M-ESTIMATORS IN BUSINESS STATISTICS

Dehnel, G ażyna A icle M-ESTIMATORS IN BUSINESS STATISTICS S a is ics in T ansi ion New Se ies P o ided in Coope a ion wi h: Polish S a is ical Associa ion Sugges ed Ci a ion: Dehnel, G ażyna (2016) : M-ESTIMATORS IN BUSINESS STATISTICS, S a is ics in T ansi ion New Se ies, ISSN 2450-0291, Exeley, New Yo k, NY, Vol. 17, Iss. 4, pp. 749-762, h ps://doi.o g/10.21307/s a ans-2016-050 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/207840 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ STATISTICS IN TRANSITION new se ies, Decembe 2016 749 STATISTICS IN TRANSITION new se ies, Decembe 2016 Vol. 17, No. 4, pp. 749–762 M-ESTIMATORS IN BUSINESS STATISTICS 1 G ażyna Dehnel 2 ABSTRACT Recen yea s ha e seen a dynamic de elopmen in s a is ical me hods o analysing da a con amina ed wi h ou lie s. One o he mo e impo an echniques ha can deal wi h ou lying obse a ions is obus eg ession, which ep esen s ou decades o esea ch. Un il ecen ly he implemen a ion o obus eg ession me hods, such as M-es ima ion o MM-es ima ion, was limi ed owing o hei i e a i e na u e. Wi h ad ances in compu ing powe and he g owing a ailabili y o s a is ical packages, such as R and SAS, S a a, he applicabili y o obus eg ession me hods has inc eased conside ably.The aim o he s udy is o e alua e one o hese me hods, namely M-es ima ion, using da a om a su ey o small and medium-sized businesses. The compa ison in ol es nine M-es ima o s, each based on a di e en weigh ing unc ion. The esul s and conclusions a e o mula ed on he basis o empi ical da a om he DG-1 business su ey. Key wo ds: obus eg ession, M-es ima ion, business s a is ics, ou lie s 1. In oduc ion Robus eg ession p o ides es ima o s which elimina e he in luence o ou lie s. In he li e a u e i is some imes p esen ed as a me hod designed o igno e ou lying obse a ions. I is o en con as ed wi h me hods aimed a de ec ing ou lie s. The ac is, howe e , ha bo h de ec ion and obus eg ession pu sue he same objec i es – he only di e ence is how hey a e achie ed (Rousseeuw, Le oy, 1987). In he case o de ec ion, he i s s ep in ol es iden i ying ou lie s; only hen a e da a co ec ed. In he case o obus eg ession, i s a eg ession model is i ed o mos o he da a, hen ou lie s can be de ec ed, based on esidual alues. While each o he wo app oaches has i s bene i s and d awbacks, i is he echniques o obus es ima ion ha ha e been a ac ing g owing in e es ecen ly. A numbe o app oaches o obus eg ession ha e been p oposed in an a emp o imp o e i s pe o mance. The s a ing poin o he wo k was O dina y Leas Squa es (OLS). One o i s i s modi ica ions was M-es ima ion, which is 1 The p ojec is inanced by he Polish Na ional Science Cen e, decision DEC- 2015/17/B/HS4/00905. 2 Poznan Uni e si y o Economics, Depa men o S a is ics. E-mail: [email p o ec ed]. 750 G. Dehnel: M-es ima o s in … cha ac e ized by a low b eakdown poin by high e iciency. The nex de elopmen was S-es ima ion and LTS-es ima ion, wi h a high b eakdown poin . The g oup o he la es me hods includes MM-es ima ion. The MM-es ima o was he i s es ima o wi h a high b eakdown poin and high e iciency unde no mal e o (S ombe g, 1993). Some o hese me hods we e de eloped in he 1970s and 80s o he 20 h cen u y, bu because hey ely on compu a ionally demanding i e a i e p ocedu es, hey had limi ed applica ions. Nowadays a numbe o s a is ical packages a e a ailable, such as R o SAS, S a a, which acili a e he implemen a ion o obus eg ession me hods (Ve a di, C oux, 2009). The g owing in e es in echniques o obus eg ession is also due o he ac ha hei applica ion, unlike o he me hods, does no equi e ea lie de ec ion o ou lie s. When choosing obus eg ession me hods, one should keep in mind he p ope ies o pa icula me hods, which ha e been iden i ied in he li e a u e (Holland, Welsch 1977), (Hube , 1981), (Hampel e al., 1986), (Chen, Yin, 2002). One o he la es me hods is MM-es ima ion. I is a combina ion o wo di e en me hods: e icien es ima ion o an M-es ima o and S-es ima ion o a LTS- es ima ion wi h a high b eakdown poin . Hence, he ul ima e quali y o es ima ion depends on he quali y o each o he wo app oaches. In each app oach i is necessa y o make addi ional decisions abou he choice o pa ame e s and unc ions. The p esen a icle is limi ed o an analysis o he p ope ies o one o hese app oaches – M-es ima ion. The s udy was aimed a e alua ing p ope ies o M-es ima o s, whe e di e en weigh ing unc ions we e used. The e alua ion was based on an empi ical s udy using da a on small and medium-sized en e p ises in he anspo sec ion o he classi ica ion o economic ac i i ies (NACE Re .2). 2. M-es ima ion The class o M-es ima o s is a gene aliza ion o Maximum likelihood ype es ima o s (MLE). M-es ima o s a e classi ied as pa o obus eg ession es ima o s o he so-called 1s gene a ion. I is a g oup which is cha ac e ized by a low b eakdown poin in he case o x-ou lie s. The M-es ima o was in oduced by Hube in 1964 (Hube , 1964). I is a obus equi alen o he app oach ep esen ed by he leas squa es me hod (Chen, 2007). The loss unc ion o he leas squa es me hod is eplaced by ano he loss unc ion ρ(·), which is less sensi i e o ex eme esidual alues           n i i Ms 1 min a g ˆ   (1) whe e:  - loss unc ion s - scale pa ame e Xθy ii  . STATISTICS IN TRANSITION new se ies, Decembe 2016 751 To ensu e obse ed esponse alues ha e compa able a ia ion wi h espec o he alues o he dependen a iable, esiduals a e s anda dized using a dispe sion measu e s. The use o a classic measu e o dispe sion o he pu poses o s anda diza ion, in he p esence o ou lie s, esul s in o e es ima ed alues. Fo his eason, s anda d de ia ion is eplaced wi h o he measu es o dispe sion, such as median absolu e de ia ion (MAD) o in e qua ile ange (IQR). The objec i e unc ion mee s he ollowing condi ions (Banaś, Ligas, 2014): • non-nega i i y • equals ze o when i s a gumen equals ze o (   00   ) • is symme ic (e en unc ion) (     ii   ), • mono onici y in i (     ji   o ji  ). Assuming he scale pa ame e s is known, an es ima e o he es ima o M  is ob ained by sol ing a sys em o p equa ions wi h espec o ec o  exp essed as a p oduc o independen a iables and pa ial de i a i es o he  unc ion: 0 1 1             i n i p kkikix s xy  (2) whe e:  - in luence unc ion, a de i a i e o  unc ion p – he numbe o a iables x. In addi ion o he loss unc ion, he in luence unc ion is ano he one ha cha ac e izes M-es ima o s. I helps o assess how a single obse a ion a ec s he alue o he es ima o . Equa ion (2) is ypically sol ed by means o i e a i ely eweigh ed leas squa es (IRLS) wi h weigh s gi en by he ollowing o mula (T zpio , 2013):                    s xy s xy w p kkiki p kkiki i 11  (3) whe e: i w - weigh ing unc ion. The weigh ing unc ion i w , which is a a io o he in luence unc ion and he esidual, is he hi d unc ion which cha ac e izes M-es ima ion. The weigh ing unc ion mee s he ollowing condi ions (Banaś, Ligas, 2014): • is con inuous, symme ic, • dec eases when he esidual inc eases, • is equal o one when i s a gumen is ze o, • dec eases o ze o o he a gumen inc easing o +/- in ini y. The alues o weigh s depend on wha unc ion  is chosen o co espond wi h unc ion  . In he li e a u e o he subjec many a ia ions o M-es ima o s 752 G. Dehnel: M-es ima o s in … a e sugges ed, wi h di e en a ian s o unc ion  (Fai , 1974), (Holland, Welsch 1977), (Hube , 1981), (Hampel e al., 1986), (Chen, Yin, 2002), (Banaś, Ligas, 2014). Cu es o he weigh ing unc ions a e di e en , bu in M-es ima ion one always a emp s o minimize o elimina e he in luence o ou lie s. Fo his eason, all he p oposed weigh ing unc ions cu o o educe he in luence o la ge esiduals on he es ima ion o unc ion pa ame e s and/o scale pa ame e . The selec ion o unc ion  is made depending on wha weigh we wan o assign o ou lie s, among o he hings. In o de o desc ibe some o hei p ope ies, e alua ion and use ulness, and hei in luence on es ima ion esul s, nine di e en weigh ing unc ions a e analysed in his a icle: And ews', Tukey's (bisqua e), Cauchy’s, Fai 's, Hampel's, Hube 's, logis ic, Talwo h's and Welsch's weigh ing unc ions (see Fig. 1). Weigh ing unc ions ha e uning ac o s, which can be modi ied. Using uning ac o s, i is possible o educe he impac o ou lie s wi h la ge esiduals, bu his is achie ed a he cos o educing he es ima o e iciency. In he s udy desc ibed in his a icle, he uning ac o s o he weigh ing unc ion we e se in such a way as o ensu e 95% e iciency o es ima es o M-es ima o s, see Table 1. And ews Tukey Cauchy o c=2 Fai Hampel Hube STATISTICS IN TRANSITION new se ies, Decembe 2016 753 logis ic Talwo h Welsch Figu e 1. Weigh ing unc ions o M-es ima o Sou ce: Based on SAS INSTITUTE INC. (2014). The ini ial alue o 0 ˆ  is es ima ed based on he OLS me hod. In each i e a ion , one uses alues o esidual and weigh s ob ained in i e a ion -1 un il con e gence is achie ed (Alma, 2011). A e each i e a ion, i is also necessa y o conduc s anda diza ion. In p ac ice he scale pa ame e s is unknown. One simple and e y esis an possibili y in hese cases is o use he median absolu e de ia ion es ima o (Hube , 1964, Ripley, 2004; T zpio , 2013) Ano he possibili y is o es ima e scale s in an MLE-like way (Venables, Ripley, 2002). Table 1. Tuning ac o s o he weigh ing unc ion Weigh ing unc ion Tuning ac o s a, b, c And ews' 1.339 Bisqua e 4.685 Cauchy's 2.385 Fai 's 1.4 Hampel's 4, 2, 8 Hube 's 1.345 Logis ic 1,205 Talwo h's 2.795 Welsch's 2,985 Sou ce: Based on SAS INSTITUTE INC. (2014). 754 G. Dehnel: M-es ima o s in … The M-es ima o is only esis an o ou lie s in he y-di ec ion; i is no esis an o le e age poin s. This a ec s he ange o po en ial applica ions. Hence, he es ima o is equen ly used bu only in si ua ions whe e le e age poin s a e no a p oblem. I s b eakdown poin is no high and is equal o 1/n. The M-es ima o is condi ional bias – condi ional on he p opo ion o he ou lie in he sample (Cox e al., 1995). 3. E alua ion o es ima es ob ained in he empi ical s udy A p elimina y e alua ion o es ima es ob ained using M-es ima o s was conduc ed in e ms o he goodness o i o he model, ep esen ed by he coe icien o de e mina ion. The obus e sion o he coe icien o de e mina ion is de ined as:                          s y s xy s y R i T iii ˆ ˆ ˆ ˆ ˆ ˆ 2       (4) whe e  is he loss unc ion o he obus es ima e,  ˆ is he obus loca ion es ima o , and s ˆ is he obus scale es ima o in he ull model. P ope ies o he es ima o s analysed in he s udy we e e alua ed using he boo s ap me hod. 1000 i e a ions o d awing samples we e made, which we e hen used o calcula e: • Rela i e es ima ion e o (REE)           d b ddb d d dYE YY YE YVa YCV ˆ ˆˆ 999 1 ˆ ˆ ˆ 2 1000 1 ,     (5) • Mean absolu e ela i e bias (ARB)      1000 1 , ˆ 1000 1 ˆ bd ddb dY YY YARB (6) • Rela i e oo mean squa e e o (RMSE)     d bddb dY YY YRMSE     1000 1 2 , ˆ 1000 1 ˆ (7) 4. The desc ip ion o he s udy The empi ical s udy was based on in o ma ion om o icial s a is ics collec ed in a business su ey known as DG-1. I is he la ges su ey in Polish sho - e m STATISTICS IN TRANSITION new se ies, Decembe 2016 755 business s a is ics. I collec s da a om businesses employing o e 9 people. The su ey collec s da a om all medium-sized and la ge en e p ises and a 10% sample o small businesses. I is conduc ed on a mon hly basis. I s objec i e is o collec up- o-da e in o ma ion abou basic indica o s o economic ac i i y o en e p ises. In he empi ical s udy only da a abou small and medium-sized companies we e used (wi h he numbe o employees anging om 10 o 250), which conduc ed hei business ac i i ies in Decembe 2011. In he model conside ed in he s udy e enue was he dependen a iable. Independen a iables came om an adminis a i e egis e . Th ee independen a iables we e used in he model: p o i , cos and he numbe o employees. A 10% sample o small and medium-sized companies om he DG-1 su ey was ea ed as he gene al popula ion. The domain o s udy was c ea ed by c oss-classi ying he adminis a i e di ision in o p o inces wi h he NACE ca ego y o business ac i i y. Resul s o he s udy we e limi ed o domains included in one NACE sec ion: anspo . The sec ion was chosen on he basis o he assessmen o he goodness o i o he eg ession model o he empi ical da a. The main mo i a ion o he choice o he sec ion was o ensu e ha domains i con ained we e cha ac e ized by he p esence o ou lie s, which conside ably educed he quali y o he classical model o eg ession (see. Table 2). The i s s age o he analysis in ol ed assessing he dis ibu ion o businesses in e ms o a iables included in he model. Values o he basic desc ip i e s a is ics o all he a iables we e cha ac e ized by high a iabili y and s ong asymme y. In he case o he a iable 'Re enue', he coe icien o a ia ion amoun ed o as much as 405%, while skewness was as high as 5.63. Table 2. S a is ical cha ac e is ics o he dis ibu ion o he Re enue a iable (in housand PLN by p o ince and sec ion 'T anspo ', 2011 P o ince CV(%) Skewness R2 Pe cen age o ou lie s (%) N Dolnośląskie 90 1.24 0.537 7.1 28 Kujawsko-Pomo skie 100 1.45 0.945 16.7 24 Lubelskie 231 4.22 0.995 12.0 25 Lubuskie 303 3,99 0.139 15.0 20 Łódzkie 106 2.47 0.991 6.7 30 Małopolskie 327 5.62 0.999 11.8 34 Mazowieckie 405 5.63 0.999 6.8 73 Opolskie 89 0.91 0.984 28.6 14 Podka packie 72 0.27 0.982 18.8 16 Podlaskie 286 3.45 0.999 8.3 12 Pomo skie 138 2.38 0.980 3.0 33 Śląskie 261 5.55 0.992 9.4 64 Świę ok zyskie 143 2.52 0.998 13.0 23 Wa mińsko-Mazu skie 65 0.76 0.892 15.4 13 Wielkopolskie 98 1.87 0.993 8.2 49 Zachodniopomo skie 138 2.21 0.970 13.3 30 Sou ce: Own calcula ions based on DG1 su ey. 756 G. Dehnel: M-es ima o s in … In addi ion, S uden 's - es and Cook's D con i med he p esence o ou lie s. These p ope ies indica ed he need o he use o obus eg ession me hod. The pe cen age sha e o ou lie s, as well as he alue o he coe icien o de e mina ion R2 a e p esen ed in Table 2. The assessmen o hese wo pa ame e s indica es ha he pe cen age sha e o ou lie s is no co ela ed wi h he alue o he coe icien o de e mina ion. In he case o bo h mo e and less nume ous sec ions, e en a ela i ely la ge numbe o ou lie s does no necessa ily ha e a nega i e impac on he model i . On he o he hand, indi idual ou lie s may ha e a la ge in luence on he quali y o he model, o he impac o ou lie s depends no only on hei numbe bu also on hei ype (ou lie s in he x- di ec ion, ou lie s in he y-di ec ion) and hei dis ance om ypical obse a ions. The g aphic p esen a ion showing he ela ionship be ween he ype o ou lie s and he model quali y only shows domains wi h he lowes alues o he coe icien o de e mina ion, ha is o p o inces o Dolnośląskie, Lubuskie, Wa mińsko-Mazu skie, Zachodniopomo skie (see. Fig. 2) Lubuskie R2= 0.139 N=20 Dolnośląskie R2= 0.537 N=30 Wa mińsko-Mazu skie R2= 0.892 N=13 Zachodniopomo skie R2= 0.970 N=20 Figu e 2. Ou lie and Le e age diagnos ic o anspo in selec ed p o inces Sou ce: Calcula ions based on he DG1 su ey and he ax egis e o Decembe 2011.