators for n...
10. Multiple Choice Question It is known that a particular company produces products of which 30% are defective. We select items at random and identify it as being defective or not. Calculate the probability that the sixth selected item will be the 3rd defective. A. 0.38282 B. 0.09261 C. 0.24518 D. 0.75494 E. none of the above 11. Multiple Choice Question Let X1,..., X10 be a random sample from a population with mean y and variance o?. Consider the following...
, X,.), the minimum of the sample. mean θ. Let X(1)-min( X1.x2, (a) show that the estimator θ nx(1) is an unbiased estimator of θ. (hint: you were asked to derive the distribution of Xa for a random sample from an exponential distribution on assignment 2 -you may use the result). (b) X, the sample mean, is also an unbiased estimator of o. Which of the unbiased Suppose you observe the following random sample from the population: Calculate point estimates...
Multiple Choice Question Let X1, ..., X10 be a random sample from a population with mean y and variance o?. Consider the following estimators for je: X1 +...+ X10 Ô, 3X1 - 2X3 +3X10 10 2 Are these estimators unbiased (i.e. is their expectation equal to j)?|| A. Both estimators are unbiased. C. Only the second is unbiased. E. Insufficient information. B. Both estimators are biased. D. Only the first is unbiased.
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
(a) Are they unbiased estimators for µ?
(b) Compute the MSE for all the 4 estimators.
(c) Which one is the best estimator for µ? Why.
PLEASE answer all parts, thanks
Let X1, X2, ..., X, be and i.i.d. sample from some distribution with mean y and variance o? Let us construct several estimators for . Let îi = X, iz = X1, A3 = (X1 + X2)/2, W = X1 + X2 (a) Are they unbiased estimators for ?...
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