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This is a very simple sample problem from ISI MStat PSB 2018 Problem 9. It is mainly based on estimation of ordinary least square estimates and Likelihood estimates of regression parameters. Try it!

Suppose satisfies the regression model,

for

where are fixed constants and are i.i.d. errors, where and are unknown parameters.

(a) Let denote the least squares estimate of obtained assuming . Find the mean squared error (MSE) of in terms of model parameters.

(b) Obtain the maximum likelihood estimator of this MSE.

Normal Distribution

Ordinary Least Square Estimates

Maximum Likelihood Estimates

These problem is simple enough,

for the given model, for .

The scenario is even simpler here since, it is given that , so our model reduces to,

, where and 's are i.i.d.

now we know that the Ordinary Least Square (OLS) estimate of is

(How ??) where is the (generally) the OLS estimate of , but here is known, so,

again,

, hence is a biased estimator for with .

So, the Mean Squared Error, MSE of is,

+

[ as, it follows clearly from the model, and 's are non-stochastic ] .

(b) the last part follows directly from the, the note I provided at the end of part (a),

that is, and we have to find the Maximum Likelihood Estimator of and and then use the inavriant property of MLE. ( in the MSE obtained in (a)). In leave it as an Exercise !! Finish it Yourself !

Suppose you don't know the value of even, What will be the MSE of in that case ?

Also, find the OLS estimate of and you already have done it for , so now find the MLEs of all and . Are the OLS estimates are identical to the MLEs you obtained ? Which assumption induces this coincidence ?? What do you think !!

This is a very simple sample problem from ISI MStat PSB 2018 Problem 9. It is mainly based on estimation of ordinary least square estimates and Likelihood estimates of regression parameters. Try it!

Suppose satisfies the regression model,

for

where are fixed constants and are i.i.d. errors, where and are unknown parameters.

(a) Let denote the least squares estimate of obtained assuming . Find the mean squared error (MSE) of in terms of model parameters.

(b) Obtain the maximum likelihood estimator of this MSE.

Normal Distribution

Ordinary Least Square Estimates

Maximum Likelihood Estimates

These problem is simple enough,

for the given model, for .

The scenario is even simpler here since, it is given that , so our model reduces to,

, where and 's are i.i.d.

now we know that the Ordinary Least Square (OLS) estimate of is

(How ??) where is the (generally) the OLS estimate of , but here is known, so,

again,

, hence is a biased estimator for with .

So, the Mean Squared Error, MSE of is,

+

[ as, it follows clearly from the model, and 's are non-stochastic ] .

(b) the last part follows directly from the, the note I provided at the end of part (a),

that is, and we have to find the Maximum Likelihood Estimator of and and then use the inavriant property of MLE. ( in the MSE obtained in (a)). In leave it as an Exercise !! Finish it Yourself !

Suppose you don't know the value of even, What will be the MSE of in that case ?

Also, find the OLS estimate of and you already have done it for , so now find the MLEs of all and . Are the OLS estimates are identical to the MLEs you obtained ? Which assumption induces this coincidence ?? What do you think !!

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