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# Restricted Regression Problem | ISI MStat 2017 PSB Problem 7 This problem is a regression problem, where we use the ordinary least square methods, to estimate the parameters in a restricted case scenario. This is ISI MStat 2017 PSB Problem 7.

## Problem

Consider independent observations from the regression model where and are scalar covariates, and are unknown scalar
coefficients, and are uncorrelated errors with mean 0 and variance . Instead of using the correct model, we obtain an estimate of by minimizing Find the bias and mean squared error of .

### Prerequisites

• Ordinary Least Square Method
• Minimizing the Square Loss Error Function
• Multiple Regression
• Mean Square Error = .
• Bias

## Solution

It is sort of a restricted regression problem because maybe we have tested the fact that . Hence, we are interested in the estimate of given . This is essentially the statistical significance of this problem, and we will see how it turns out in the estimate of . Let's minimize by differentiating w.r.t and equating to 0.   From, the given conditions, . .

Since, are constant, . .

Thus, observe that the more is close to 0, the more bias is close to 0.

From, the given conditions, ~ Something ). ~ Something .  Observe, that even the MSE is minimized if .

This problem is a regression problem, where we use the ordinary least square methods, to estimate the parameters in a restricted case scenario. This is ISI MStat 2017 PSB Problem 7.

## Problem

Consider independent observations from the regression model where and are scalar covariates, and are unknown scalar
coefficients, and are uncorrelated errors with mean 0 and variance . Instead of using the correct model, we obtain an estimate of by minimizing Find the bias and mean squared error of .

### Prerequisites

• Ordinary Least Square Method
• Minimizing the Square Loss Error Function
• Multiple Regression
• Mean Square Error = .
• Bias

## Solution

It is sort of a restricted regression problem because maybe we have tested the fact that . Hence, we are interested in the estimate of given . This is essentially the statistical significance of this problem, and we will see how it turns out in the estimate of . Let's minimize by differentiating w.r.t and equating to 0.   From, the given conditions, . .

Since, are constant, . .

Thus, observe that the more is close to 0, the more bias is close to 0.

From, the given conditions, ~ Something ). ~ Something .  Observe, that even the MSE is minimized if .

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### 6 comments on “Restricted Regression Problem | ISI MStat 2017 PSB Problem 7”

1. SOMJIT ROY says:

I guess the estimate of Beta 1 here is slightly incorrect .

2. akshay says:

beta 1 estimate is wrong dude !!!

3. akshay says:

beta 1 estimate is wrong dude.

4. Giorno Giovana says:

Mistake in taking derivative..Whole solution gone wrong..Yare yare daze

5. Giorno Giovana says:

Mistake in taking derivative. Whole solution gone wrong..

6. Rajeev Kumar says:

Yes srijit da!!
Estimate value of β1 is not true

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