Suppose we observe independent pairs (Xi,Yi) where each (Xi,Yi) has a uniform distribution in the circle...
Suppose we observe independent pairs (Xi,Yi) where
each (Xi,Yi) has a uniform distribution in the circle of unknown
radius θ and centered at (0,0) in the plane.
Show that for the method of moments estimator and the maximum
likelihood estimator, it is the case that the distribution of ˆθ/θ
does not depend on θ. Explain why this means we can write
MSEθ(ˆ θ) = θ2*MSE1(ˆ θ) where here the subscript θ means “under
the assumption that the true value is θ,” so MSE1 denotes the mean
square error under the assumption that θ = 1. From this, explain
why it suffices that we compare the two estimators when θ = 1
Homework Answers
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Suppose we observe independent pairs (Xi,Yi) where
each (Xi,Yi) has a uniform distribution in the circle...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
2 Suppose that we observe the continuous random variable X (X1,.., Xn) with state space S,...
2 Suppose that we observe the continuous random variable X (X1,.., Xn) with state space S, whose distribution we do not know but we are assuming that its p.d.f. belongs to a known family of distributions {fe;Be Θ). We construct an estimator for the unknown parameter θ(X) (a) Explain why it is wrong to write E(ex) and correct it. 12 marks (b) Explain the difference between pdf and likelihood function. [1 mark] (c) Explain the different between estimate and estimator....
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...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distr...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
2 Suppose that we observe the continuous random variable X (X1,.., Xn) with state space S, whose distribution we do not know but we are assuming that its p.d.f. belongs to a known family of distributions {fe;Be Θ). We construct an estimator for the unknown parameter θ(X) (a) Explain why it is wrong to write E(ex) and correct it. 12 marks (b) Explain the difference between pdf and likelihood function. [1 mark] (c) Explain the different between estimate and estimator....
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...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
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