Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible size of error) associated with the estimate, e.g., Var( ) for a ubiased estimator, or M.S. E. E[( ) ], or the amout of iformatio this estimate is based o. Remember that the amout of iformatio icreases with sample size. Oe way to assess the ucertaity of a estimator is to defie its estimate via a iterval 1, where, ( 1, ) are the lower ad upper limits of a suitably defied iterval based o the poit estimator. o The radom iterval ( 1, ), obtaied via the samplig distributio of the poit estimator, covers the ukow parameter with a specified coverage probability (1 ); i.e., P (1 ). 1 What do the uderlied phrases mea? How does oe iterpret the above probability statemet? The iterval ( 1, ) varies from sample to sample. For a give radom sample, the statemet ( 1, ), i.e., the observed iterval ( ( x,, x ), ( x,, x )), 1 1 1 cotais (covers) the true value of, may be True/False. [Biary outcome] Sice the value of is ukow, we have o way of kowig whether or ot this statemet is true for a give sample. But amog all future realizatios of this iterval, approximately 100(1 )% will cotai the true value of. Thus, if (1 ) is fairly large, this statemet will be true for most of the samples. I this sese, oe has a high degree of cofidece that the true value of is i the observed iterval. Of course, there is o guaratee that a give iterval covers the true value. o Defiitio: For a specified value of (1 ), the iterval ( ( x,, x ), ( x,, x )) 1 1 1 is called a 100 (1 ) % cofidece iterval estimate for.
o Defiitio: The value (1 ) is called the Degree of Cofidece. o Defiitio: The edpoits ( 1, ) are called the lower ad upper cofidece limits. Sice there are may poit estimators for a give parameter, ayoe ca be used to obtai a iterval estimator with a give degree of cofidece. o Which oe should be used? o Choice based o desirable statistical properties. o Commet: For a real valued parameter, (, ) cotais the true value with degree of cofidece 1.0. But, is this iformatio meaigful? Why ot Would you wat to use a iterval based o oly oe observatio? Why ot? Desirable statistical properties of a iterval estimator: o It should use all the iformatio i the data. How do we achieve this goal? Use sufficiet statistics! o For a give degree of cofidece (1 ) Shortest legth iterval may be useful. Iterval legth ( 1, ) may be a radom! Shortest I what sese? Shortest Expected legth. o Give two ubiased estimators, which oe should be used to obtai a iterval estimator? Ay suggestios! Key Idea: Use the samplig distributio of the poit estimator (or its large sample approximatio) to defie the iterval estimator with the specified cofidece coefficiet:
Example: Iterval estimator of the Mea of the Normal populatio, with kow variace, X N(, ) What poit estimator would you choose to develop the iterval estimator? o For this problem, X is MVUE, MLE, ad MOM-Good Choice o From Theorem 8.4, the exact samplig distributio of X : X N (, / ). X o Therefore, Z ~ N(0,1) [Stadardize X Z] / o For a fixed, let z / be the value such that PZ [ z /] 1. Therefore, the area uder the stadard ormal desity i the iterval ( z /, ) is /. How does oe obtai this value for a give? Table III, Page 574, provides area uder the stadard ormal pdf from 0 to z, for z=0(.01)3.9,4,5,6. How does the value of z / chage as icreases? o Note that the followig iequalities are all equivalet: X Z z z X z / / / / o Therefore, PZ [ z/ ] P X z/ 1. This statemet provides a probabilistic assessmet of the size of maximum error, ( X ) with a specified probability, leadig to the followig result.
o Theorem 11.1. Give a radom sample of size from a Normal populatio X N(, ), with kow variace, if oe uses the mea X as a estimator of the populatio mea, the with probability (1 ), the absolute error will be less tha { z / }. Examples: o (i) o (ii) o (iii) o (iv) 4, 16 64, 16 1600, 16 64, 1 What patters do you see? o Effect of icreasig o Effect of decreasig o Effect of icreasig variace
To obtai a iterval estimator, covert the above probability statemet to a iterval estimate of : P X z = P[ z X z ] / / / For large, the iterval ( s s X z/, X z/ ) is a approximate 100(1- )% CI for the populatio mea. ( X z/, X z/ ) with degree of cofidece (1- ), or ( X z/, X z/ ) is a 100(1- )% Cofidece Iterval (CI). Examples: o (i) 4, 16 o (ii) 64, 16 o (iii) 1600, 16 o (iv) 64, 1 Extesios: o What to do if the uderlyig populatio is o-ormal? Need more data! The sample size large ( > 30) [Hit: Use Cetral Limit Theorem]. Fact: For large, X ~ N(0,1). / Nothig ew, except ow the samplig distributio of X is approximately ormal. What if variace is ukow? Replace by the sample variace o Fact: For large, X ~ N(0,1). s/ s. [Eve for o-ormal populatio]
Situatio ot covered yet: small ( 30), ukow variace. Assume Normal distributio i the uderlyig populatio. Note that X ( X )/ /. s/ s / This is a ratio of two radom variables: o Z i the umerator, ad s / i the deomiator. Chapter 8: Sectio 8.5 The t-distributio Theorem 8.1. If Y ad Z are idepedet radom variables, with Z Y ad Z N(0,1), the the distributio of T is give by Y / 1 ( 1)/ u f( u) 1, u. It is called the t distributio with d.f. Proof (Page 78-79). Uses chage of variables techiques from Sectio 7.3. Start with the joit distributio of (y,z) to joit distributio of (y,t), fidig the Jacobia of the trasformatio ad the itegrate out y. Note: Those iterested i reviewig chage of variables topic may wat to complete the proof. More iterestig historical fact: A special case of this theorem, discussed i Theorem 8:13 below, was origially derived by W.S. Gossett, a employee of Guiess Brewig Co., which did t allow employees to publish research. He published uder the Pseudo-ame Studet. o It is called Studet s t distributio (S was his middle iitial)
Theorem 8.13. If X ad S are the sample mea ad variace of a RS of size from a ormal populatio, N(, ), the the samplig distributio of T X s / is the t-distributio with ( 1) d.f. Proof: Kow that Z N(0,1), (from Theorem 8.4). ( 1) S o Theorem 8.11 proves that the distributio of Y, as well as the fact that Y ad Z are idepedetly distributed. o Now use Theorem 8.1. Fact: The t-desity is symmetric aroud, goes to zero as u, but much more slowly tha the Normal desity. o For 1, it is Cauchy Desity, ad as v, it approaches ormal desity. o For 30, just use Table III for ormal desity. Used a lot i practice, so Tables of area uder the curve, for 1, 9 were made available. o [Table IV, page 575] provides t,. The iterval ( s s X t/, X t/ ) is a exact 100(1- )% CI for. o The differece from large case: I small sample case, istead of z-tables, we use t-tables for a give ( 1) d.f. Examples: Exercise 11.31 Expoetial: 1 f x e x x/ ( ), 0. 1 Uiform: f( x ),0 x Geeral method of costructig CI is called Pivot-method. o Fid a fuctio of the data ad the parameter, whose distributio does ot deped o the populatio parameter. For Example, o Z or T for meas from ormal populatios or for large. What about Expoetial distributio, Uiform distributio.