Non-parametric bootstrap mean squared error estimation for M-quantile estimates of small area means, quantiles and poverty indicators *

Transcription

1 Non-parametrc bootstrap mean squared error maton for M-quantle mates of small area means quantles and poverty ndcators * Stefano Marchett 1 Monca Prates 2 Nos zavds 3 1 Unversty of Psa e-mal: stefano.marchett@for.unp.t 2 Unversty of Psa e-mal: m.prates@ec.unp.t 3 Unversty of Southampton e-mal: n.tzavds@soton.ac.u Abstract Small area maton s conventonally concerned wth the maton of small area averages and totals. More recently emphass has been also placed on the maton of poverty ndcators and of ey quantles of the small area dstrbuton functon usng robust models for example the M-quantle small area model Chambers and zavds (2006. In parallel to pont maton Mean Squared Error (MSE maton s an equally crucal and challengng tas. However whle analytc MSE maton for small area averages s possble analytc MSE maton for quantles and poverty ndcators s extremely dffcult. Moreover one of the man crtcsms of the analytc MSE mator for M-quantle mates of small area averages proposed by Chambers and zavds (2006 and Chambers et al. (2009 s that t can be unstable when the area-specfc sample szes are small. In ths paper we propose a non-parametrc bootstrap framewor for MSE maton for small averages quantles and poverty ndcators mated wth the M-quantle small area model. Because the small area statstcs we consder n ths paper can be expressed as functonals of the Chambers-Dunstan mator of the populaton dstrbuton functon the proposed non-parametrc bootstrap presents an extenson of the wor by Lombarda et al. (2003. Alternatve bootstrap schemes based on resamplng emprcal or smoothed resduals are consdered. Emphass s also placed on second order propertes of MSE mators wth results suggng that the bootstrap MSE mator s more stable than correspondng analytc MSE mators. he proposed bootstrap s evaluated n a seres of smulaton studes. Keywords: M-quantles; dstrbuton functon; bootstrap. 1. Small area maton wth the M-quantle model In what follows we assume that a vector of p auxlary varable x s nown for each populaton unt n small area =1 m and that values of the varable of nter y are avalable from a random sample s that ncludes unts from all the small areas of nter. We denote the populaton sze sample sze sampled part of the populaton and non sampled part of the populaton n area respectvely by N n s and r. * Wor supported by the proect SAMPLE Small Area Methodology for Poverty and Lvng Condton Estmates awarded by the European Commsson n the 7thFP.

2 We assume that the sum over the areas of N and n s equal to N and n respectvely. We further assume that condtonal on covarate nformaton for example desgn varables the samplng desgn s gnorable. A recently proposed approach to small area maton s based on the use of a quantle/m-quantle regresson model Chambers and zavds (2006. he classcal regresson model summarses the behavour of the mean of a random varable y at each pont n a set of covarates x. Instead quantle regresson summarses the behavour of dfferent parts (e.g. quantles of the condtoned dstrbuton of y at each pont n the set of the x s. In the lnear case quantle regresson leads to a famly of hyper-planes ndexed by a real number q "(01. For a gven value of q the correspondng model shows how the qth quantle of the condtonal dstrbuton of y vares wth x. For example for q = 0.1 the quantle regresson hyperplane separates the lower 10% of the condtonal dstrbuton from the remanng 90%. Let us for the moment and for notatonal smplcty drop subscrpt. Suppose that (x y =1 n denotes the observed values for a random sample consstng of n unts where x are row p-vectors of a nown desgn matrx X and y s a scalar response varable correspondng to a realsaton of a contnuous random varable wth unnown contnuous cumulatve dstrbuton functon F. A lnear regresson model for the qth condtonal quantle of y gven x s Q y (q x = x "(q. An mate of the qth regresson parameter "(q s obtaned by mnmzng n %[ y " x #(q{(1 " qi(y " x #(q $ 0 + qi(y " x #(q > 0 }]. =1 Quantle regresson presents a generalzaton of medan regresson and expectle regresson Newey and Powell (1987 a quantle-le generalzaton of mean regresson. M-quantle regresson Chambers and Breclng (1988 ntegrates these concepts wthn a framewor defned by a quantle-le generalzaton of regresson based on nfluence functons (M-regresson. he M-quantle of order q for the condtonal densty of y gven the set of covarates x f(y x s defned as the soluton MQ y (q x;" of the matng equaton # " q { y $ MQ y (q x;" } f (y xdy = 0 where " q denotes an asymmetrc nfluence functon whch s the dervatve of an asymmetrc loss functon " q. A lnear M-quantle regresson model y gven x s one where we assume that MQ y (q x ;" = x # " (q (1.1 and mates of " # (q are obtaned by mnmzng (. (1.2 n #" q y $ x % & (q =1 Dfferent regresson models can be defned as specal cases of (1.2 by varyng the specfcatons of the asymmetrc loss functon " q. hroughout ths paper we wll tae the lnear M-quantle regresson model to be defned by when " q s the Huber loss functon Chambers and Breclng (1988. Settng the frst dervatve of (1.2 equal to zero leads to the followng matng equatons n #" q (r q x = 0 =1

3 where r q = y " x # $ (q " q (r q = 2"(s #1 r q { qi(r q > 0 + (1 # qi(r q $ 0 } and s > 0 s a sutable mate of scale. For example n the case of robust regresson s = medan r q Snce the focus of our paper s on M-type maton we use the Huber Proposal 2 nfluence functon "(u = ui(#c $ u $ c + c% sgn(u. Provded that the tunng constant c s strctly greater than zero mates of " # (q are obtaned usng teratve weghted least squares (IWLS. 1.1 Estmators of small area averages Chambers and zavds (2006 extended the use of M-quantle regresson models to small area maton. Followng ther development (see also Koc et al these authors characterze the condtonal varablty across the populaton of nter by the M-quantle coeffcents of the populaton unts. For unt wth values y and x ths coeffcent s the value " such that MQ y (" x ;# = y. he M-quantle coeffcents are determned at the populaton level. Consequently f a herarchcal structure does explan part of the varablty n the populaton data then we expect unts wthn clusters (domans defned by ths herarchy to have smlar M-quantle coeffcents. When the condtonal M-quantles are assumed to follow the lnear model (1.1 wth " # (q a suffcently smooth functon of q Chambers and zavds (2006 sugged a plug n (nave mator of the average value of y n area (* m ˆ MQ = N y + x ˆ # $ (% ˆ * ' ' - =1...m (1.3 + * &s &r.* where " ˆ s an mate of the average value of the M-quantle coeffcents of the unts n area. he area-specfc M-quantle coeffcents " ˆ can be vewed as pseudo-random effects. Emprcal wor ndeed ndcates that the area-specfc M- quantle coeffcents are postvely and hghly correlated wth the mated random area-specfc effects obtaned wth the ned error regresson small area model. Chambers and zavds (2006 also observed that the nave M-quantle mator (1.3 can be based especally n the presence of heterosedastc and/or asymmetrc errors. hs observaton motvated the wor n zavds et al. (2010. In partcular these authors proposed a bas adusted M-quantle mator for the small area average that s derved by usng an mator of the fnte populaton dstrbuton functon such as the Chambers-Dunstan mator Chambers and Dunstan (1986. he Chambers- Dunstan mator of the small area dstrbuton functon s of the form F ˆ CD (t = N I(y # t + n I(x ˆ $ % (& ˆ + ( (( + e # t. * + 's 'r 's -.. Estmates of " and " # ($ are obtaned followng Chambers and zavds (2006 and e = y " x ˆ # $ (% ˆ are model resduals. he M-quantle bas-adusted mator of the average of y n small area s then defned as ˆ m CD = +# y df ˆ CD ' $ "# (y = N ( * & %s y + & %r y ˆ + (1" f & %s e + - (1.4

4 where f = n N s the samplng fracton n area and y ˆ = x ˆ " # ( $ ˆ. Analytc MSE maton for M-quantle mators of small area averages s descrbed n Chambers and zavds (2006 and Chambers et al. (2009. In partcular Chambers et al. (2009 proposed an analytc mean squared error mator that s a frst order approxmaton to the mean squared error of mator (1.4. Emprcal studes show that ths analytc MSE mator s bas robust aganst msspecfcaton of the model Chambers et al. (2009. However ts man crtcsm s that t can be unstable especally wth small area-specfc sample szes. 1.2 Estmators of small area poverty ndcators and quantles Although small area averages are wdely used n small area applcatons relyng only on averages may not be very nformatve. hs s the case for example n economc applcatons where mates of average ncome may not provde an accurate pcture of the area wealth due to the hgh wthn area nequalty. Our goal n ths secton s to also express quantles and specfc poverty ndcators as functonals of the Chambers- Dunstan mator of the populaton dstrbuton functon. Wth regards to the maton of small area quantles an mate of quantle " for small area s the value q ˆ ( ; " obtaned by a numercal soluton to the followng matng equaton q ˆ ( ;$ df ˆ CD %"# (t = $. (1.5 Estmatng poverty ndcators at dsaggregated geographcal levels s also mportant. In ths paper we focus on the maton of the ncdence of poverty or Head Count Rato (HCR and of the Poverty Gap (PG. Denotng by t the poverty lne dfferent poverty ndcators are defned by usng F " = t # y " $ ' & I( y % t ( * t =1...N. In partcular settng α = 0 defnes the HCR whle settng α = 1 defnes the PG. he populaton dstrbuton functon n small area F " can then be decomposed as follows & #1 F " = N (% F " + % F " + '( $s $r * +. Hence one approach for matng the HCR n small area s by usng the Chambers- Dunstan mator of the dstrbuton functon and the M-quantle model for predctng for out of sample unts as follows F ˆ 0 = N + * + ( 's I(y # t + n (( 'r 's Smlarly an mator of the poverty gap for area F ˆ t " y 1 = N t " x ˆ $ % (& ˆ " e + ( I(y t # t + n (( * + 's 'r 's t I(x ˆ $ % (& ˆ + e # t I(x ˆ $ % ( ˆ. -.. (1.6 & + e # t. -.. (1.7

5 In practce the HCR and PG for area can be mated by usng a Monte Carlo approach. he maton procedure s as follows: 1. Ft the M-quantle small area model (2.1 usng the raw y s sample values and obtan mates " ˆ and ˆ " # ( $ ˆ. 2. Draw an out of sample vector usng y * = x ˆ " # ( $ ˆ + e * % r where e * " r s a vector of sze N " n drawn from the emprcal dstrbuton functon of the mated M-quantle model. 3. Repeat the process H tmes. Each tme combne the sample data and out of sample data for matng F 0 and F 1 4. Average the results over H smulatons. he M-quantle approach for matng poverty ndcators s smlar n sprt to the EBP approach proposed by Molna and Rao (2010. Note for example that y * " r s generated usng x ˆ " # ( $ ˆ.e. from the condtonal M-quantle model where " ˆ plays the role of the area random effects n the M-quantle modellng framewor. 2. Non-parametrc bootstrap MSE maton All small area target parameters we presented n Secton 1 have been expressed as functonals of the Chambers-Dunstan mator of the populaton dstrbuton functon. Unle MSE maton for small area averages analytc MSE maton for small area poverty ndcators and quantles s complex. In ths secton we present a nonparametrc bootstrap framewor for MSE maton of small area parameters mated wth the M-quantle model and the Chambers-Dunstan mator. Let us start wth the M-quantle small area model y = x " # ($ + e where " # ($ s the unnown vector of M-quantle regresson parameters for the unnown area-specfc M-quantle coeffcent " and e s the unt level random error term wth dstrbuton functon G for whch no explct parametrc assumptons are beng made. Usng the sample data we obtan mates " ˆ ˆ " # ( $ ˆ and mated model resduals. he target s to mate the small area fnte populaton dstrbuton functon or to be more precse a functonal of ths dstrbuton functon " by usng the Chambers-Dunstan mator and the M-quantle small area model F ˆ CD = N I(y # t + n ˆ G (t " x ˆ $ % (& ˆ + ( (. * + 's 'r -. (1.8 where G ˆ (u s the emprcal dstrbuton G ˆ (u = n % I(e # u of the model resduals e. Usng (1.8 we obtan mates of the small area target parameters we presented n Secton 2 whch we collectvely denote by ˆ ". Gven an mator G ˆ (u of the dstrbuton of the resduals G(u = Pr(" # t a bootstrap populaton consstent wth the M-quantle small area model " * = { y * x } can be generated by samplng from G ˆ (u to obtan e * $s

6 y * = x ˆ " # ( $ ˆ * + e =1...N =1...m. For defnng G ˆ (u we consder two approaches: (1 samplng from the emprcal dstrbuton functon of the model resduals or (2 samplng from a smoothed dstrbuton functon of the model resduals. For each of the two above mentoned approaches samplng can be done n two ways namely by samplng from the dstrbuton of all resduals wthout condtonng on the small area (uncondtonal approach or by samplng from the dstrbuton of the resduals wthn small area (condtonal approach. he emprcal dstrbuton of the resduals for the uncondtonal approach s m G ˆ (t = n %% I(e " e s # t (1.9 =1 $s where e s s the sample mean of the resduals e whle for the condtonal approach the emprcal dstrbuton s m G ˆ (t = n %% I(e " e s # t =1 $s where e s s the sample mean of the resduals n area. he correspondng smoothed mators of the dstrbuton of the resduals for the uncondtonal and the condtonal approaches are respectvely and m { } G ˆ (t = n $ $ K h (t " e + e s (1.10 =1 #s { } G ˆ (t = n $ K h (t " e + e s #s where h > 0 (or h s a smoothng parameter and K s the dstrbuton functon correspondng to a bounded symmetrc ernel densty. Hence there are four possble approaches for defnng e *. We sugg however usng the uncondtonal emprcal or smoothed approach. he reason s that n applcatons of small area maton samplng from the condtonal dstrbuton would rely on potentally a very small number of data ponts whch can cause G ˆ (t to be unstable. Let us now defne the fnte dstrbuton functon for the bootstrap populaton as follows & F * (t = N (% I(y * # t + % I(y * # t + '( $s $r * +. he bootstrap populaton dstrbuton functon can be mated by selectng a wthout replacement sample from the bootstrap populaton and by usng the Chambers- Dunstan mator F ˆ *CD (t = N I(y * # t + % G ˆ * (t " x ˆ & * ' ( ˆ + % ( *. * + $s $r -. (1.11 where ˆ " * # ( $ ˆ * are bootstrap sample mates of the M-quantle model parameters and G ˆ * = n % I(e * # u. Usng (1.11 we obtan bootstrap mates ˆ " * of the $s bootstrap populaton small area parameters " *.

7 he steps of the bootstrap procedure are as follows: Startng from sample s selected from a fnte populaton " wthout replacement we ft the M-quantle small area model and obtan mates of " and " # ($ whch are used to compute the model resduals. We then generate B bootstrap populatons " *b usng one of the prevously descrbed methods for matng the dstrbuton of the resduals G(u. From each bootstrap populaton " *b we select L bootstrap samples usng smple random samplng wthn the small areas and wthout replacement n a way such that n * = n. Usng the bootstrap samples we obtan mates of ". Bootstrap mators of the bas and varance of the mated target small area parameter ˆ " derved from the dstrbuton functon n area are defned respectvely by ( Bas(ˆ " = B #1 L #1 *bl *b $ $ ˆ " #" B L b =1 l =1 B L *bl " ( 2 Var(ˆ " = B #1 L #1 *bl $ $ ˆ # ˆ " b =1 l =1 where " *b *bl s the small area parameter of the bth bootstrap populaton ˆ " s the small area parameter mated by usng (1.11 wth the lth sample of the bth bootstrap *bl populaton and ˆ " = L #1 L *bl $ ˆ " l =1. he bootstrap MSE mator of the mated small area target parameter s then defned as MSE( ˆ " = Var( ˆ " + Bas( ˆ " 2. ( A smulaton study In ths secton we use model-based Monte-Carlo smulatons to emprcally evaluate the performance of the bootstrap MSE mator (1.12 when used to mate the MSE of the M-quantle mators of (a the small area average (1.4 (b the small area medan (1.5 (c the head count rato (HCR (1.6 and (d the poverty gap (PG (1.7. Moreover snce analytc MSE maton for M-quantle mates of small area averages s possble the proposed bootstrap MSE mator s also contrasted to the correspondng analytc MSE mator (see Chambers et al both n terms of bas and stablty. In what follows subscrpt dentfes small areas =1 m and subscrpt dentfes unts n a gven area =1 n. Populaton data " = (xy n m=30 small areas are generated usng y =11" x + # + e where " " 2 (1 e " 2 (6 and x N( µ 1 wth µ U[811]. For each Monte Carlo smulaton a wthn small areas random sample s selected from the correspondng generated populaton. he total populaton sze s N=2820 wth areaspecfc populaton szes rangng between 50 " N 50 and the total sample sze s n=282 wth area-specfc sample szes rangng between 5 " n 5. Usng the sample data we obtan pont mates of small area averages wth (1.4 of the 0.50 percentle of the dstrbuton of y wth (1.5 and of the HCR and PG wth (1.6 and (1.7 respectvely. For small area averages MSE maton s performed usng both the analytc MSE (Chambers et al and the bootstrap MSE mator (1.12. For mators of small area percentles and poverty ndcators MSE maton s

8 performed usng the bootstrap MSE mator (1.12. We run n total H=500 Monte- Carlo smulatons. For bootstrap MSE maton we used one bootstrap populaton B=1 from whch we drew 400 bootstrap samples L=400. Fnally we present results only for the smoothed uncondtonal bootstrap scheme. he results are reported n able 1. able 1. Results from evaluatng the bootstrap MSE mator Averages smoothed approach Mn. 1 st Qu. Medan Mean 3 rd Qu. Max. rue Estmated(Analytc Estmated(Bootstrap Rel. Bas(%(Analytc Rel. Bas(%(Bootstrap RMSE(Analytc RMSE(Bootstrap HCR smoothed approach rue Estmated Rel. Bas(% RMSE PG smoothed approach rue Estmated Rel. Bas(\% RMSE Q50 smoothed approach rue Estmated Rel. Bas(% RMSE he results we present here ndcate that the bootstrap MSE can be relably used for matng the MSE of M-quantle small area averages percentles and poverty ndcators. In partcular for small area averages we should note that these results sugg that the analytc and the bootstrap MSE mators trac very well the emprcal MSE and have on average reasonably low relatve bas. However the bootstrap MSE mator appears to be notably more stable. For the two poverty measures and the medan the mated RMSE also tracs well the entre dstrbuton of the emprcal RMSE. 4. R functons for pont and MSE maton R functons that mplement small area maton wth the M-quantle model are avalable upon requ from the authors. In partcular functon mq.sae produces M- quantle mates of small area averages usng (1.4 and MSE maton usng the analytc MSE mator proposed by Chambers et al.(2009. Functon mq.sae.quant produces M-quantle mates of the small area quantles of the dstrbuton of y usng (1.5 and bootstrap MSE maton usng MSE mator (1.12. Functon

9 mq.sae.poverty produces M-quantle mates of the small area HCR and PG usng respectvely (1.6 and (1.7 and bootstrap MSE maton usng MSE mator (1.12. Optons for usng emprcal or smoothed condtonal and uncondtonal resduals for generatng the bootstrap populaton are avalable. 5. Conclusons In ths paper we propose the use of non-parametrc bootstrap for matng the MSE for small area averages quantles and poverty ndcators mated wth the M- quantle model and the Chambers-Dunstan mator. Gven that analytc MSE maton for quantles and poverty ndcators s dffcult the proposed MSE mator provdes one practcal approach for MSE maton of complex small area statstcs. As llustrated n the emprcal secton the proposed bootstrap MSE mator approxmates well the true MSE error of the target parameters. In addton these results show that bootstrap MSE maton s notably more stable than correspondng analytc maton. he practcal mplementaton of the maton procedures we descrbe n ths paper s asssted by the avalablty of R functons. References Aragon Y. Casanova S. Chambers R. Leoconte E Condtonal orderng usng nonparametrc expectles. Journal of Offcal Statstcs 21 (4. Bowman A. Hall P. Prvan Bandwdth selecton for the smoothng of dstrbuton functons. Bometra Breclng J. and Chambers R. (1988. M-quantles. Bometra Chambers R. Chandra H. zavds N On robust mean squared error maton for lnear predctors for domans. Worng Paper 11 Unversty of Wollongong Australa. Chambers R. Dorfman A. Peter H Propertes of mators of the fnte populaton dstrbuton functon. Bometra 79 ( Chambers R. Dunstan Estmatng dstrbuton functon from survey data. Bometra Chambers R. zavds N M-quantle models for small area maton. Bometra 93 ( Foster J. Greer J. horbece E A class of decomposable poverty measures. Econometrca Hayfeld. Racne J. S Nonparametrc econometrcs: he np pacage. Journal of Statstcal Software 27 (5. URL Koener R. Bassett G Regresson quantles. Econometrca Koc P. Chambers R. Breclng J. Beare S A measure of producton performance. Journal of Busness and Economc Statstcs 15 ( L Q. Racne J Nonparametrc econometrc: theory and practce. Prnceton Unversty Press. Lombarda M. Gonzalez-Mantega W. Prada-Sanchez J Bootstrappng the chambers-dunstan mate of fnte populaton dstrbuton functon. Journal of Statstcal Plannng and Inference

10 Molna I. Morales D. Prates M. zavds N. (Eds Fnal small area maton development and smulaton results. No. Delverable D12 and D16 - S.A.M.P.L.E. proect. European Unon - 7th Framewor Programme. Molna I. Rao J Small area maton of poverty ndcators. he Canadan Journal of Statstcs. Newey W. Powell J Asymmetrc least squares maton and tng. Econometrca 55 ( R Development Core eam R: A Language and Envronment for Statstcal Computng. R Foundaton for Statstcal Computng Venna Austra. Rao J Small Area Estmaton. New Yor:Wley. Royall R. Cumberland W Varance maton n fnte populaton samplng. Journal of the Amercan Statstcal Assocaton Snha S. Rao J Robust small area maton. he Canadan Journal of Statstcs 37 ( zavds N. Marchett S. Chambers R Robust maton of small area means and quantles. Australan and New Zeland Journal of Statstcs 52 (