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Statistical Assumptions for SLRPowerPoint Presentation

Statistical Assumptions for SLR

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Statistical Assumptions for SLR

- The assumptions for the simple linear regression model are:
1) The simple linear regression model of the form Yi = β0 + β1Xi +εi

where i = 1, …, n is appropriate.

2) E(εi)=0

2) Var(εi) = σ2

3) εi’s are uncorrelated.

STA302/1001 week 2

Properties of Least Squares Estimates

- Estimate of β0and β1 – functions of data that can be calculated numerically for a given data set.
- Estimator of β0and β1 – functions of the underlying random variables.
- Recall: the least-square estimators are…
- Claim: The least squares estimators are unbiased estimators for β0and β1.
- Proof:…

STA302/1001 week 2

Estimation of Error Term Variance σ2

- The variance σ2 of the error terms εi’s needs to be estimated to
obtain indication of the variability of the probability distribution of Y.

- Further, a variety of inferences concerning the regression function and
the prediction of Y require an estimate of σ2.

- Recall, for random variable Z the estimates of the mean and variance of Z based on n realization of Z are….
- Similarly, the estimate of σ2 is
- S2 is called the MSE – Mean Square Error it is an unbiased estimator of σ2 (proof later on).

STA302/1001 week 2

Normal Error Regression Model

- In order to make inference we need one more assumption about εi’s.
- We assume that εi’s have a Normal distribution, that is εi ~ N(0, σ2).
- The Normality assumption implies that the errors εi’s are
independent (since they are uncorrelated).

- Under the Normality assumption of the errors, the least squares
estimates of β0and β1 are equivalent to their maximum likelihood

estimators.

- This results in additional nice properties of MLE’s: they are
consistent, sufficient and MVUE.

STA302/1001 week 2

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