STAT 371 F18 Chap 4 part 1.pdf - Chapter 4 Multiple Linear...

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Chapter 4Multiple Linear Regression
ā€¢We now consider the general linear model:? = š›½0+ š›½1?1+ š›½2?2+ ā‹Æ + š›½š‘?š‘+ šœ€ā€¢We will discuss how to estimate the model parameters,š›½0, š›½1, š›½2, ā€¦ , š›½š‘, and how to test various hypotheses about them.ā€¢To start, suppose we have information onncases, or subjectsš‘– = 1, 2, ā€¦ , š‘›ā€¢Let??be the observed response value for subjectš‘–and let??1, ??2, ā€¦ , ??š‘be the values on the explanatory or predictorvariables.
ā€¢Recall, the values of theppredictor variables are treated as fixedconstants; however, the responses are subject to variability.ā€¢Hence the model forsubjectš’Šiswritten as??= š›½0+ š›½1??1+ š›½2??2+ ā‹Æ + š›½š‘??š‘+ šœ€?ā€¢Also assume, as before thatšœ€?is a random variable having a mean of0 and constant variance,šœŽ2. We suppose that they are normallydistributed and that errors for different cases (šœ€?, šœ€?) are assumedindependent.ā€¢Recall these points also imply that the responses?1, ?2, ā€¦ , ?š‘›areindependent normal random variables with meanšø??= šœ‡?andvarianceš‘‰š‘Žš‘Ÿ??= šœŽ2.
This model can be expressed in vector form, we write??= ??ā€²šœ· + šœ€?where??ā€²=1??1ā‹Æ??š‘andšœ· =š›½0š›½1ā‹®š›½š‘
Combining the vectors we obtain the following model:?1?2ā‹®?š‘›=1?11ā‹Æ?1š‘1ā‹®ā‹Æā‹Æā‹±ā‹±ā‹®ā‹®1ā‹Æā‹Æ?š‘›š‘š›½0š›½1ā‹®š›½š‘+šœ€1šœ€2ā‹®šœ€š‘›š’š = š‘‹šœ· + šā€¢š’šis known as the response vectorā€¢X is a non-random matrix of our explanatory (predictor) variables known asthe design matrixā€¢Ī²is a vector of unknown parametersā€¢šis a random vector of errors
ā€¢

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Term
Spring
Professor
AHMED
Tags
Linear Regression, Normal Distribution, Regression Analysis, Variance, Maximum likelihood, Multivariate normal distribution, variance covariance matrix

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