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The Cramer Rao bound can also be used to bound the variance of biased estimators of given bias.
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Suppose that ôn, and 2 are estimators of the parameter 6. We know that Elên,) =...
7:30 Suppose that e, az and , are estimators of We know that E(e)-E(6) ed, E(a-e...
7:30 Suppose that e, az and , are estimators of We know that E(e)-E(6) ed, E(a-e v(61) -14, V(62)-12 and E(63 -0) 8. (a) Which do you prefer between the first two estimators? b) Compare these three estimators. Which do you prefer? Why?
4. (a) Let Xi,X ,x, be n observations from an N(u2) distribution, and define the estimators (i) Determine whether T and T2 are unbiased estimators of u. 4 points (ii) Compute the variances Var(Ti)...
4. (a) Let Xi,X ,x, be n observations from an N(u2) distribution, and define the estimators (i) Determine whether T and T2 are unbiased estimators of u. 4 points (ii) Compute the variances Var(Ti), and Var(T2). Which is the better estimator T or T2 -and why? [2 points] Determine the maximum likelihood estimator of μ. (iii) [5 points) (b) A manufacturer is testing the performance of two products, A and B. At each of 20 field sites, product A and...
Part IV: SHORT ANSWER Question 4: 6 marks) Suppose we have two estimators of the population...
Part IV: SHORT ANSWER Question 4: 6 marks) Suppose we have two estimators of the population parameter V:. Ely) = y +8/n? v (ų)= 03/13 and E(ū)= y +9/n? viņ)= 202/12 Determine the bias, if any, of each estimator. (ii) Determine the MSE. Which estimator is preferred? (iii) Determine if the estimators are consistent. Explain.
Estimator properties: 6 Estimators properties 6.1 Exercise 1 In order to estimate the average number of...
Estimator properties:
6 Estimators properties 6.1 Exercise 1 In order to estimate the average number of hours that children spend watching tv, a Bernoulli sample of size n = 5 children was selected from a primary school. Let X be the variable that represents the hours spent watching tv, let E(X)-μ the parameter to estimate and var(X-σ2 the variance. Compare the following two proposed estimators Τι 1. Compare the two estimators for u on the basis of their bias 2....
You observe sarriples X1....,x. Ber(@) where 0 € (0,1) is an unknown parameter. Suppose that is...
You observe sarriples X1....,x. Ber(@) where 0 € (0,1) is an unknown parameter. Suppose that is much larger than 1 so we have access to many samples from the specified distribution. Consider three candidate estimators for 0. . X.-1x . 0.5 In the next three questions, you will consider potential strengths and weaknesses of these estimators. In this particular section (just for now), the phrase "efficiently computable" refers to the existence of an explicit formula. More precisely, here we say...
PROBLEM 3 Let X1, X2, ..., Xn be a random sample from the following distribution -...
PROBLEM 3 Let X1, X2, ..., Xn be a random sample from the following distribution - 5) +1 if 0 <r <1 fx(2) = 10 0. 0.w.. where @ € (-2, 2) is an unknown parameter. We define the estimate ēn as: ô, = 12X – 6 to estimate . (a) Is ên an unbiased estimator of e? (b) Is Ôn a consistent estimator of e?
Two samples Xi and X2 are taken from an exponential random variable X with unknown parameter...
Two samples Xi and X2 are taken from an exponential random variable X with unknown parameter θ; that is. r (a r >0. We propose two estimators for θ in the forms 4 In terms of unbiasedness and minimum variance, which one is the better of the two?
2. Suppose we have the simple regression model Y =a+8X:+E, and their OLS coefficient estimators a...
2. Suppose we have the simple regression model Y =a+8X:+E, and their OLS coefficient estimators a and b. Answer the following questions. (a) Suppose we multiply X, by 1/2 for all i and do the OLS estimation again using X as the regressor (the independent variable). What will be your new estimators, denoted by ă (intercept) and b (slope)? Compare them with the original OLS estimators a and b, respectively (b) Compare Var[b] and Var[b]. Are they the same or...
. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show...
. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show that theestimator θ t914(1-t)a, is also an unbiased estimator of θ for any value of the constant t. Suppose V[6]:ơİ and v[62] of. Ifa, anda,are independent, find an expression for V[d] in terms of t, σ' and σ Find the value of t that produces an estimator of the form 6 ะเอิ,+(1 that has the smallest possible variance. (Your final answer will be in...
Question 4 0.8 pts Suppose Ô is an estimator for the population parameter 4. Which of...
Question 4 0.8 pts Suppose Ô is an estimator for the population parameter 4. Which of the following is true for us to Say that Ô is an unbiased estimate of e? Elên) = 0 None of the options must be true Elê) = Elê) < Elo), Question 5 0.8 pts Which of the following is a threat to validity of the instrumental variables method?
7:30 Suppose that e, az and , are estimators of We know that E(e)-E(6) ed, E(a-e v(61) -14, V(62)-12 and E(63 -0) 8. (a) Which do you prefer between the first two estimators? b) Compare these three estimators. Which do you prefer? Why?
4. (a) Let Xi,X ,x, be n observations from an N(u2) distribution, and define the estimators (i) Determine whether T and T2 are unbiased estimators of u. 4 points (ii) Compute the variances Var(Ti), and Var(T2). Which is the better estimator T or T2 -and why? [2 points] Determine the maximum likelihood estimator of μ. (iii) [5 points) (b) A manufacturer is testing the performance of two products, A and B. At each of 20 field sites, product A and...
Part IV: SHORT ANSWER Question 4: 6 marks) Suppose we have two estimators of the population parameter V:. Ely) = y +8/n? v (ų)= 03/13 and E(ū)= y +9/n? viņ)= 202/12 Determine the bias, if any, of each estimator. (ii) Determine the MSE. Which estimator is preferred? (iii) Determine if the estimators are consistent. Explain.
Estimator properties:
6 Estimators properties 6.1 Exercise 1 In order to estimate the average number of hours that children spend watching tv, a Bernoulli sample of size n = 5 children was selected from a primary school. Let X be the variable that represents the hours spent watching tv, let E(X)-μ the parameter to estimate and var(X-σ2 the variance. Compare the following two proposed estimators Τι 1. Compare the two estimators for u on the basis of their bias 2....
You observe sarriples X1....,x. Ber(@) where 0 € (0,1) is an unknown parameter. Suppose that is much larger than 1 so we have access to many samples from the specified distribution. Consider three candidate estimators for 0. . X.-1x . 0.5 In the next three questions, you will consider potential strengths and weaknesses of these estimators. In this particular section (just for now), the phrase "efficiently computable" refers to the existence of an explicit formula. More precisely, here we say...
PROBLEM 3 Let X1, X2, ..., Xn be a random sample from the following distribution - 5) +1 if 0 <r <1 fx(2) = 10 0. 0.w.. where @ € (-2, 2) is an unknown parameter. We define the estimate ēn as: ô, = 12X – 6 to estimate . (a) Is ên an unbiased estimator of e? (b) Is Ôn a consistent estimator of e?
Two samples Xi and X2 are taken from an exponential random variable X with unknown parameter θ; that is. r (a r >0. We propose two estimators for θ in the forms 4 In terms of unbiasedness and minimum variance, which one is the better of the two?
2. Suppose we have the simple regression model Y =a+8X:+E, and their OLS coefficient estimators a and b. Answer the following questions. (a) Suppose we multiply X, by 1/2 for all i and do the OLS estimation again using X as the regressor (the independent variable). What will be your new estimators, denoted by ă (intercept) and b (slope)? Compare them with the original OLS estimators a and b, respectively (b) Compare Var[b] and Var[b]. Are they the same or...
. Suppose that 6, and o2 are both unbiased estimators of e. a) b) e) Show that theestimator θ t914(1-t)a, is also an unbiased estimator of θ for any value of the constant t. Suppose V[6]:ơİ and v[62] of. Ifa, anda,are independent, find an expression for V[d] in terms of t, σ' and σ Find the value of t that produces an estimator of the form 6 ะเอิ,+(1 that has the smallest possible variance. (Your final answer will be in...
Question 4 0.8 pts Suppose Ô is an estimator for the population parameter 4. Which of the following is true for us to Say that Ô is an unbiased estimate of e? Elên) = 0 None of the options must be true Elê) = Elê) < Elo), Question 5 0.8 pts Which of the following is a threat to validity of the instrumental variables method?
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