Hypothesis Testing. H 0 : θ 1 1. H a : θ 1 1 (but > 0... required in distribution) Simple Hypothesis - only checks 1 value

Transcription

1 Hyothesis estig ME's are oit estimates of arameters/coefficiets really have a distributio Basic Cocet - develo regio i which we accet the hyothesis ad oe where we reject it H - reresets all ossible values of oit estimate wo regios - H - ull hyothesis regio where we accet the hyothesis H a - alterative hyothesis regio where we reject Rules - o Overla - H H a Covers Everythig - H H a H Eamle - from midterm questio wat to test the hyothesis that H : H a : but >... required i distributio Simle Hyothesis - oly checks value H : H a : but >... required i distributio... this is comosite alterative Comosite Hyothesis - H : H a : < Accetig ull Hyothesis - if H reject if H a Errors - statistically seakig we ca ever be % sure ye Error - chace that we observe H a eve though H is true α Pr[reject H H true] ye Error - chace that we observe H eve though H a is true β Pr[fail to reject H H false] Eamle - suose that the life of a light bulb is draw from a eoetial distributio f e-// >, > F - e-/ > > there are two tyes of light bulbs ordiary ad log life H :, hours H a :, hours Accet H if < d, where is the lifetime i hours of the light bulb that you test reject H accet H a if > d, where d is some set umber of hours e.g., d 6,. Ca't be % sure because eoetial is ifiite distributio so there's always a chace that a ordiary bulb lasts more tha 6, hours ye I error ad there's a chace that a log life bulb lasts less tha 6, hours ye II error α - F6, - - e-6/ % chace that we say it's a log life bulb whe it's really a ordiary bulb β F6, - e-6/.5... so 5% of the time we'll wrogly label log life bulbs as ordiary bulbs Critical Regio - regio where we reject H Eamle - from revious eamle, > 6, is critical regio evel of Sigificace - robability of makig a tye error should choose what you're willig to accet before cruchig the data rade-off Problem - if we lower the chace of tye error, we icrease the chace of tye II error i.e. lower ower of test of 8

2 og Ru - level of sigificace tells you that if you coduct the test may times, that ercetage of them will have a tye I error e.g., 5% sigificace, for every tests, you'll eect to get false rejectio Power of a est - - Pr[ye error] Pr[reject H H false] wat to be as high as ossible Eamle - from light bulb eamle, Power etremely low rade-off Problem - would like to maimize ower ad miimize level of sigificace, but imrovig oe is doe at the eese of the other from light bulb eamle, to icrease sigificace, we eed to icrease d to icrease ower, we eed to decrease d other solutio is to use more light bulbs Best est - withi a class of tests, o is said to be "best" if it has maimum ower amogst all tests with level of sigificace size less tha or equal to some give value Cosistet est - ower goes to oe as samle size grows to ifiity Secific ests Assumtios for R,, & - stadard regularity coditios iformatio matri is osigular kow distributio or desity that geerates the data i.e., f Geeral - wat to test whether i the kow distributio we kow what the likelihood ad log-likelihood fuctios are just wat to test whether or ot the arameter value is ikelihood Ratio est - based o ook at differece betwee l l H : simle hyothesis H a : comosite alterative Reject H if this differece is "big" accet H if the differece is "small" l l l Uder geeral coditios: R l l χ df equals umber of arameters Eamle - cosider observatios form a Beroulli rv t with robability ad with robability - we wish to test H : e.g., / H a : l l t l t [ t l t l ] t l l ME to get l t t t t t t of 8

3 l l R l l l l [ ] l l χ Suose you toss a coi times ad observe 8 heads. hat is the test statistic for H : / i.e., have a fair coi?,.8,.5.8. R.8l. l Critical value χ for.95 oe tailed is.8 reject H agrage Multilier est - looks at sloe of log likelihood at H : simle hyothesis H a : comosite alterative If sloe is "small" accet H if it's "big" reject H based o fact that at, sloe of l is i.e., score Set u as otimizatio: Ma l subject to agragia: l ', where is vector of shadow rices First Order Coditios: l l From first coditio: S score Distributio of the score fuctio has mea zero ad variace I d S, I, lim... wave magic wad est Statistic: S I S χ k umber of arameters k Eamle - Cosider the revious eamle of Beroulli trials Remember that l l l ad ME is l S l I l E E E E of 8

4 For Beroulli trials E I S I S Suose ad.8 ad.5 assumig fair coi Critical value χ for.95 oe tailed is.8 do't reject H ald est - looks at distace betwee ad if distace is "small" accet H if it's "big" reject H based o asymtotic ormality of the ME d I,, lim... wave magic wad est Statistic: I χ k k umber of arameters Eamle - same as before l l l ad ME is S ad I I Suose ad.8 ad.5 assumig fair coi Critical value χ for.95 oe tailed is.8 reject H ote - ikelihood ratio test does't accout for variace or iformatio matri agrage multilier ad ald tests do but R is geerally easiest to do of 8

5 estig Restrictios o Subset of Parameters Uder H : k, o k restricted uder H k urestricted uder H et the ME of for ad uder H be so o k, whereas k use urestricted ME ikelihood Ratio est - R l l χ k agrage Multilier est - S I S χ k S [ S ] I Choose the uer left-had corer of the iverse of the iformatio matri I I I, uer left-had corer of iverse is [ I ] II I I I S [ I I I I ] S χ, with I ij evaluated at k ald est - o I I I I o χ [ ] k Eamle - from H5 roblem, test H :.6 H :.6 f,, where [ ] f i i ad [ ] i i i i i, where j ote: E j i j j i j ote: for simlicity substitute ji i, j,, l l l l l 5 of 8

6 6 of 8 Score, S l l l Stadard MEs: solve, S for lots of comlicated algebra will show that j j, j,, Restricted MEs: lug i.6 for ad resolve for o-restricted arameters ad l.6 l.6 Solve first oe for... Substitute ito the secod oe ad solve for Go back ad solve Comute all the values we'll eed for the test statistics / / / 556 / / /

7 /.8 / /. / S I [ I I I ] I I [ ] [ ] [ ] [ I I I ] I [ ] 6.88 [ ] [ 6.] Critical Value χ.8 ikelihood Ratio est - R l l 5l.6 8l.79 l.676 l.5 5l l.9 l.87 l <.8 do't reject H. he data seems to agree with.6 agrage Multilier est - S [ ] I II I S [ 77.5][.][ 77.5].59 <.8 do't reject H. he data seems to agree with.6 7 of 8

8 ald est - o o [ I I I I ] o [.5665] - [.6] [.5] [.5][ 6.][.5].5 <.8 do't reject H. he data seems to agree with.6 8 of 8