MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
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1 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable. Now as we eter the realm of Mathematical Statistics, we will explore how to use a sample of values of a radom variable to estimate the parameters of the probability distributio of the radom variable, specifically we will use sample statistics to estimate populatio parameters Readigs: Pruim, Chapter 4 Learig Objectives: 1. Be able to explai the purpose of a estimator. Be able to explai why precisio ad accuracy are importat to a estimator 3. Be able to determie, based o a the samplig distributio of a estimator a. If the estimator is accurate, or ubiased b. If the estimator is precise, or has a small variace c. The Mea Square Error the estimator 4. Be able to calculate ad iterpret a cofidece iterval for a estimator a. Iterpret the cofidece level of a cofidece iterval b. Be able to calculate ad iterpret a large sample cofidece itervals for the populatio mea ad proportio 5. Be able to determie if a give estimator is a. Efficiet compared to aother estimator b. Cosistet c. Sufficiet i. Use either the Factorizatio Theorem or the defiitio to determie sufficiet statistics of a distributio ii. Be able to determie ubiased fuctios of the sufficiet statistics to estimate the parameters of iterest d. Use the Rao-Blackwell Theorem to determie if a estimator is the Miimum Variace Ubiased Estimator 6. Be able to calculate the method of momets estimator(s) for the parameters of iterest 7. Be able to calculate the maximum likelihood estimator(s) for the parameters of iterest a. Be able to state the properties of the MLE ad adjust the MLE so that it is ubiased 8. Be able to determie the samplig distributio of a give sample statistic icludig its a. Expected Value b. Variace c. Distributio Type 9. Be able to use the Cetral Limit Theorem to determie the probability of observig a specified sample mea from a give populatio 10. Be able to explai the Cetral Limit Theorem ad importace of the Cetral Limit Theorem to makig iferece about a populatio based o a sample Page 1 of 11
2 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters How do we kow if a statistic is a good estimator of a parameter? We will use our kowledge of probability, radom variables ad their distributios, expectatios ad momet-geeratig fuctios, ad the samplig distributios of statistics to evaluate whether a statistic is a good estimate of a parameter. Parameter θ=g(all Y) Populatio Y Statistic =g(y 1,...,Y ) Sample Y 1,...,Y Defiitio: A estimator is a rule that tells how to calculate the value of a estimate based o the measuremets cotaied i a sample. Ofte deoted as ˆ Defiitio: A Simple Radom Sample is a process that selects a sample of size from a populatio of size N i such a way that each of the samples of size is equally likely to be selected from the populatio ( idepedet ad idetically distributed radom variables) Example: Cosider a sample from a ormal distributio with mea ad variace. Which kow statistic do you thik is a good estimator of? Which kow statistic do you thik is a good estimator of? Page of 11
3 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Methods to Fid Estimators: The Method of Momets Recall the defiitio of the k th momet a probability distributio is () k k Y where Y is a radom variable that follows the give probability distributio I practice, we are of course dealig with a sample from the distributio ad may ot kow the origial distributio from which to determie the momets. Defiitio: If Y,, 1 Y are idepedet ad idetically distributed (iid) radom variable from the a distributio, the k th sample momet is defied as 1 k ˆ k Yi i1 We ca view ˆk as a estimate of k ad because the populatio momets k are fuctios of the populatio parameters, we ca use the sample momets ˆk to estimate the populatio parameters. Method of Momets: Choose as estimates those values of the parameters that are solutios of the equatios ˆ 1 1, ˆ,, ˆk k, where k is the umber of parameters to be estimated. Example: Cosider a Poisso distributio with parameter. Recall that ()Y. Use the method of momets to determie a estimator of. Page 3 of 11
4 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Example: A radom sample of observatios Y,, 1 Y are idepedetly ad idetically distributed with a uiform desity fuctio over the iterval (0,) where is ukow. Use the method of momets to estimate the parameter. Example: A radom sample of observatios Y,, 1 Y are idepedetly ad idetically distributed with a gamma distributio with parameters ad. Fid the method of momets for the ukow parameters ad. Samplig Distributios of Estimators: The probability distributio of a statistic or estimator is called the samplig distributio. Page 4 of 11
5 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Bias ad Mea Square Error There are other methods of fidig estimators for parameters, but how do we assess the utility of the estimator? Cosider a sharpshooter shootig at a target. How do you determie if the sharpshooter is a good shot? Describe the abilities of the three sharpshooters who created the targets below usig the terms accuracy ad precisio. Which sharpshooter is the best? A Measure of Accuracy i a Estimator: Bias We desire to fid a estimator that is accurate i its estimatio of our parameter of iterest. To defie accuracy for a estimator, we must defie the bias of a estimator as it relates to the parameter we wish to estimate. Defiitio: The bias of a poit estimator ˆ is give by B()() ˆ ˆ Example: Cosider the graph to the right. We took 1000 samples of size 100 from a distributio with 0. We calculated two differet estimators: ˆ 1 Yi 100 i ˆ Yi i1 We averaged the differet estimators to simulate the expected value of the estimators- which estimator is biased? Which is ubiased? Estimatio of Mea Trial Page 5 of 11
6 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Defiitio: Let ˆ be a poit estimator for a parameter. The ˆ is a ubiased estimator if () ˆ. Otherwise ˆ is said to be biased. Example: Let fuctio. 1 f () y Prove that the sample mea is a ubiased estimator of. Y,..., 1 Y be a radom sample from a expoetial distributio with the desity y/ e, y 0 Example: Determie which of the followig is a ubiased estimator of variace, where,, Y1 Y is a radom sample with () Y i ad Var() Y i. 1 1 S () Yi Y or S () Yi Y 1 i 1 i1 Page 6 of 11
7 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters A Measure of Precisio i a Estimator: Variace of a Estimator While beig ubiased is a good characteristic of a poit estimator, it is just oe quality. We also desire that our estimator is precise i its estimatio of our parameter. We measure precisio based o the variability we would observe i our estimators, or the variace of the estimator. Example: Sketch a samplig distributio for two ubiased estimators, oe with a large variace ad oe with a small variace. Which oe would you choose to be your poit estimator? Example: Cosider the followig two estimators for the mea of a distributio. Which is the more precise estimator? ˆ 1 1 Yi or ˆ Y i i1 i1 Balacig Precisio ad Accuracy: Mea Square Error The goal for ay estimator is that we miimize both the bias ad the variace of the estimator. To do this, we use a measure of the estimator called the mea square error. Defiitio: The mea square error of a poit estimator ˆ MSE()()() ˆ ˆ Var ˆ B ˆ is the expected value of ˆ Page 7 of 11
8 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Example: If X is a biomial distributio with parameters ad p, ad let ˆp X estimator of p. Fid the MSE() p ˆ. be a Example: Suppose we have a sample X,, 1 X from a distributio with E() X i X ad aother sample Y 1,, Y from a distributio with E() Y i X. Show that ˆ X Y is a ubiased estimator of X Y. Fid the MSE() ˆ. Page 8 of 11
9 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Limit Theorems We have determied the samplig distributios of the sample statistics whe we have a sample ormally distributed radom variables. Evetually, we will show that we ca use a ormal distributio to make iferece about a sample mea o matter what the distributio of the radom variable, if the sample size is large eough. First, though, we have a little more theory. Theorem (Law of Large Numbers): Let Y,, 1 Y be a sequece of idepedet ad idetically distributed radom variables, with () Y i ad Var() Y i. Let The, for ay 0, Y 1 i 1 Y. 0 P Y as i Before we reach our fial destiatio, the Cetral Limit Theorem we eed a bit more iformatio. Here are two defiitios ad a theorem to help us. Defiitio: Let Z be a sequece of radom variables, if P Z 0 as for ay 0, where is some scalar, the Z is said to coverge i probability to. Rewrite the above defiitio i more plai laguage. Defiitio: Let Y1, Y, be a sequece of radom variables with cumulative distributio fuctios F1, F,... ad let Y be a radom variable with distributio fuctio, F. We say that Y coverges i distributio to Y if lim()() F y F y at every poit at which F is cotiuous. Rewrite the above defiitio i more plai laguage. Page 9 of 11
10 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Theorem (Cotiuity Theorem): Let F be a sequece of cumulative distributio fuctios with the correspodig momet-geeratig fuctio m. Let F be a cumulative distributio fuctio with the momet-geeratig fuctio m. If m ()() t m t for all t i a ope iterval d cotaiig zero, the F ()() x F x at all cotiuity poits of F as. Cetral Limit Theorem Here we are, at the peak of our build up to the oe of the most importat foudatioal cocepts of statistical aalysis: Theorem (Cetral Limit Theorem): Let Y,, 1 Y be a sequece of idepedet radom variables havig a mea () Y i ad Var() Y i. Defie d The Z N (0,1) as. Z Y Proof: Page 10 of 11
11 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Example: The distributio of the umber of childre per woma i the Uited States is skewed right with a mea of childre per woma ad a stadard deviatio of. Suppose a athropologist travels to Salt Lake City, Utah ad radomly samples 100 wome ad foud a mea of 5 childre per woma. Does the birth rate i Salt Lake City differ from the atioal average more tha would be expected by radom chace? Example: Workers employed i a large service idustry have a average wage of $7.00 per hour with a stadard deviatio of $0.50. I idustry of 64 workers have a average wage of $6.90 per hour. Is it reasoable to assume that the wage rate of the ethic group is equivalet to that of a radom sample of workers from those employed i the service idustry? Page 11 of 11
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