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Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 6QF Matrix Solutions to Normal Equations. Direct (Kronecker) Products. (Left) Direct Product. Some Properties. assuming all operations are valid. Solving the Normal Equations.

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Statistical Analysis

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### Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 6QF

Matrix Solutions to Normal Equations

### Direct (Kronecker) Products

(Left) Direct Product

Some Properties

assuming all operations are valid

### Solving the Normal Equations

Single-Factor, Balanced Experiment

yij = m + ai + eij i = 1, ..., a; j = 1, ..., r

Matrix Formulation

y = Xb + e

y = (y11 y12 ... y1r ... ya1 ya2 ... yar)’

Show

### Properties of X’X

• Symmetric

• Same rank as X

• Has an inverse (X’X)-1 iff X has full column rank

### Solving the Normal Equations

Residuals

Least Squares

Solution: Solve the Normal Equations

### Solving the Normal Equations

Normal Equations

Problem: X’X is Singular, has no inverse

Show

### Generalized Inverse: G or

A

Definition

AGA = A

not unique

Moore-Penrose Generalized Inverse

• AGA = A

• GAG = G

• AG is symmetric

• GA is symmetric

unique

Some Properties

if A has full row rank, G = A’(AA’)-1

if A has full column rank, G = (A’A)–1A’

Common Notation

A

Theorem: For any , X’X X’ = X’

(X’X)

(X’X)

### Solving the Normal Equations

Normal Equations

Solutions

• Every solution to the normal equations corresponds to a generalized inverse of X’X

• Every generalized inverse of X’X solves the normal equations

### A Solution to the Normal Equations

One Generalized Inverse

Verification

### A Solution to the Normal Equations

Corresponds to the solution to the normal equations

with the constraint m = 0 imposed

### Assignment

• Find another generalized inverse for X’X in a one-factor balanced experiment

• Verify that it is a generalized inverse

• Solve the normal equations using the generalized inverse

• Determine what constraint on the model parameters correspond to the generalized inverse