1. 1 Sampling Models for the Population Mean Ed Stanek UMASS Amherst

2. 2 Basic Problem (Population Mean) Population Data ListingLatent Value Rose Lily Daisy What is ?

3. 3 Basic Problem (Population Mean) Some Notation Population ListingLatent Value Rose Lily Daisy Label Set of Subjects in the Population Listing Latent Values Assumption: Response is equal to the latent value for the subject. There is no measurement error. Using vector notation: Using set notation:

4. 4 Sampling Model Select a simple random sample without replacement of size n –Define an estimator that is a linear function of the sample data –Require the estimator to be unbiased –Determine coefficients that minimize the variance (over all possible samples) Best Linear Unbiased Estimator (BLUE)

5. 5 Sampling Model Select a simple random sample without replacement p * =1 p*=2 p*=3 p*=4 p*=5 p*=6 All possible Permutations of subjects OrderPotential Response R L D R D L L R D L D R D R L D L R Probability of Permutation for all Listing

6. 6 Sampling Model Select a simple random sample without replacement p * =1 p*=2 p*=3 p*=4 p*=5 p*=6 All possible Permutations of latent values Potential Response

7. 7 Permutation All possible Permutation OrderPotential Response Data Remainder Sampling Model Select a simple random sample without replacement

8. 8 Represent the Population as a Vector of Random Variables The random variables are indexed by their position- not the label for the subject in a position subject The subject corresponding to a random variable can not be identified Permutation Data Remainder Position i=1 Sampling Model Select a simple random sample without replacement Sample Size: n=1 Permutation Data Remainder Sample Size: n=2

9. 9 Sampling Model Define the Target Linear combination of Population Random Variables: Special case: Mean (Parameter) May be a Parameter May be a Random variable Special case: Latent value for Randomly Selected Subject

10. 10 Sampling Model Expected Value Data Expected Value Under SRS w/o Rep: Linear Link Function

11. 11 Sampling Model Variance Data Variance Term due to finite population correction factor where

12. 12 Sampling Model Expected Value and Variance Reference Sets Reference Set: The set of possible values that sample random variables can have with positive probability Expectation is evaluated over a reference set Data Example: If Reference set for

13. 13 Sampling Model Expected Value and Variance: Reference Sets Data Reference set for

14. 14 Sampling Model Expected Value and Variance Reference Sets Data Reference set for Example when Sets of possible latent values If Reference set for

15. 15 Sampling Model Expected Value and Variance Reference Sets vs Sequence Data Example when Reference Set for L R L D R L R D D L D R DR DL RL Permutation (sequences) p * =1 p*=2 p*=3 p*=4 p*=5 p*=6 Reference Sequence for

16. 16 Sampling Model Expected Value and Variance Reference Sets vs Sequence Data Example when Reference Set : Reference Sequence : Used in Random Permutation Model Sufficient, assuming order doesn’t matter

17. 17 Sampling Model Determining the BLUE for Linear Estimator: Question: What should a be so that the estimator is unbiased and has minimum variance? Target: where data

18. 18 Sampling Model Determining the BLUE for Unbiased Constraint Unbiased requirement: Implies that

19. 19 Sampling Model Determining the BLUE Minimizing the Variance Variance Unbiased Constraint Lagrangian Function to Minimize with Respect to a

20. 20 Sampling Model Determining the BLUE Minimizing the Variance Solving the Estimating Equations where

21. 21 Sampling Model Determining the BLUE Minimizing the Variance Solving the Estimating Equations Let

22. 22 Sampling Model Determining the BLUE of Using and so that where

23. 23 Sampling Model Determining the BLUE of Now where and As a result

24. 24 Sampling Model Determining the BLUE of Now whereand Since