NOTES ON ESTIMATION AND CONFIDENCE INTERVALS. 1. Estimation

Size: px

Start display at page:

Download "NOTES ON ESTIMATION AND CONFIDENCE INTERVALS. 1. Estimation"

Arleen Flowers
5 years ago
Views:

1 NOTES ON ESTIMATION AND CONFIDENCE INTERVALS MICHAEL N. KATEHAKIS 1. Estimatio Estimatio is a brach of statistics that deals with estimatig the values of parameters of a uderlyig distributio based o observed/empirical data. The parameters describe the physical law that geerates the observed values of data. A estimator is a fuctio of the observable sample data that is used to estimate a ukow populatio parameter; a estimate is the result from the actual applicatio of the fuctio to a particular set of data. For example, to estimate the proportio p of a populatio of voters who will vote for a particular cadidate. That proportio is the uobservable parameter; the estimator i p X/ is based o a radom sample of voters. Ofte, may estimators are possible for a give parameter. Some are better tha others. The mai criteria used to choose oe estimator.over others are the ubiasedess, Cosistecy, Efficiecy ad Robustess. These properties are illustrated i the examples below. Bias - Mea Squared Error ad Variace of a Estimator For a poit estimator θ of a parameter θ, The error of θ is θ θ. The bias of θ is defied as B( θ) E( θ) θ. θ is a ubiased estimator of θ if ad oly if B( θ) 0 for all θ, or, equivaletly, if ad oly if E( θ) θ for all θ. The mea squared error of θ is defied as Note that MSE( θ) E[( θ θ) ]. MSE( θ) V ar( θ) + (B( θ)), i.e. mea squared error variace + square of bias. Biased estimator is a statistic whose expectatio differs from the value of the quatity beig estimated. 1

2 MICHAEL N. KATEHAKIS Stadard Error of a estimator θ is the square root of the variace of θ), i.e. the stadard deviatio of θ. Remark. The term bias is used for two differet cocepts. A biased sample is a statistical sample i which some members of the populatio are more likely to be chose i the sample tha others. A biased estimator is oe that for some reaso o average over- or uderestimates the quatity that is beig estimated. I the 1936 US presidetial electio polls, the Literary Digest held a poll that forecast that Alfred M. Lado would defeat Frakli Delao Roosevelt by 57% to 43%. George Gallup, usig a much smaller sample (300,000 rather tha,000,000), predicted Roosevelt would wi, ad he was right. What wet wrog with the Literary Digest poll? They had used lists of telephoe ad automobile owers to select their sample. I those days, these were luxuries, so their sample cosisted maily of middle- ad upperclass citizes. These voted i majority for Lado, but the lower classes voted for Roosevelt. Because Digest s sample was biased towards wealthier citizes, their forecast was icorrect, for the electio, eve though it did correctly predict the proportio of voters i the middle- ad upper-class that did vote for Lado! Example 1. Cosider a radom sample X 1,..., X of i.i.d. observatios from the Normal distributio with mea µ variace σ, where 3. Two estimators form the mea µ are: X X i /, ad X X i /. We have: (1) Both are ubiased : () Oly X is a cosistet estimator, EX µ, EX µ. X µ, for large. (3) The efficiecy of X is V ar(x ) σ / is better (smaller is better) tha that of X which is V ar(x ) σ /. Example. Cosider a radom sample X 1,..., X of i.i.d. observatios from the Normal distributio with mea µ variace σ, where 3. Two estimators for the variace σ 1 S (X i X ), 1 1 (X i X ). The σ is a biased estimator of σ with bias: B( σ )) E σ σ σ /, while the bias of S is B(S ) 0, i.e., S is ubiased. However, σ has a smaller variace, better efficiecy tha S. Ideed,

3 NOTES ON ESTIMATION AND CONFIDENCE INTERVALS 3 we have: E σ 1 σ V ar( σ ) ( 1) σ 4 ES 1 1 σ σ V ar(s ) ( 1) ( 1) σ4 ( 1) σ4. The above properties ca be show (Cochra s theorem i math statistics) usig the fact that (X i X) follows the χ 1 distributio. For this distributios is is kow that Eχ 1 1 ad V ar(χ 1 ) ( 1). Remark A direct proof of the ubiasedess of S ad biasedess of σ is as follows ( ( ) E X i X) E X i X ( ) E (X i µ) (X µ) { E ( (X i µ) ) ( ) E (X i µ)(x µ) + E ((X µ) )} σ 1 {σ σ ( 1)σ ( 1)σ E ((X i µ)(x j µ)) + 1 j1 + σ } j1 k1 σ E ((X j µ)(x k µ)) Usig the above we get the followig: ES 1 1 E (X i X) E σ 1 E (X i X) ( 1)σ 1 ( 1)σ σ. σ.

4 4 MICHAEL N. KATEHAKIS. Cofidece itervals - Iterval estimatio I Iterval estimatio we use sample data to calculate a iterval of probable values of a ukow populatio parameter. The most prevalet forms of iterval estimatio are cofidece itervals. A cofidece iterval (CI) for a populatio parameter is a iterval betwee two umbers with a associated probability p 1 α. They are computed from a radom sample of a uderlyig populatio, such that if the samplig was repeated umerous times ad the cofidece iterval recalculated from each sample accordig to the same method, a proportio p of the cofidece itervals would cotai the populatio parameter i questio..1. Simple examples. Suppose X 1,..., X are a idepedet sample from a ormally distributed populatio with mea ad variace. Let The ad X (X X )/, S 1 1 ( Xi X ). Z X µ σ/, T X µ S/ have respectively a Normal distributio with mea 0 ad variace 1, ad a Studet s t-distributio with 1 degrees of freedom. Hece if σ is kow ad if σ is ot kow P [ z α/ Z z α/ ] 1 α, P [ t 1,α/ T t 1,α/ ] 1 α. We ca use the above to get cofidece itervals for the ukow parameter µ as follows: Hece, 1 α cofidece itervals for µ are: X z α/ σ/ µ X + z α/ σ/, if σ is kow X t 1,α/ S/ µ X + t 1,α/ S/, if σ is ot kow.

5 NOTES ON ESTIMATION AND CONFIDENCE INTERVALS 5.. The process of costructig cofidece itervals cosists of two steps. 1. Idetify a test statistic that has a kow distributio, idepedet of the parameter of iterest, see Z ad T above.. Compute the cofidece regio which is a set of all sample values A of the test statistic T S such that P ( T S A) 1 α. 3. Rephrase the relatio ito a relatio: T S A L papameter of iterest U Where L ad U are respectively the lower ad the upper limits of the cofidece iterval. I the simple examples above we have if σ is kow A [ z α/, z α/ ] ad L X z α/ σ/, U X + z α/ σ/. If σ is ot kow A [ t 1,α/, t 1,α/ ], L X t 1,α/ S/, U X t 1,α/ S/. We ext give a comprehesive list of test statistics. Table 1. Sigle Populatio meas Name Formula Assumptios Oe-sample z-test z X µ 0 σ Two-sample z-test z (X 1 X ) (µ 1 µ ) σ 1 + σ m Normal distributio, or 30, ad σ kow. Normal distributio ad idepedet observatios, σ 1 ad σ kow. Oe-sample t-test t X µ 0 S, Normal populatio, or 30, ad σ ukow. df 1

6 6 MICHAEL N. KATEHAKIS Table. Variaces Name Formula Assumptios Oe-sample χ test χ 1 ( 1)S σ Normal distributio, or 30, ad σ kow uder H 0. F -test S1 S F ( 1, m 1) Normal distributio ad idepedet. Table 3. Tests for Proportios. Name Formula Assumptios Oe-proportio z-test p p z, p(1 p) p 10 ad (1 p) 10. Two-proportio z-test z ( p 1 p ) (p 1 p ) p(1 p)( 1 + 1m ), p 1 6 ad (1 p 1 ) 6. equal variaces p X 1 + X + m mp 6 ad m(1 p ) 6. ( p 1 p ) (p 1 p ) Two-proportio z-test z p1 (1 p 1 ) + p, p 1 6 ad (1 p 1 ) 6. (1 p ) m uequal variaces mp 6 ad m(1 p ) 6.

7 NOTES ON ESTIMATION AND CONFIDENCE INTERVALS 7 Table 4. Other Commo test statistics. Name Formula Assumptios Paired t-test t D d 0, S d 30 ad σ ukow df 1 D i X i Y i Two-sample t (X 1 X ) (µ 1 µ ), Normal populatios, or + m > 40, pooled t-test S p m Sp ( 1)S 1 + (m 1)S, + m ad σ 1 σ, where σ 1 σ, are ukow. df + m Two-sample t (X 1 X ) (µ 1 µ ) S 1 + S m upooled t-test df ( 1)(m 1) (m 1)c + ( 1)(1 c ), Normal populatios, or + m > 40, idepedet observatios ad σ 1 σ are ukow. c S1 S1 + S m or df mi{, m}

8 8 MICHAEL N. KATEHAKIS 3. Importat Distributios Studet s t-distributio Studet s t distributio is the distributio of the radom variable T 1 which is the best that we ca do whe we do ot kow σ. where T 1 X µ S /, S 1 1 (X i X ). The formula for the probability desity fuctio of the T distributio is Γ( + 1 ) f(x) Γ( 1 ( x ) ) π where is the shape parameter ad Γ is the gamma fuctio. The formula for the gamma fuctio is Γ(a) 0 t a 1 e t dt. As icreases, the t-distributio approaches the stadard ormal distributio. The mea, ad variace, of the t-distributio are: ad ET µ 0 σ (T ). The chi-square distributio. A statistic χ that results whe idepedet variables Z i with stadard ormal distributios are squared ad summed, has the χ distributio with degrees of freedom. χ The formula for the probability desity fuctio of the chi-square distributio is Z i,

9 NOTES ON ESTIMATION AND CONFIDENCE INTERVALS 9 f(x) e x/ x / 1, where x 0,, / Γ(/) where is the shape parameter ad Γ is the gamma fuctio. The formula for the gamma fuctio is defied above. I a testig cotext, the chi-square distributio is treated as a stadardized distributio (i.e., o locatio or scale parameters). However, i a distributioal modelig cotext (as with other probability distributios), the chi-square distributio itself ca be trasformed with a locatio parameter, µ, ad a scale parameter, sigma. Properties (1) µ Eχ () Media /3 for large, (3) Mode, (4) Stadard Deviatio σ(χ ), (5) Coefficiet of Variatio: c v µ σ /. Sice the chi-square distributio is typically used to develop hypothesis tests ad cofidece itervals ad rarely for modelig applicatios, we omit ay discussio of parameter estimatio. Commets The chisquare distributio is used i may cases for the critical regios for hypothesis tests ad i determiig cofidece itervals. Two commo examples are the chi-square test for idepedece i a RxC cotigecy table ad the chi-square test to determie if the stadard deviatio of a populatio is equal to a pre-specified value. γ( F (x), x ) Γ( ) where x 0, where Γ the gamma fuctio defied above ad γ is the icomplete gamma fuctio. The formula for the icomplete gamma fuctio is F Distributio γ(a, x) x 0 t a 1 e t dt A statistic F,m that is the ratio of two statistics with chi-square distributios with degrees of freedom ad m, respectively, where each chi-square statistic has first bee divided by its degrees of freedom, i.e., F,m χ / χ m /m,

10 10 MICHAEL N. KATEHAKIS has the F distributio with degrees of freedom ad m, o domai [0, ). The formula for the probability desity fuctio f (, m)(x) of F,m is give by Γ( + m ) / m m/ f,m (x) Γ( x/ 1 )Γ(m ), where x 0, (m + x) (+m)/ ad ad m are the shape parameters. I a testig cotext, the F distributio is also treated as a stadardized distributio (i.e., o locatio or scale parameters). Properties (1) E(F,m ) m m, () V ar(f,m ) (m) ( + m ) (m ) (m 4). Refereces 1. Averill M. Law ad W. David Kelto (1999). Simulatio Modelig ad Aalysis (3rd ed). McGraw-Hill Higher Educatio, New York, NY.. Mauel Lagua ad Joha Marklud (004). Busiess Process Modelig, Simulatio, ad Desig, Pretice Hall, Eglewood Cliffs, 3. NJ. Abramowitz, M. ad Stegu, I. A. (Eds.) (197).. Hadbook of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables, 9th pritig. New York: Dover, pp David, F. N. (1949). The Momets of the z ad F Distributios. Biometrika 36, Press, W. H.; Flaery, B. P.; Teukolsky, S. A.; ad Vetterlig, W. T. (199). Icomplete Beta Fuctio, Studet s Distributio, F-Distributio, Cumulative Biomial Distributio. 6. i Numerical Recipes i FORTRAN: The Art of Scietific Computig, d ed. Cambridge, Eglad: Cambridge Uiversity Press, pp Spiegel, M. R. (199). Theory ad Problems of Probability ad Statistics. New York: McGraw-Hill, pp Departmet of Maagemet Sciece ad Iformatio Systems, Rutgers Busiess School, Newark ad New Bruswick, 180 Uiversity Aveue, Newark, NJ address: mk@rci.rutgers.edu

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1 Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for