22 Nonparametric Methods.

Transcription

1 22 oparametrc Methods. I parametrc models oe assumes apror that the dstrbutos have a specfc form wth oe or more ukow parameters ad oe tres to fd the best or atleast reasoably effcet procedures that aswer specfc questos regardg the parameters. If the assumptos are volated our procedures mght become faulty. Ofte the procedures are stll vald eve f they are ot the most effcet, ad these are the stable or robust stuatos. Sometmes we could be way off. Let us dscuss ths by meas of two examples. We have observatos x 1,,x from some populato ad we wat to test that the mea s 0. If we assume that the observatos come from the ormal populato wth mea µ ad ukow varace σ 2, we would aturally use the t test. The statstc would be P x t = x s 1 P x2 where x s the sample mea ad s 2 s the sample varace x2.the stastc t has a t dstrbuto wth 1 degrees of freedom. Far large t s early ormal wth mea 0 ad varace 1. If realty the observatos came from a expoetal dstrbuto wth desty ae ax dx for x 0, whle for small ts ologer dstrbuted as a t asymptotcally t s stll dstrbuted lke a stadard ormal wth mea 0 ad varace 1. Usg the sample mea to do the t-test s robust for questos regardg the populato mea. Let us look at the problem testg that the varace s 1. If we use the statstc based o the sample varace s 2 ad do the χ 2 test, whch s atural for the ormal model, asymptotcally U = s2 ( 1) 2 wll be stadard ormal. But f the model were expoetal ad the observatos are draw from e x dx, although E[s 2 ]= 1, ts varace s dfferet ad t s oly V = s2 ( 1) 5 that s asymptotcally ormal. We are way off. The oparametrc odels avods these ssues ad makes o assumpto or atleast oly very geeral assumptos cocerg the model. For stace f 49

2 we wat to test that x 1,...,x are draw from a populato wth meda 0, we do t smply by coutg the umber of the umber of observatos that are above 0. Ths radom varable X, s a Bomal wth probablty 1 ad 2 far large X 2 4 s asymptotcally ormal. Oe farly geeral assumpto that s ofte made s that the probablty dstrbuto from whch the samples are draw are cotuous.e. the dstrbuto fucto F (x) =P [X x] s a cotuous fucto of x. The t s easy to check that the radom varable Y = F (X) whch les betwee 0 ad 1 has the uform dstrbuto o [0, 1]. To see ths let us suppose for smplcty that F s strctly creasg. The P [F (X) y] =P [X F 1 (y)] = F (F 1 (y)) = y provg that the dstrbuto of Y = F (X) s uform. If we have observatos ad we wat to test f F s the true uderlyg dstrbuto we may wat to compare the emprcal dstrbuto F (x) = [# : x x] wth F (x) ad use the Kolmogorov=-Smrov statstc D = sup F (x) F (x) x It turs out that f we employthe trasformato F (x )=y ad calculate D = sup [# : y y] y 0 y 1 The dstrbuto of D, uder the assumpto that the observatos come from F sthesameasthatofd uder the assumpto that y come from the uform dstrbuto o [0, 1]. The asymptotcs of ths statstc has bee worked out. The dstrbuto of D(t) = [ [# : y t] t ] 50

3 s asymptotcally ormal wth varace t(1 t). Oe ca see ths easly from thefactthat[# : y t] s a bomal B(, t). The jot dstrbuto of {D(s),D (t)} s bvarate ormal wth covarace m(t, s) ts. Fromthese cosderatos oe ca deduce that asymptotcally the dstrbuto of D s that of sup Z(t) 0 t 1 where Z(t) s a ormal radom fucto wth mea zero ad covarace m(s, t) st. If oe wats to test f two sets of samples x 1,x 2,...,x ad y 1,y 2,...,y m come from the same populato F agast the alteratve whle the x scome from F,they s come from a shfted dstrbuto F (x a) for some a>0. A test called rak test s used for ths. Let us group the + m observatos ad arrage them creasg order. The ras of the y saresomeumbers 1 k 1 k 2 k m + m. Uder the ull hypothess we expect them to uformly spaced [0,m+ ], whle uder the alteratve they should buch up to the rght ed. We wat to use the statstc U,m = k ad compute ts mea ad varace. It s kow that V,m = U,m E[U,m ] Var U,m s asymptotcally ormal. Let us compute the mea ad varace. The followg trck s ofte useful smlar cotexts. Let us defe Z =1fthe-th smallest observato s a y ad 0 otherwse. The k = jz j Let us compute E[Z j ]ade[z Z j ]. E[Z ]= m + m s the probablty that the j-th observato s a y. ote that uder the ull hypothess they are all from the same populato so that all possble 51

4 arragemet have the same probablty. Smlarly E[Z Z j ]= E[U,m ]= ( + m) m(m 1) ( + m)( + m 1) [ j j ] = + m +1 2 Var U,m =Var[Z j ][ j j 2 ]+Cov [Z j Z k ][ j k jk] m The varace of Z j s computed easly to be betwee Z ad Z j equals wth = m + m(m 1) ( + m)( + m 1) m 2 ( + m) 2 = m (+m) 2 = m whle the covarace [ ] (m 1) ( 1)m The varace ca ow be computed as Var U m, = m [ ( + 1)(2 +1) m ( ) 2 ] = m ( + 1)(2 +1) m ( +1) (3 +2) 6 12 m( +1) = 12 Fally suppose we have a fte populato from whch we draw a sample wthout replacemet. The populato s a 1,...,a ad we draw a sample x 1,x 2,...,x of sze. We wat to compute the mea ad varace of the sample mea x. It s better to work wth S = a j Z j where Z j s 1 f a j s cluded the sample. E[Z j ]= Var [Z ( ) j]= 2 ( 1) Cov [Z Z j ]= ( 1) 2 ( ) = 2 From ths t s easy to deduce that E[ x] =ā = aj 52

5 ad Var x = 1 2 [ ( ) 2 = = Var a a 2 1 (a ā) 2 ( ) ] a a j,j 53