Research Article New Bandwidth Selection for Kernel Quantile Estimators

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1 Hidawi Publishig Cororatio Joural of Probability ad Statistics Volume, Article ID 3845, 8 ages doi:.55//3845 Research Article New Badwidth Selectio for Kerel Quatile Estimators Ali Al-Keai ad Kemig Yu Deartmet of Mathematical Scieces, Bruel Uiversity, Uxbridge UBB 3PH, UK Corresodece should be addressed to Ali Al-Keai, magaja@bruel.ac.uk Received 8 August ; Revised 6 Setember ; Acceted October Academic Editor: Jubi B. Gao Coyright q A. Al-Keai ad K. Yu. This is a oe access article distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited. We roose a cross-validatio method suitable for smoothig of kerel quatile estimators. I articular, our roosed method selects the badwidth arameter, which is kow to lay a crucial role i kerel smoothig, based o ubiased estimatio of a mea itegrated squared error curve of which the miimisig value determies a otimal badwidth. This method is show to lead to asymtotically otimal badwidth choice ad we also rovide some geeral theory o the erformace of otimal, data-based methods of badwidth choice. The umerical erformaces of the roosed methods are comared i simulatios, ad the ew badwidth selectio is demostrated to work very well.. Itroductio The estimatio of oulatio quatiles is of great iterest whe oe is ot reared to assume a arametric form for the uderlyig distributio. I additio, due to their robust ature, quatiles ofte arise as atural quatities to estimate whe the uderlyig distributio is skewed. Similarly, quatiles ofte arise i statistical iferece as the limits of cofidece iterval of a ukow quatity. Let X,X,...,X be ideedet ad idetically distributed radom samle draw from a absolutely cotiuous distributio fuctio F with desity f. Further, let X X X deote the corresodig order statistics. For << aquatile fuctio Q is defied as follows: Q ) if { x : F x }..

2 Joural of Probability ad Statistics If Q deotes th samle quatile, the Q x where deotes the itegral art of. Because of the variability of idividual order statistics, the samle quatiles suffer from lack of efficiecy. I order to reduce this variability, differet aroaches of estimatig samle quatiles through weighted order statistics have bee roosed. A oular class of these estimators is called kerel quatile estimators. Parze roosed a versio of the kerel quatile estimator as below: Q K ) [ i/ ) K h t dt ]X i.. i / From. oe ca readily observe that Q K uts most weight o the order statistics X i, for which i/ is close to. I ractice, the followig aroximatio to Q K is ofte used: ) [ Q AK ) ] K h i/ X i..3 Yag 3 roved that Q K ad Q AK are asymtotically equivalet i terms of mea square errors. Similarly, Falk 4 demostrates that, from a relative deficiecy ersective, the asymtotic erformace of Q AK is better tha that of the emirical samle quatile. I this aer, we roose a cross-validatio method suitable for smoothig of kerel quatile estimators. I articular, our roosed method selects the badwidth arameter, which is kow to lay a crucial role i kerel smoothig, based o ubiased estimatio of a mea itegrated squared error curve of which the miimisig value determies a otimal badwidth. This method is show to lead to asymtotically otimal badwidth choice ad we also rovide some geeral theory o the erformace of otimal, data-based methods of badwidth choice. The umerical erformaces of the roosed methods are comared i simulatios, ad the ew badwidth selectio is demostrated to work very well.. Data-Based Selectio of the Badwidth Badwidth lays a critical role i the imlemetatio of ractical estimatio. Secifically, the choice of the smoothig arameter determies the tradeoff betwee the amout of smoothess obtaied ad closeess of the estimatio to the true distributio 5 Several data-based methods ca be made to fid the asymtotically otimal badwidth h i kerel quatile estimators for Q AK give by.3. Oe of these methods use derivatives of the quatile desity for Q AK. Buildig o Falk 4, Sheather ad Marro gave the MSE of Q AK as follows. If f is ot symmetric or f is symmetric but /.5, AMSE Q AK ) ) 4 μ k [ Q )] h 4 )[ Q )] R K [ Q )] h,.

3 Joural of Probability ad Statistics 3 where R K uk u K u du, μ k u K u du ad K is the atiderivative of K. If Q > the h ot α K β Q /3,. where α K R K /μ k /3, β Q Q /Q /3. There is o sigle otimal badwidth miimizig the AMSE Q AK whe F is symmetric ad.5. Also, If q, we eed higher terms ad the AMSE Q AK ca be show to be AMSE Q AK ) ) 4 ) h 4[ Q )] μ k h [ Q )] q ) ht tk t j t dt,.3 where j t t xk x dx, see Cheg ad Su 6. I order to obtai h ot we eed to estimate Q q ad Q q. It follows from.3 that the estimator of Q q ca be costructed as follows: ) q AK Q ) [ i AK X i K a ) )] i K a..4 Joes 7 derived that the AMSE q AK as AMSE q )) a 4 AK 4 μ k [ q )] [ )] q a K y ) dy..5 By miimizig.5, we obtai the asymtotically otimal badwidth for Q AK : [[ Q a )] K y ) ] /5 dy ot [ Q )]..6 μ k To estimate Q q i., we emloy the kow result Q ) d AK d Q ) AK a [ ) i / X i K a )] i/ K,.7 a

4 4 Joural of Probability ad Statistics ad it readily follows that a ot [ [ 3 Q )] K ] /7 x dx.8 [ Q )] μ k which reresets the asymtotically otimal badwidth for Q AK. By substitutig a a ot i.4 ad a a ot i.7 we ca comute h ot. 3. Cross-Valdatio Badwidth Selectio Whe measurig the closeess of a estimated ad true fuctio the mea itegrated squared MISE defied as MISE h E { Q ) Q )} d 3. is commoly used as a global measure of erformace. The value which miimises MISE h is the otimal smoothig arameter, ad it is ukow i ractice. The followig ASE h is the discrete form of error criterio aroximatig MISE h : ASE h { ) )} i i Q Q. 3. The ukow Q is relaced by Q ad a fuctio of cross-validatory rocedure is created as: { ) )} i i Q i Q, 3.3 where Q i i/ deotes the kerel estimator evaluated at observatio x i, but costructed from the data with observatio x i omitted. The geeral aroach of crossvalidatio is to comare each observatio with a value redicted by the model based o the remaider of the data. A method for desity estimatio was roosed by Rudemo 8 ad Bowma 9. This method ca be viewed as reresetig each observatio by a Dirac delta fuctio δ x x i, whose exectatio is f x, ad cotrastig this with a desity estimate based o the remaider of the data. I the cotext of distributio fuctios, a atural characterisatio of each observatio is by the idicator fuctio I x x i whose exectatio is F x. This imlies that the kerel method for desity

5 Joural of Probability ad Statistics 5 estimatio ca be exressed as f x K h x x i, 3.4 whe h K h x x i δ x x i. The kerel method for distributio fuctio F x ) x xi W, 3.5 h where W is a distributio fuctio, h is the badwidth cotrols the degree of smoothig. Whe h ) x xi W I x x i, h 3.6 where I x x i is the idicator fuctio {, if x xi, I x x i, otherwise. 3.7 Now, from.3 whe h Q AK ) δ i )X i, 3.8 ad thus a cross-validatio fuctio ca be writte as CV h { ) i i δ X i Q i d. 3.9 )} The smoothig arameter h is the chose to miimise this fuctio. By subtractig a term that characterise the erformace of the true we have H h CV h { ) )} i i δ X i Q d 3.

6 6 Joural of Probability ad Statistics which does ot ivolve h. By exadig the braces ad takig exectatio, we obtai H h { Q i ) ) ) ) ) )} i i i i i i δ X i Q i δ X i Q Q d. 3. Whe the th order statistic x is asymtotically ormally distributed x AN Q ) ) ), [ f Q ))], [ E{H h } E E{H h } E{H h } E { Q i i ) i δ )} d ) ], i Q [ { )} i i E Q i δ )] d, i Q { ) )} i i Q Q d, X i ) i Q Q i i ) ) ) i i δ X i Q ) { )} ) ) i i i E Q i δ Q 3. where the otatio Q i/ with ositive subscrit deotes a kerel estimator based o a samle size of. The roceedig argumets demostrate that CV h rovides a asymtotic ubiased estimator of the true MISE h curve for a samle size. The idetity at 3. strogly suggests that crossvalidatio should erform well. 4. Theoretical Proerties From 3., we ca write MISE h bias Q K d var Q K d. Sheather ad Marro have show that ) bias ) Q K h μ k Q ) h ). 4. while Falk 4, age 63 roved that ) var ) Q K )[ Q )] R K [ Q )] ) h h. 4.

7 Joural of Probability ad Statistics 7 O combiig the exressios for bias ad variace we ca exress the mea itegrated square error as MISE h 4 h4 μ k [ R K Q )] ) d h h 4 h, 4.3 [ Q )] ) [ d Q )] d ad for C Q d, C R K Q d ad C 3 μ k Q d the MISE ca be exressed as MISE h C C h ) 4 C 3h 4 h 4 h. 4.4 Therefore, the asymtotically otimal badwidth is h C /3, where C {C /C 3 } /3. We ca see from 3. that H h may be a good aroximatio to MISE h or at least to that fuctio evaluated for a samle of size rather tha. Additioally,thisistrueif we adjusted H h by addig the quatity J { Q ) Q )) E Q ) Q )) }. 4.5 This quatity is demea ad does ot deed o h which makes it attractive for obtaiig a articularly good aroximatio to MISE h. Theorem 4.. Suose that Q is bouded o, ad right cotiuous at the oit, ad that K is a comactly suorted desity ad symmetric about. The, for each δ, ε, C >, H h J MISE h { 3/ h 3/ / h 3) δ} 4.6 with robability, uiformly i h C δ,as. A outlie roof of the above theorem is i the aedix. From the above theorem, we ca coclude that miimisatio of H h roduces a badwidth that is asymtotically equivalet to the badwidth h that miimises MISE h. Corollary 4.. Suose that the coditios of revious theorem hold. If ĥ deotes the badwidth that miimises CV h i the rage h C δ, for ay C>ad ay ε /3, the ĥ h 4.7 with robability as.

8 8 Joural of Probability ad Statistics Q with true quatile = Boxlot of mea squared error = a b Q with true quatile = Boxlot of mea squared error = c d with true quatile = 5.3 Boxlot of mea squared error = 5 Q e f Figure : Left ael: lots of the quatile estimators for method solid lie, method dotted lie, ad true quatile dashed lie for differet samle sizes ad for data from a ormal distributio. Right ael: box lots of mea squared errors for the quatile estimators for method ad method for differet samle sizes. 5. A Simulatio Study A umerical study was coducted to comare the erformaces of the two badwidth selectio methods. Namely, the method reseted by Sheather ad Marro ad our roosed method.

9 Joural of Probability ad Statistics 9 Q with true quatile = Boxlot of mea squared error = a b Q with true quatile = Boxlot of mea squared error = c d Q with true quatile = Boxlot of mea squared error = 5 e f Figure : Left ael: lots of the quatile estimators for method solid lie, method dotted lie ad true quatile dashed lie for differet samle sizes ad for data from a exoetial distributio. Right ael: box lots of mea squared errors for the quatile estimators for method ad method for differet samle sizes. I order to accout for differet shaes for our simulatio study we cosider a stadard ormal, Ex, Log-ormal, ad double exoetial distributios ad we calculate 8 quatiles ragig from.5 to.95. Through the umerical study the Gaussia kerel was used as the kerel fuctio. Samle sizes of, ad 5 were used, with simulatios i each case. The erformace of the methods was assessed through the

10 Joural of Probability ad Statistics 6 5 with true quatile = Boxlot of mea squared error = 4.5 Q a b Q with true quatile = Boxlot of mea squared error = c d Q with true quatile = Boxlot of mea squared error = e f Figure 3: Left ael: lots of the quatile estimators for method solid lie, method dotted lie ad true quatile dashed lie for differet samle sizes ad for data from a Log-ormal distributio. Right ael: box lots of mea squared errors for the quatile estimators for method ad method for differet samle sizes. mea squared errors criterio MSE.MSE h E{ Q Q }. Ad the relative efficiecy R.E [ )] MISEMethod hmethod,ot R.E ). 5. MISE Method hmethod,ot

11 Joural of Probability ad Statistics 3 Q 3 with true quatile = Boxlot of mea squared error = a b 3 with true quatile =.4.3 Boxlot of mea squared error = Q c d Q 3 with true quatile = Boxlot of mea squared error = e f Figure 4: Left ael: lots of the quatile estimators for method solid lie, method dotted lie ad true quatile dashed lie for differet samle sizes ad for data from a double exoetial distributio. Right ael: box lots of mea squared error for the quatile estimators for method ad method for differet samle sizes. Further, for comariso uroses we refer to our roosed method ad that of Sheather ad Marro as method ad method resectively. a Stadard ormal distributio see Table ad Figure. b Exoetial distributio see Table ad Figure. c Log-ormal distributio see Table 3 ad Figure 3. d Double exoetial distributio see Table 4 ad Figure 4.

12 Joural of Probability ad Statistics Table : Mea squared errors results for badwidth selectio methods for differet samle sizes ad for data from a ormal distributio. 5.5 method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method We ca comute ad summarize the relative efficiecy of h Method,ot for the all revious distributios i Table 5. From Tables,, 3, ad4, for the all distributios, it ca be observed that i 5.3% of cases our method roduces lower mea squared errors, slightly wis Sheather-Marro method. Also, from Table 5 which describes the relative efficiecy for h Method,ot we ca see h Method,ot more efficiet from h Method,ot for all the cases excet the stadard ormal distributio cases with, 5 ad double exoetial distributio cases with 5.

13 Joural of Probability ad Statistics 3 Table : Mea squared errors results for badwidth selectio methods for differet samle sizes ad for data from a exoetial distributio. 5.5 method method e 5. method method e 4.5 method method e 4. method method e 4.5 method method e 4.3 method method e 4.35 method method e 4.4 method method e 3.45 method method e 3.55 method method e 3.6 method method e 3.65 method method e 3.7 method method e 3.75 method method e 3.8 method method e 3.85 method method e.9 method method e.95 method method e So, we may coclude that i terms of MISE our badwidth selectio method is more efficiet tha Sheather-Marro for skewed distributios but ot for symmetric distributios. 6. Coclusio I this aer we have a roosed a cross-validatio-based-rule for the selectio of badwidth for quatile fuctios estimated by kerel rocedure. The badwidth selected by our

14 4 Joural of Probability ad Statistics Table 3: Mea squared errors results for badwidth selectio methods for differet samle sizes ad for data from a Log-ormal distributio. 5.5 method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method roosed method is show to be asymtotically ubiased ad i order to assess the umerical erformace, we coduct a simulatio study ad comare it with the badwidth roosed by Sheather ad Marro. Based o the four distributios cosidered the roosed badwidth selectio aears to rovide accurate estimates of quatiles ad thus we believe that the ew badwidth selectio method is a ractically useful method to get badwidth for the quatile estimator i the form.3.

15 Joural of Probability ad Statistics 5 Table 4: Mea squared errors results for badwidth selectio methods for differet samle sizes ad for data from a double exoetial distributio. 5.5 method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method method Table 5: The relative efficiecy R.E of h Method,ot. Stadard ormal dist. Exoetial dist. Log ormal dist. Double exoetial dist

16 6 Joural of Probability ad Statistics Aedix Ste. Let H S S, where S i Q i ) Q ) ), S i ) i δ X i Q )) ) ) ) Q i Q. A. Ste. With D i K h i/ X i Q ad D i δ i/ X i Q S S Q ) Q )) D i ), Q ) Q )) Q ) Q )) A. D i D ) i. Ste 3. This ste combies Stes ad to rove that H { } Q ) Q )) D ) i { } Q ) Q )) Q ) Q )) D i ) D i ). A.3 Ste 4. This ste establishes that { E Q ) Q )) } { E Q ) Q )) Q ) Q ))} { E 3 D ) } i var D i D ) ) i 3). h 8), A.4 Ste 5. This ste combies Stes 3 ad 4, cocludig that H Q ) Q )) Q ) Q )) μ h 3/ h 4), A.5 where μ h E D i D i.

17 Joural of Probability ad Statistics 7 Let U ξ, for a radom variable U U ad a ositive sequece ξ ξ E U ) ξ ). A.6 Ste 6. This ste otes that Q Q S T, where S g ) X i,x j, T g X i,x i, i / j g ) ) ) } i i j j X i,x j {K h X i δ )X i }{K h X j δ )X j d, A.7 ad that S S S g, where S i / j{ g Xi,X j ) g X i g Xj ) g }, S ) { } g X i g, g x E { g x, X },g E { g X }. A.8 Ste 7. Shows that E{g X,X }, E{g X,X } h 3,E{g X } h 6 var{t} 3,E S h 3 ad E S h 6. Ste 8. This ste combies the results of Stes 5, 6, 7, obtaiig H Q ) Q )) E T ) g μ h 3/ h 3/ / h 3) E Q ) Q )) μ h 3/ h 3/ / h 3). A.9 Ste 9. This ste otes that μ h h ad E Q ) Q )) E Q ) Q )) E Q ) Q )) μ h. A. Ste. This ste combies Stes 8 ad 9, establishig that H Q ) Q )) E Q ) Q )) E Q ) Q )) 3/ h 3/ / h 3). A.

18 8 Joural of Probability ad Statistics This meas that E{H J MISE h } 3/ h 3/ / h 3). A. Refereces S. J. Sheather ad J. S. Marro, Kerel quatile estimators, Joural of the America Statistical Associatio, vol. 85, o. 4,. 4 46, 99. E. Parze, Noarametric statistical data modelig, Joural of the America Statistical Associatio, vol. 74,. 5 3, S.-S. Yag, A smooth oarametric estimator of a quatile fuctio, Joural of the America Statistical Associatio, vol. 8, o. 39,. 4, M. Falk, Relative deficiecy of kerel tye estimators of quatiles, The Aals of Statistics, vol., o.,. 6 68, M. P. Wad ad M. C. Joes, Kerel Smoothig, Chama ad Hall, Lodo, UK, M. Y. Cheg ad S. Su, Bdwidth selectio for kerel quatile estimatio, Joural of Chies Statistical Associatio, vol. 44, o. 3,. 7 95, 6. 7 M. C. Joes, Estimatig desities, quatiles, quatile desities ad desity quatiles, Aals of the Istitute of Statistical Mathematics, vol. 44, o. 4,. 7 77, M. Rudemo, Emirical choice of histograms ad kerel desity estimators, Scadiavia Joural of Statistics, vol. 9, o., , A. W. Bowma, A alterative method of cross-validatio for the smoothig of desity estimates, Biometrika, vol. 7, o., , 984.

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