Statistics 350 Lecture 14. Today Last Day: Matrix results and Chapter 5 Today: More matrix results and Chapter 5 Please read Chapter 5.

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Presentation transcript:

Statistics 350 Lecture 14

Today Last Day: Matrix results and Chapter 5 Today: More matrix results and Chapter 5 Please read Chapter 5

Multivariate Normal Distribution Suppose have a random vector Y, with elements Y i E(Y)=  2 (Y)=  =

Multivariate Normal Distribution Suppose have a random vector Y, with elements Y i Y is said to have a multivariate normal distribtuion if it has density The density for each Y i is

Multivariate Normal Distribution and Simple Linear Regression For the simple linear regression model, Y has a multivariate normal distribution:

Multivariate Normal Distribution and Simple Linear Regression Distribution of b:

Multivariate Normal Distribution and Simple Linear Regression Suppose we could choose the X i ’s where we observe the Y’s What does covariance matrix tell us about the choice of X’s

Fitted Values and Residuals Fitted values: Hat matrix: Properties of the hat-matrix

Fitted Values and Residuals Residuals: Covariance of residuals Estimated covariance:

Fitted Values and Residuals Geometric interpretation of residuals:

Fitted Values and Residuals Geometric interpretation of residuals:

Analysis of Variance Results SSTO SSE

Analysis of Variance Results SSR

Analysis of Variance Results Quadratic Forms: