STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. Comments:
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1 Recall: STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Commets:. So far we have estimates of the parameters! 0 ad!, but have o idea how good these estimates are. Assumptio: E(Y x)! 0 +! x (liear coditioal mea fuctio) Data: (x, y ), (x, y ),, (x, y ) Least squares estimator: ˆ E (Y x) ˆ " 0 + ˆ " x, where ad ˆ " SXY ˆ " 0 y - ˆ " x " ( x i - x ) " x i ( x i - x ). If our data were the etire populatio, we could also use the same least squares procedure to fit a approximate lie to the coditioal meas. 3. If we have a simple radom sample from the populatio ad also assume e x (equivaletly, Y x) is ormal with costat variace, the the least squares estimates are the same as the maximum likelihood estimates of! 0 ad!. SXY " ( x i - x ) (y i - y ) " ( x i - x ) y i
2 3 4 Properties of ˆ " 0 ad ˆ " : ) ˆ " SXY " (x i y i c i y i where c i (x i (x i y i Thus: If the x i 's are fixed (as i the blood lactic acid example, or i ay example if we coditio o x, x,, x ), the ˆ " is a liear combiatio of the y i 's. Note: Here we wat to thik of each y i as a observatio from a radom variable Y i with distributio Y x i. Thus we may say that as a radom variable (i.e., lookig at the samplig distributio of ˆ " ) ˆ " " c i (Y x i ) " c i Y i I other words, the radom variable ˆ " is a liear combiatio of the radom variables Y x i. Sayig that the observatios y i are idepedet is the same as sayig that the radom variables Y i are idepedet. I this case, we ca coclude: If each Y x i is ormal, the ˆ " is also ormal. If the Y x i 's are ot ormal but is large, the ˆ " is approximately ormal. This will allow us to do iferece o ˆ ". (Details later.)
3 5 6 ) " c i " (x i (x i 0 (as see earlier) 3) " x i c i " x i (x i. x i (x i 4) ˆ " 0 y - ˆ " x " y i - " c i y i x ( " c ix )y i, also a liear combiatio of the y i 's. Hece: Remark: Recall the somewhat aalogous properties for the residuals e ˆ i. 5) The sum of the coefficiets i (4) is ( " c i x ) ( ) " x c i ( ) " x 0.
4 7 8 Samplig distributios of ˆ " 0 ad ˆ " : Cosider x,, x as fixed (i.e., coditio o x,, x ). Model Assumptios: ("The" Simple Liear Regressio Model Versio ). E(Y x)! 0 +! x (liear coditioal mea fuctio). (NEW) Var(Y x) (costat variace) (Equivaletly, Var(e x) ) 3. (NEW) y,, y are idepedet observatios. (idepedece) The ew assumptio meas we ca cosider y,, y as comig from idepedet radom variables Y,, Y, where Y i has the distributio of Y x i. Commet: We do ot assume that the x i 's are distict. If, for example, x x, the we are assumig that y ad y are idepedet observatios from the same coditioal distributio Y x. Sice Y,, Y are radom variables, so is ˆ " -- but it depeds o the choice of x,, x, so we ca talk about the coditioal distributio ˆ " x,, x. Expected value of ˆ " (as the y i 's vary): E( ˆ " x,, x ) E( " c i Y i x,, x ) "c i E(Y i x,, x ) "c i E(Y i x i ) (sice Y i depeds oly o x i ) "c i (! 0 +! x i ) (model assumptio)! 0 "c i +! "c i x I! 0 0 +!! Thus: ˆ " is a ubiased estimator of!.
5 9 0 Variace of ˆ " (as the y i 's vary): Var( ˆ " x,, x ) Var( " c i Y i x,, x ) "c i Var(Y i x,, x ) "c i Var(Y i x i ) (Y i depeds oly o x i ) "c i "c i " (x i % " $ $ & ( ' (x ( ) i x ) " $ SD( ˆ " ) " (defiitio of c i ) For short: Var( ˆ " ) " Commets: Aalogy to samplig stadard deviatio for a mea y : SD(estimator) populatio stadard deviatio somethig Here, "somethig" -- more complicated tha "somethig" (for y ). Recall: For y, as becomes larger, SD( y ) gets smaller. Aalogous reasoig for SD( ˆ " ): (Recall: " ( x i - x ) ) If the x i 's are far from x (i.e., spread out), is, so SD( ˆ " ) is. If the x i 's are close to x (i.e., close together), is, so SD( ˆ " ) is. Thus if you are desigig a experimet, choosig the x i 's to be from their mea will result i a more precise estimate of ˆ ". (Assumig all the model coditios fit!)
6 Expected value ad variace of ˆ " 0 : Use the formula ˆ " 0 ( " c x )y i i to show (calculatios left to the iterested studet): E( ˆ " 0 )! 0 (So ˆ " 0 is a ubiased estimator of! 0.) Var ( ˆ " 0 ) " & % (, + x $ ' Aalyzig the variace formula: so SD( ˆ " 0 ) " + x A larger x gives a variace for ˆ " 0. % Does this agree with ituitio? Covariace of ˆ " 0 ad ˆ " : Similar calculatios (left to the iterested studet) will show Thus: Cov( ˆ " 0, ˆ " ) " x ˆ " 0 ad ˆ " are ot idepedet (except possibly whe ) % Does this agree with ituitio? The sig of Cov( ˆ " 0, ˆ " ) is opposite that of x. % Does this agree with ituitio? A larger sample size teds to give a variace for ˆ " 0. The variace of ˆ " 0 is (except whe x < ) tha the variace of ˆ ". % Does this agree with ituitio? The spread of the x i 's affects the variace of ˆ " 0 i the same way it affects the variace of ˆ ".
7 3 4 Estimatig " : To use the variace formulas above for iferece, we eed to estimate ( Var(Y x i ), the same for all i). Plausible reasoig: If we had lots of observatios y i, y i,...,y im from Y x i, the we could use the uivariate stadard deviatio m (y m " i j " y i ) j of these m observatios to estimate. ( y i sample mea of y i, y i,...,y im, the best estimate of E(Y x i ) just usig y i, y i,...,y im ). If we do't have lots of y's from oe x i, we might take E ˆ (Y x i ) as our best estimate of E(Y x i ) ad try [y i " E ˆ (Y x i )] " e ˆ " i " RSS. However (just as i the uivariate case, we eed a deomiator - to get a ubiased estimator for ), a legthy calculatio (omitted) shows: E(RSS x,, x ) (-) (The expected value is over all samples of the y i 's with the fixed x i 's.) Thus we use the estimate ˆ " " RSS to get a ubiased estimator for : E( " ˆ x,, x ). [If you like to thik heuristically i terms of losig oe degree of freedom for each calculatio from data ivolved i the estimator, this fits: Both ˆ " 0 ad ˆ " eed to be calculated from the data to get RSS.]
8 5 Stadard Errors for ˆ " 0 ad ˆ " : Usig ˆ " RSS " as a estimate of i the formulas for SD( ˆ " 0 ) ad SD( ˆ " ) gives the stadard errors ad s.e. ( ˆ " ) ˆ " s.e.( ˆ " 0 ) " ˆ + x as estimates of SD( ˆ " ) ad SD( ˆ " 0 ), respectively.
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