A C T A U N I V E R S I T A T I S L O D Z I E N S I S
F O L IA O E C O N O M IC A 194, 2005
Tomasz Jurkiewicz* , K rzyszto f Najman*
P R O P O SIT IO N OF APPLYING tf-M EA N S C LASSIFIC ATIO N M E T H O D AND T H E SO M TYPE N EU R A L NETW ORK
TO IM PROVE TH E EFFICIENCY OF SM ALL DO M A IN S ESTIM ATION IN A REPRESENTATIVE ST U D Y
O F SM A LL A N D M E D IU M -SIZ E D E N T ER PR ISE S
Abstract
T h e p ro b lem o f a to o sm all n u m b er o f observ atio n s o f a sam ple, rep resen tin g a defined d o m ain o f a p o p u la tio n m ay be solved inter alia th an k s to the a p p lic atio n o f estim a to rs which w ould use in fo rm a tio n a b o u t o th er co m p o n en ts o f the sam ple (derived fro m outside the defined p a rt o f the p o p u latio n ) to estim ate p aram ete rs in a given s u b p o p u la tio n (sm all area, d o m ain ). O ne o f e stim atio n m eth o d s fo r sm all d o m ain s - the synthetic e stim atio n - assum es, th a t th e d istrib u tio n o f the studied sm all d o m ain is identical w ith the d istrib u tio n o f the w hole p o p u la tio n . T h is assu m p tio n rem ain s usually unfulfilled, in p a rtic u la r in case o f specific d o m ain s, w h at results in large estim ation errors.
T h e a u th o rs p resen t a p ro p o sitio n o f tw o-stage estim atio n process. In the first stage, using the S O M -ty p e n e u ral n etw o rk s and using the /с-m eans classification m eth o d the sim ilarity o f c o m p o n en ts belonging to th e sm all d o m ain w ith th e c o m p o n e n ts b elonging to th e rem ain in g p a rt o f the sam ple is d eterm in ed . T h e second step consists in using th e in fo rm a tio n only from those d o m ain s, w hich are sim ilar to the studied sm all d o m ain with th e help o f a p p ro p ria te ly co n stru ed w eights. A u th o rs p resen t the results o f th e above p ro ced u re in the analysis o f the building in d u stry o n the basis o f a representative study o f sm all an d m edium -sized enterprises. T h ey h av e also u n d e rta k en a n a tte m p t to estim ate the e rro rs o f the sy n th etic estim ation m eth o d m odified in such a way.
Key words: sm all d o m ain estim atio n , classification m eth o d s, n e u ral netw o rk s.
I. IN T R O D U C T IO N
T h e process o f econom ic and social developm ent results i.a. in a grow ing dem and fo r statistical inform ation . One o f effective ways o f satisfying th at d em and are representative studies. Because o f o rg a n isatio n al and financial
c o n stra in ts those studies, how ever, arc no t able to supply credible d a ta for a m ore detailed division o f the population into su b p o p u latio n s (dom ains of studies). T o o small a num ber o f observations com ing from a particular dom ain m ay be an obstacle in applying certain statistical conclusion generating techniques o r lead to considerable errors o f estim ation (B racha, 1996).
O ne o f possible ways o f solving th a t problem is the constru ctio n o f such estim ators, which could use in form ation a b o u t o th e r co m po nen ts of a sam ple, nam ely those com ing from outside a p artic u la r p a rt o f the po p u latio n or add itio n al inform ation from outside o f the sam ple to estim ate p aram eters o f a defined su b p o p u latio n (small area, dom ain).
T h e “ sm all d o m a in ” (small area) is defined as a do m ain o f studies, for which in fo rm atio n is essential from the po in t o f view o f the d a ta user, and it is n o t possible to acquire th a t info rm ation using th e direct estim ation m ethod because the size o f the sam ple is to o small or when the in form ation acquired with indirect m eth ods is m ore credible. T h ere is n o reason for which the scope o f statistics o f sm all areas should be confined to territorial (a d m in istratio n ) units. F ro m a m ethodological p oin t o f view it does n o t m ak e any difference w hether we consider a su b p o p u latio n of one territory or a su b p o p u latio n isolated according to any o th er m etho d.
T he principal aim o f the p aper is an attem p t to determ ine th e qualities of a synthetic estim ator after a m odification consisting in using only the inform a tion a b o u t com ponents sim ilar to the ones found in the small d o m ain in the estim ation process. T he parallel aim o f the study is to em pirically verify the m odified synthetic estim ato r while studying a concrete sam ple.
II. E S T IM A T O R S O F S M A L L D O M A IN S
T he essence o f indirect estim ation consists in “ borrow ing the inform ation” to strengthen the estim ation in the dom ain being o f interest to the statistician. In case o f a representative study it is possible to use the follow ing sources o f ad d itio n al d a ta (Jurkiewicz, 2001):
- o th e r dom ain s in the sample;
- in fo rm atio n a b o u t the n u m b er o f p artic u la r s tra ta an d th e num ber o f d o m ain s in the studied p opulation;
- in fo rm atio n ab o u t the values o f an add itio nal variable in a sam ple, strongly related to the studied variable and at least as credible as the variable in question;
in fo rm atio n ab o u t values o f an ad dition al variable in the studied popu lation;
T h e direct estim ato r o f an unknow n p aram eter G Y d in a sm all dom ain is the H o rv itz-T h o m p so n estim ator, know n as the expansion estim ator. It uses only the d a ta ab o u t random ly draw n com ponents o f a sam ple belonging to the sm all dom ain. T h e I I T estim ator is, how ever, unbiased, but bccausc o f the sm all size o f the sam ple its variance is usually high. T h a t estim ator will have the follow ing form for the p ro p o rtio n param eter:
where kd an d nd sym bolise the num ber o f elem ents distinguished in the d o m ain d and the size o f the small dom ain d correspondingly.
S ynthetic estim ation constitutes one o f the first p ro p o sitio n s o f solving the principal problem o f estim ation for small do m ain s, which stem s from the insufficient size o f a sam ple. T o this end an assum p tio n is m ade th at the stru ctu re o f the studied population in the sm all d om ain an d outside o f it is uniform , w hat allows to use the info rm atio n from the whole sam ple to estim ate the value for the dom ain. T his assu m ption m ay be limited in som e cases to th e sim ilarity o f only certain param eters in the po pu lation and in the d o m ain . F o r instance, the basis for co n stru ctio n o f the com m on synthetic estim ato r is the assum ption th a t the m eans o f the studied feature in the po p u latio n and in the dom ain do n o t essentially differ. F o r the p ro p o rtio n the estim ato r adop ts the form o f the follow ing statistics:
к
з у п Р л — n ( 2 )
w here к and n d en o te the n um ber o f elem ents distinguished in the sam ple and the size o f the whole sam ple correspondingly.
W hile applying the synthetic estim ation it is very im p o rta n t to pay careful a tte n tio n to the problem o f efficiency o f the ad o p ted m odel. T he fu rth e r the assum ptions laying at the base o f the estim ation are from the reality, the m o re biased will be the estim ators. It has to be b o rn e in m ind at the sam e tim e, th a t firstly, the bias m ay be o f considerable size, and secondly, it is in no way taken into account in fo rm ulae for m ean square errors and estim ato rs o f errors.
III. M O D IF IE D S Y N T H E T IC E S T IM A T O R (M E S )
T h e assum ption a b o u t the com patibility o f stru ctu res o f the popu lation and the d o m ain rem ains usually unfulfilled, in p artic u la r in case o f specific dom ains, w hat results in large estim ation errors. T he solution to the problem
m ay be to strengthen the estim ation process by m odifying the estim ator with info rm atio n from co m pon ents or d om ains sim ilar to the studied one.
T h e proposed procedure o f estim ation is carried o u t in tw o stages. T he first step consists in estab lish ing, w hich co m p o n en ts o r d o m ain s are sim ilar to the studied one. W eights for add ition al in fo rm atio n arc cal culated in relatio n to the degree o f sim ilarity. T h u s d a ta from similar co m p o n en ts will imply a relatively high value o f the weight, while d ata from d ista n t co m ponents will have a relatively low er weight or will no t be tak en into account a t all. T he p ro p o rtio n estim ato r will a d o p t the fol low ing form :
kd - n u m b er o f elem ents distinguished in th e sam ple belonging to the do m ain,
nd - size o f the sam ple in the dom ain,
W; - weights for the com ponents from outside the sm all do m ain, y t - values o f the studied zero-one feature.
T he establishm ent o f the sim ilarity o f the studied feature to o th er features in the p o p u latio n m ay be carried o u t i.a. using the m etho d o f m ultid im en sional analysis. In the present pap er the m eth od o f g ro u p in g k -m eans was used. A s an alternative m etho d o f classification the neural netw o rk o f the Self O rganizing M ap (SO M ) type was used (K o ho nen, 1997), and then on the acquired neural m ap the grouping was carried o u t according to the /с-m eans m ethod.
T h e n um ber o f classes in the grouping process was established using as the criterio n the value o f the D avies-B oulding clustering evalu atio n index in the form (D avis, B oulding, 1979):
M S E P d — (3)
nd+ X>,
where:
с where:
S, - stan d ard deviation in the i-th class, M y - distance betw een classes,
T he DB index is based on the q u o tien t o f v ariatio n w ithin the class and the distance betw een classes. T he establishm ent o f the op tim um nu m b er o f classes consists in the calculation o f the value o f the index fo r all variants o f the n um ber o f classes and selecting the v aria n t w ith th e m in im um value o f the DB index.
W hile establishing the weights for com ponen ts from outside the small do m ain an assum ption was m ad e th a t th e w eight should be in direct p ro p o rtio n to the percentage share o f units from th e sm all d o m ain , which were found in the given class. T he w eight m ay be w ritten as:
Пц
where:
ndi - n u m b er o f com ponents belonging to th e d o m ain d w hich were fo u n d in the class i,
у - stan d ard isin g coefficient from the range (0, 1) defining the m axim um value o f th e weight.
F o r instance, if in the z'-th class twice as m an y co m p o n en ts were found th a n in the j'-th class, then all com p onents from outside the sm all dom ain in the i-th class will have the sam e weight and it will be a w eight twice as high as the one used for com ponents from the j-th class.
It is w o rth to pay atten tio n to one o f the ad vantages o f the M E S estim ato r, which consists in th e possibility o f using in fo rm atio n derived from outside the study. N am ely, while establishing th e sim ilarity between d o m ain s it is possible to use d a ta from com pletely different, e.g. earlier studies o r the available inform ation ab o u t the p op u latio n . In such case it is also possible to calculate the estim ations o f p aram eters fo r a dom ain, w hich is n o t represented in the sam ple.
IV . E V A L U A T IO N O F P R O P E T IE S O F T H E M E S E S T IM A T O R
T o evaluate the M E S estim ator the b o o tstra p m eth o d was used. In subsequent repetitions 224 com ponents were draw n independently at ran do m , considering co m po nents, th a t were fou nd originally in the sam ple as the
p o p u latio n in question. 1000 sim ulations were m ade. F o r each sim ulation g rou ping w ith the use o f the /e-means m ethod for 5 classes and 20 iterations was m ade and grouping with the use o f the SOM neural netw ork was carried ou t, assum ing the 12 x 12 neurones with the “ b ubble” neighbourhood function and the num ber o f clusters established from the (2, 9) range on the basis o f the DB index. T he above assum ptions were optim al for the d a ta from the original sample. Searching o f optim um p aram eters o f grouping fo r each b o o tstra p sam ple m ight im prove final results o f estim ation, but because o f a long period o f each sim ulation it was decided to retain uniform param eters in all sim ulations.
T o evaluate the properties o f estim ators o f the S Y d p aram eter in this study the m ean bias o f estim ato r in all s experim ents was used, calculated according to the follow ing form ula:
Í ( P f , i - & Y d)
B I A S r = — ---100 (6) s
where:
P/,i — the value o f the / - t h estim ato r in the i-th experim ent, & Y d - the real value o f p ro p o rtio n o f the featu re Y in d o m ain d. T he second elem ent o f the evaluation was the (square) ro o t o f the m ean sq u are erro r, calculated according to the follow ing form ula:
l t ( P / . i - & Y < ) 2
s qr (MS Ef ) = y — --- 100. (7)
T h e studied characteristics were the structu ral indices, th a t is why the bias and the m ean e rro r were expressed in percentage term s for th e sake of transparency.
A fter the experim ent the value o f the third relative m om ent was calculated, th a t is the m easures o f the skewness o f d istrib u tio n o f the acquired values o f estim ations and the fo u rth relative m om ent, being the m easu re o f flatness o f th e distrib u tio n .
V. A R E P R E S E N T A T IV E S I U D Y O F S M A L I. E N T E R P R IS E S IN T H E P O M E R A N IA N V O IV O D S H IP
T h e study o f the small business sector in the P om eran ian and the Lublin voivodships was carried ou t by an intern atio n al team o f scientists1. T he studied p o p u la tio n was m ad e o f sm all en terp rises in th e P o m e ran ia n voivodship em ploying betw een 10 and 49 people registered in th e R E G O N register on 30th Ju n e 1999. Some sectors were excluded from th e po p u latio n , such as th e E secto r - pow er gen eratio n in d u stry as well as public ad m in istratio n , h ealth services and education.
T he size o f the sample for the Pom eranian voivodship was calculated at the level o f 237 enterprises, i.e. about 5% o f the studied population. A questionna ire construed for the sake o f the study included 58 questions an d was divided into six sections. T h e sam ple received as a result o f interview s included 239 co m ponents. F o r the sake o f the present p ap e r we excluded 15 com ponents, which did n o t m eet certain criteria set at the m om en t o f designing the study project. These were firms, which - according to their R E G O N num ber carried activities in other areas th an selected for the project and firms, which failed to give answ ers to m any questions included in the questio nnaire.
T h e building sector is one o f the m ost essential sectors o f any econom y. Very often th e financial results and the level o f o u tp u t o f th a t sector are considered as the b arom eter o f the econom y. In p u blication s ab o u t the econom ic situ atio n changes in the level o f o u tp u t fo r th e w hole econom y are given to g eth er w ith inform ation ab o u t the level o f o u tp u t o f the c o n stru c tio n and building assem bly industry (Acs, 1996).
In the studied g ro u p o f 224 enterprises in the P o m e ran ia n voivodship 19 com panies (8 .5 % ) belong to the building sector (E K D co de beginning from 45). T his n u m b er is far insufficient for a credible d escriptio n o f the co n stru c tio n sector w ith the use o f direct estim ators. I t results from the potentially very high value o f the average e rro r o f estim atio n which m ay even reach th e level o f 11.5%. T h u s the description o f th a t sector should be based on o th e r m eth o d s o f estim ation, giving m o re credible results. One o f tho se possibilities is to consider th a t sector as a sm all d o m ain an d to apply the m eth o d s o f estim ation used for small do m ains.
VI. R E S U L T S O F T H E ST U D Y
In the T ab le 2 values o f the M S E ro o t fo r estim ations o f exem plary six variables are given:
- percentage o f firm s which have been established since 1994,
- firm s perceiving the advantage over their co m p etito rs in attractiveness o f th eir p ro ducts,
- firm s perceiving the advantage over their com p etito rs in high quality o f their o u tp u t,
- firm s, th a t incurred capital investm ent o utlays in 1999,
- firm s perceiving their chance in high skills o f th eir em ployees, - firm s perceiving their chance in good know ledge o f the m ark et. T he tw o last variables were characterized with relatively close levels in the popu latio n and in the d om ain. T he first four variables were characterized with quite high a difference between the value in the d o m ain and in the p o p u latio n reaching in the case o f the second feature th e level o f over 20 percentage points.
It m ay be perceived th a t even if the variance o f the M E S estim ato r (Table 1) is m uch lower in relation to the H T estim ato r, yet because o f the bias the m ean square erro r is usually larger. Only in the case o f estim ations o f the last tw o last variables the M E S estim ato r app ears to be m ore effective, but only in the case o f p aram eter у sm aller th an 0.5.
Table 1. V ariance o f estim ato rs (ro o t) using the neural n etw o rk s o f the SO M type depending o n the m axim um weight
У M E S p i M E S p 2 M E S p , M E S p Ą M E S p s M E S p A 1 3.89% 5.06% 5.18% 5.98% 5.24% 4.88% 0.5 3.99% 4.90% 5.28% 5.78% 5.19% 4.86% 0.3 4.28% 5.07% 5.65% 5.86% 5.40% 5.06% 0.2 4.68% 5.49% 6.18% 6.17% 5.79% 5.41% 0.1 5.60% 6.66% 7.48% 7.22% 6.89% 6.33% H T 8.59% 10.64% 11.72% 11.50% 10.83% 9.73%
T able 2. M e a n sq u are e rro r (ro o t) o f e stim ato rs usin g the SO M type n eu ral n e tw orks d ep en d in g on the m axim um weight
V M E S p t M E S Pl M E S p 3 M E S P t M E S p s M E S p 6 1 17.73% 15.81% 21.29% 13.30% 11.05% 11.40% 0.5 15.74% 13.95% 18.94% 11.86% 9.84% 10.19% 0.3 13.74% 12.10% 16.56% 10.40% 8.62% 8.96% 0.2 11.89% 10.42% 14.35% 9.04% 7.49% 7.80% 0.1 8.55% 7.41% 10.32% 6.53% 5.42% 5.68% H T * 7.5% 9.0% 8.2% 9.5% 9.4% 9.8% syn* 12.4% 12.2% 20.6% 17.0% 3.3% 3.8% * A p p ro x im ate values.
T able 3 . M ean sq u a re e rro r (ro o t) o f e stim ato rs using the g ro u p in g m eth o d o f /c-mcans d ep en d in g o n the m axim um weight
У M E S p , M E S p t M E S p 3 M E S Pi M E S p s M E S p , 1 17.12% 14.11% 24.22% 14.12% 10.83% 11.28% 0.5 15.47% 12.76% 21.87% 12.78% 9.82% 10.21% 0.3 13.74% 11.34% 19.40% 11.35% 8.76% 9.08% 0.2 12.07% 9.98% 17.03% 9.98% 7.73% 7.99% 0.1 8.90% 7.37% 12.53% 7.35% 5.74% 5.91%
T able 4. D ifference in e rro rs o f e stim ato rs calculated a cc o rd in g to the SO M m ethod and estim a to rs calculated with the use o f the /<-means m eth o d
У M E S p , M E S p 2 M E S p t M E S p s M £ S p fi 1 0.61% 1.70% -2 .9 3 % -0 .8 2 % 0.22% 0.12% 0.5 0.27% 1.18% -2 .9 3 % -0 .9 1 % 0 .02% -0 .0 2 % 0.3 0.00% 0.76% -2 .8 4 % -0 .9 5 % -0 .1 3 % -0 .1 3 % 0.2 -0 .1 8 % 0.44% -2 .6 8 % -0 .9 4 % -0 .2 3 % -0 .1 9 % 0.1 -0.35% 0.04% -2 .2 1 % -0 .8 2 % -0 .3 2 % -0 .2 3 %
G raph 1. V alues o f estim a to rs using the n e u ral n e tw o rk s o f th e SO M type and th e m eth o d o f g ro u p in g o f /с-m eans d e p en d in g on the m axim um w eight
C o m p arin g results acquired while using tw o so different m eth o d s of g ro uping it m ay be said th a t the calculated values o f estim ated param eters d o no t differ to o m uch (cf. G ra p h 1). T he highest observed value am ou nts to 4.8 percentage points for the fifth variable at the p aram eter y = 1. T he effectiveness o f estim ators rem ains as well at a sim ilar level, although in the case o f the third variable the m ethod o f g ro up in g fc-means appeared to be definitely less effective.
Exam ples o f received d istributions o f M E S estim ato rs fo r various values o f the estim ated param eter pd. were presented in G ra p h 2. T h e d istrib ution s o f estim ato rs arc characterised with relatively norm al flatness, aĄ in m ost cases was close to zero and in a great m ajority o f cases had a positive value, which m eans th a t the d istributions o f estim ators arc slim m er th an the no rm al d istrib u tio n . F o r the flattest d istrib u tio n the value o f a4 was equal to ab o u t -0.2. T h e acquired d istribu tions were also approxim ately sym m etrical, while the value o f the asym m etry increased in line with the decrease o f the p aram eter y. Besides th a t, at relatively high values o f the p aram eter у the distrib u tio n s could be considered as no rm al (x2 test at the division into 18 classes). C ertain disto rtio n s visible in the grap h result from a small n u m b er o f repetitions o f the sim ulation.
G raph 2. E xem plary d istrib u tio n s o f m odified synthetic estim ato rs and ap p ro x im atio n o f the n o rm al d istrib u tio n
VII. CONCLUSIONS
A pplication o f the M E S m odified synthetic estim ato r seems to be a good alternativ e to the estim ation o f param eters o f distribu tion s in small dom ains, in p artic u la r in those dom ains, which ra th e r significantly differ from the p o p u latio n . It is characterised w ith a relatively low v ariatio n, even if its bias m ay be quite considerable, in a vast m ajority o f cases it is sm aller th an the bias o f the synthetic estim ator. T h e d istrib u tio n o f the estim ato r in m any cases m ay be considered as norm al o r elose to norm al.
T h e choice o f th e m eth o d o f g ro u p in g seems to be o f secondary im po rtan ce, even if differences in effectiveness m ay be observed, the values o f estim ation o f p aram eters rem ain, how ever, a t a sim ilar level.
A n im p o rtan t issue is the establishm ent o f the way o f weighing additional inform ation. T h e change in param eter y, defining the m axim um value o f the weight resulted in quite m eaningful changcs b o th in the estim ation of p aram eters and the effectiveness o f estim ators. In the p aper weights related to the n u m b er o f appearances o f com pon ents from the sm all d o m ain in the class were applied. It seems th a t a better solu tio n would be to establish the w eight for each observation derived from outside o f the sm all dom ain individually, on the basis o f the distance o f cach com ponent from com ponents belonging to the small dom ain. T his m ethod, how ever, requires the presence o f an a p p ro p ria te nu m b er o f com p onents from the sm all d o m ain in the sam ple.
REFERENCES
A cs Z .J . (red.) (1996), S m a ll F irm s and Econom ic G row th, vol. 1, E lg ar Publishing L td , C h elten h am , E ngland.
B rach a С. (1996), Teoretyczne podstaw y m etody reprezentacyjnej, P W N , W arszaw a.
D av is D .L ., B o ulding D .W . (1979), A cluster sep aratio n m easure, IE E E Transactions on Pattern A nalysis a n d M achine Intelligence, P A M I-l, 2, 224-227.
Jurkiew icz T . (2001), Efficiency o f sm all d o m ain e stim ato rs fo r th e p o p u la tio n p ro p o rtio n : a M o n te C arlo analysis, S tatistics in Transition, 5, 2.
Tomasz Jurkiewicz, K rzysztof Najman
PROPOZYCJA ZASTOSOWANIA METODY KLASYFIKACJI ЛГ-ŚREDNICH O R A / SIECI NEURONOWEJ TYPU SOM
DO POPRAWY EFEKTYWNOŚCI ESTYMACJI DLA MALYCII DOMEN W REPREZENTACYJNYM BADANIU
MAŁYCH I ŚREDNICH PRZEDSIĘBIORSTW Streszczenie
P roblem zbyt małej liczby obserw acji w p ró b ie, reprezentującej o k reślo n ą d o m en ę populacji, m oże być rozw iązan y m iędzy innym i poprzez zasto so w an ie tak ich e sty m ato ró w , k tó re d o szacow ania p aram etró w w określonej supopulacji (małym obszarze, dom enie) m ogłyby wykorzystać in fo rm acje o innych jed n o s tk a ch w p ró b ie, k tó re p o ch o d zą spoza określonej części populacji. Jed n a z m eto d estym acji d la m ałych dom en zw ana estym acją sy ntetyczną zak ład a , że rozkład w bad an ej m ałej d o m en ie jes t identyczny z rozkładem całej po p u lacji. Z ałożenie to pozostaje zazwyczaj niespełnione, zwłaszcza w przypadku specyficznych dom en, co skutkuje dużym i błędami estym acji.
A u to rz y p rzed sta w iają propozycję dw u etap o w eg o p ro cesu estym acji. W pierw szym etapie za p o m o c ą sieci n eu ro n o w y ch typu SO M o ra z za p o m o cą m etody klasyfikacji /c-średnich o k reśla się p o d o b ień stw a jed n o ste k należących d o m ałej d o m en y d o jed n o ste k z pozostałej części p ró b y . D ru g im k ro k ie m jes t w y k o rzy stan ie w estym acji, za p o m o c ą o d p o w ied n io sk o n stru o w an y c h wag, inform acji tylko z tych do m en , k tó re są p o d o b n e d o bad an ej małej dom en y . A u to rzy p rz ed staw iają re zu lta ty zasto so w an ia p o d a n ej p ro c ed u ry w analizie branży b udow lanej n a pod staw ie w yników reprezentacyjnego b a d an ia m ałych i średnich przedsiębiorstw . P odjęli także p ró b ę o szacow ania błędów tak zm odyfikow anej m eto d y estym acji syntetycznej.
