lecture5_s10 observed

# Lecture5_s10 - SEM with observed variables estimation Psychology 588 Covariance structure and factor models Feb 5 2010 Estimation 2 Tries to find a

This preview shows pages 1–8. Sign up to view the full content.

SEM with observed variables: estimation Psychology 588: Covariance structure and factor models Feb 5, 2010

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Estimation 2 • Tries to find a solution that best approximates ideally population covariance matrix Σ , but in reality a sample estimate S () ˆ Ff = Σθ S • The “best” approximation is defined in various ways, leading to different fitting functions • Our job is to find which is the best under what data conditions
Desirable fitting functions 3 • have the following properties: () ˆ , ˆ ,0 ˆˆ f f f f == S Σθ S S S S S is a scalar if and only if is continuous both in and • Minimizing such fitting functions provides a consistent estimator of θ (e.g., ML, ULS and GLS) --- true for all functions to be considered

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Desirable asymptotic properties of estimators 4 • Unbiased • Consistent • Efficient Note: asymptotic means N →∞ by definition but its practical meaning is “as N becomes sufficiently large” --- “how large is sufficient” will depend on many things such as complexity of the model, size of measurement errors, etc.
: ˆ N N θ θθ parameters in the population estimate of from a sample of size • If is unbiased • If is asymptotically unbiased • If is consistent --- or called is efficient if its asymptotic variance is the minimum of all consistent estimator of θ () ˆˆ , NN E = θ EN =→ θ as 1 0, PN αα −< = > θ as for any ˆ N θ ˆ plim "" =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Maximum likelihood 6 • ML assumes: ¾ Satisfactorily large sample ¾ All observed variables distributed multivariate normal --- we will consider later a relaxed alternative to this for exogenous x ¾ All observations independent and identically distributed • Minimizing its fitting function F ML maximizes joint (log) likelihood of the model parameters θ given observed data S ( ) () 1 ML ˆˆ log tr log F pq =+ + Σ S Σ S Obviously, both S and must be nonsingular for F ML to be defined ˆ Σ
How to optimize a fitting function? 7 • Find the partial derivatives w.r.t. all free model parameters a and solve --- necessary for minimization • The second-derivative matrix is positive definite (nonsingular) at the that minimizes --- sufficient for minimization • Usually, there is no closed form solution to this problem;

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 06/11/2010 for the course PSYC 588 taught by Professor Sunjinghong during the Spring '10 term at University of Illinois at Urbana–Champaign.

### Page1 / 20

Lecture5_s10 - SEM with observed variables estimation Psychology 588 Covariance structure and factor models Feb 5 2010 Estimation 2 Tries to find a

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online