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# ISI MStat PSB 2018 Problem 9 | Regression Analysis 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!

## Problem - ISI MStat PSB 2018 Problem 9

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.

### Prerequisites

Normal Distribution

Ordinary Least Square Estimates

Maximum Likelihood Estimates

## Solution :

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 !

## Food For Thought

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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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!

## Problem - ISI MStat PSB 2018 Problem 9

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.

### Prerequisites

Normal Distribution

Ordinary Least Square Estimates

Maximum Likelihood Estimates

## Solution :

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 !

## Food For Thought

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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