Scentfc Annals of Economcs and Busness 64 (), 07, 7-85 DOI: 0.55/saeb-07-00 REGIONAL INCOMES STRUCTURE ANALYSIS IN SLOVAK REPUBLIC ON THE BASIS OF EU-SILC DATA Mlan TEREK * Abstract The paper deals wth the regonal ncomes structure analyss n Slovak republc on the bass of European Unon statstcs on ncome and lvng condtons n Slovak republc data. The emprcal probablty mass functon and emprcal cumulatve dstrbuton functon s constructed wth ad of gven samplng weghts. On the bass of these functons the medan, medal, standard devaton and populaton hstogram of the whole gross household ncomes for the whole Slovak republc and separately for eght Slovak regons are estmated and compared. Keywords: regonal ncomes structure, samplng weghts, emprcal probablty mass functon, emprcal cumulatve dstrbuton functon JEL classfcaton: C83, R9. INTRODUCTION The regonal structure of ncomes n Slovak republc wll be analyzed on the bass of data from the European Unon Statstcs on Income and Lvng Condtons (EU-SILC) realzed n Slovak republc n the year 04. EU-SILC s an nstrument amng at collectng tmely and comparable cross-sectonal and longtudnal multdmensonal mcrodata on ncome, poverty, socal ecluson and lvng condtons. Ths nstrument s anchored n the European Statstcal System. The start of the EU-SILC nstrument was n 004 for the EU- 5. In Slovak republc EU-SILC s yearly realzed from the year 005. In general EU-SILC data are the data from comple survey. The survey contanng more components such as random samplng, stratfcaton, clusterng and so on s obvously called comple survey. A survey may be stratfed wth several stages of clusterng and rely on rato and regresson estmaton to adjust for other varables. In these cases samplng weghts based on aulary nformaton are commonly used to provde the correct results. The analyss of the regonal structure of ncomes n Slovak republc based on usng samplng weghts wll be studed n the paper. The usng samplng weghts n constructon * Department of Statstcs, Unversty of Economcs n Bratslava, Slovaka; e-mal: mlan.terek@gmal.com.
7 Terek, M. of emprcal probablty mass functon and emprcal cumulatve dstrbuton functon wll be descrbed. On the bass of these functons the estmaton of populaton hstogram, medan, medal and standard devaton of the whole gross household ncomes for the whole Slovak republc and separately for eght domans Slovak regons was realzed.. MATERIAL AND METHODS There are a lot of papers concernng the ncomes dstrbuton estmaton and structure analyss, studyng these problems from dfferent regards. For eample n Kloek and van Djk (978) the estmaton of ncome dstrbuton parameters s studed. The parameters of several famles of dstrbutons are estmated by means of mnmum. The approach s appled on random samples taken from Dutch ncome-earnng groups n 973. In Ghosh et al. (996) a general methodology for small doman estmaton based on data from repeated surveys s studed. The results are drectly appled to the estmaton of medan ncome of four-person famles for the 50 states and the Dstrct of Columba n the USA. In Sala--Martn (006), the world dstrbuton of ncome by ntegratng ndvdual ncome dstrbutons for 38 countres between 970 and 000 s effectuated. In Dowrck and Akmal (005) the global ncome nequalty s studed. In Wang and Woo (0) the sze and dstrbuton of hdden household ncome n Chna are analyzed. In Cowell and Flachare (007) the statstcal performance of nequalty ndces n the presence of etreme values n the data s analyzed. It s shown that these ndces are very senstve to the propertes of the ncome dstrbuton. Estmaton and nference can be dramatcally affected, especally when the tal of the ncome dstrbuton s heavy, even when standard bootstrap methods are employed. In Atknson and Salverda (005) a method for usng ncome-ta data to nvestgate the evoluton of the hghest ncomes over vrtually the entre 0 th century s developed. In Chotkapanch et al. (007) the natonal and regonal ncome dstrbutons are estmated wthn a general framework that relaes the assumpton of constant ncome wthn groups. A technque to estmate the parameters of a beta- dstrbuton usng grouped data s proposed. Dstrbutons of ncomes or wages are obvously skewed and outlers 3 are present. Then, the nterpretaton power of the mean s very small 4. Generally n such dstrbutons the mean s not consdered as approprate measure of central tendency. Then the mean ncome s not convenent measure of typcal ncome. The medan s generally consdered as good measure of central tendency n such dstrbutons because of ts stablty and robustness toward outlers. Alternatvely some non-tradtonal measures of locaton could be also nterestng as good measures of central tendency for such dstrbutons. The usng of the trmmed mean (Pegorsch, 05, p. 55), Wnsorzed mean or M-estmators s recommended 5. Interestng results provdes also tradtonal measures of central tendency appled on the data set from whch the outlers were removed 6. Sometmes the standard statstcal methods supposng the ndependence and dentc dstrbuton of observatons are appled to the data from comple surveys. In Lohr (00, pp. 87-88) s stated: When you read the paper or book n whch the authors analyze data from the comple survey, see whether they accounted for the data structure n the analyss, or whether they smply ran the raw data through non-survey statstcal package procedure and reported the results. If the latter, ther nferental results must be vewed wth suspcon.
Scentfc Annals of Economcs and Busness, 07, Vol. 64, Issue, pp. 7-85 73. Samplng weghts The samplng weghts 7 allow to construct an emprcal dstrbuton for the populaton 8. On the bass of ths dstrbuton the estmaton of medan and other quantles 9, medal, standard devaton and populaton hstogram s also possble. The samplng weghts can be calculated usng aulary nformaton. Suppose we know the sze N of fnte populaton U. Symbol denotes varable under study and also ts values, U =,,...N s the set of unt ndees n the populaton. Symbol S denotes sample from the populaton subset contanng n unts from U. Let's be the probablty that unt U wll be n random sample. Samplng weghts for any samplng desgn are defned as follows: w () Samplng weght of unt can be nterpreted as number of unts n the populaton represented by unt. The estmators of the parameters n cluster samplng, stratfed samplng, and other samplng desgns ncludng ts combnatons such as for eample multstage stratfed samplng can be epressed by samplng weghts. These weghts can be modfed n regard to nonresponse and coverage error 0. Samplng weghts for all observatons unts are equal n self-weghtng surveys. Each observed unt represents the same number of unobserved unts n the populaton. Samplng weghts are not equal for all observatons unts n non-selfweghtng surveys. If the sample s non-self-weghtng, pont estmates of means, totals and other quanttes produced by standard statstcal software wll be based. It s the case also n mentoned applcaton. The EU-SILC sample s non-self-weghtng. The capturng the structure of data s necessary n pont estmaton of populaton quanttes. The usng of samplng weghts s needed.. Estmatng an emprcal probablty mass functon and emprcal cumulatve dstrbuton functon Suppose the values for the entre populaton of N unts are known. A value of probablty mass functon (PMF) n s N p () N where N s number of unts whose value s. A value of cumulatve dstrbuton functon (CDF) n s y F p (3) y
74 Terek, M. Note that t s probablty mass functon and cumulatve dstrbuton functon of observaton from the populaton because the model-free or dstrbuton-free approach to sample survey s under consderaton. Samplng weghts allow to construct emprcal probablty mass functon and emprcal cumulatve dstrbuton functon. Emprcal probablty mass functon pˆ s defned by the sum of weghts for all observatons takng on the value dvded by the sum of all the weghts: w S ; pˆ (4) w S Emprcal cumulatve dstrbuton functon Fˆ s y Fˆ pˆ (5) y.3 Plottng data from a comple survey One from the smplest plots dsplayng the data dstrbuton s hstogram. If a sample s self-weghtng, a regular hstogram of the sample data wll estmate the populaton probablty mass functon. If a sample s non-self-weghtng the samplng weghts are used to construct a hstogram that estmates the populaton hstogram. The range of the data s dvded nto k classes wth each class havng wdth b. The heght of the hstogram n class j s Heght (j) = w u S b w where u j = f observaton s n class j and 0 otherwse. The denomnator n formula ensures that the total area under the hstogram equals. Such heghts are obvously called the denstes of relatve frequences (Wonnacott and Wonnacott, 984, p. 07). S.4 Estmatng of some populaton quanttes The populaton parameters can be calculated on the bass of probablty mass functon. For eample populaton varance s j (6) N K N p N p p K p p where s populaton mean. K N (7)
Scentfc Annals of Economcs and Busness, 07, Vol. 64, Issue, pp. 7-85 75 Any populaton quantty can be estmated from the emprcal probablty mass functon pˆ or from emprcal cumulatve dstrbuton functon Fˆ. For eample populaton varance can be estmated by: ˆ pˆ pˆ (8) K The fnte populaton medan s defned to be value ~ satsfyng F ~ f such a value ests. Otherwse, a populaton medan s any value n the nterval ~, ~, where ~ s the largest value of n the populaton wth ~ s smallest value of wth F < and F >. In general, Q s 00 p % quantle (percentle) f p F Q p = p, f such a value ests, otherwse, Q a,b where a s the largest populaton value of wth F < p p and b s the smallest value of wth F > p. If p <, Q s the smallest value of and p N f p > N, Q s the largest value of. p Populaton quantles are estmated as follows. Snce the emprcal cumulatve dstrbuton functon Fˆ s a step functon, the nterpolaton s usually needed to fnd a unque value for the quantle. Let y be the largest value n the sample for whch F ˆ y p and let y s smallest value n the sample for whch ˆ y F p. Then: Qˆ = y p + Fˆ p Fˆ y ˆ y y y F y (9) We wll formulate the relatons enablng to estmate medal wth ad of samplng weghts. Medal (Ml) s such value for whch the sum of varable values less or equal to Ml s equal to the half of varable total. It can be proven that f all values of varable are nonnegatve then: Ml Q (Dagnele, 998, p. 8). The sum of varable values for all 0, 5 observatons takng on the value we wll call the class total. The medal s calculated as medan but on the bass of class totals nstead of frequences. Emprcal probablty mass functon ˆ n ths case can be defned as: p Ml S ; pˆ Ml (0) w S w Emprcal cumulatve dstrbuton functon F Ml ˆ s then: y Fˆ Ml pˆ Ml () y
76 Terek, M. Let y be the largest value n the sample for whch Fˆ Ml y 0,5 and let value n the sample for whch Fˆ Ml y 0,5. Then the medal can be estmated by: y s smallest Mˆ l = y + 0,5 Fˆ Ml Fˆ y Ml y y y Fˆ y Ml () The medal provdes n some applcaton areas very nterestng nterpretaton possbltes. Note that estmators constructed usng ths method are not necessarly unbased or numercally stable. For eample the estmator ˆ K of the populaton varance s senstve to round off error. Despte of t, the statstcs calculated usng weghts are much closer to the populaton quanttes as n not weghtng case (Lohr, 00, p. 93). 3. ANALYSIS OF REGIONAL STRUCTURE OF INCOMES ON THE BASIS OF EU-SILC 04 DATA The analyss of regonal structure of ncomes was effectuated on the data from the survey EU-SILC realzed n Slovak republc n 04 (EUROSTAT, 007). The stratfed two-stage survey s used n Slovak republc. A stratfcaton was effected wth two stratfcaton varables regon and settlement sze. There are eght regons n Slovak republc. Bratslava, Trnava, Trenčín and Ntra n western Slovaka, Žlna and Banská Bystrca n central Slovaka, Košce and Prešov n eastern Slovaka. The survey EU-SILC 04 was effectuated on the sample of 6,00 households, 5,490 households and 3,433 ndvduals 6 and more years old were ncluded to database. Samplng weghts were calculated and modfed wth respect to nonresponse. These weghts can be used to nference about the populaton of Slovak households. Other modfed samplng weghts nvolve ndvduals. In general EU-SILC sample data are the data from non-self-weghtng survey. Data from EU-SILC 04 are concentrated n many sets. Each household has one dentfcaton number. The analyss of the whole gross household ncomes n eght domans Slovak regons was realzed. The mentoned regons correspond wth values of one from stratfcaton varables. Frstly the matchng of needed data samplng weghts and whole gross household ncomes was effected accordng to household numbers. Then the matched data were dstrbuted accordng to regons. Eght sets of data were obtaned, one for each regon. Each regon was analyzed separately. The values of the emprcal probablty mass functon were calculated accordng to (4) and on the bass of that the values of the emprcal cumulatve dstrbuton functon were calculated by relaton (5) for the whole Slovak republc and separately for each regon. The estmate of medan whole gross household ncome was calculated accordng to relaton (9) for the whole Slovak republc and separately for each regon. The estmate of populaton medan whole gross household ncome for the whole Slovak republc n the year 04 equals 3,305.83 euros. The obtaned results for regons are n Table no.. Ths Table contans also the orderng of regons accordng to the medan whole gross household ncome. Then the values of the emprcal probablty mass functon pˆ Ml were calculated accordng to (0) and on the bass of that the values of the emprcal cumulatve dstrbuton functon ˆ were calculated by relaton () for the whole Slovak republc and F Ml
Scentfc Annals of Economcs and Busness, 07, Vol. 64, Issue, pp. 7-85 77 separately for each regon. The estmate of medal whole gross household ncome was calculated accordng to relaton () for the whole Slovak republc and separately for each regon. The estmate of medal whole gross household ncome n 04 for the whole Slovak republc was 0,355.80 euros and percentage of households havng ncomes less or equal to medal n Slovak republc was 74.5%. The obtaned results for regons are n Table no.. For eample n the year 04, n regon Bratslava, the half of the ncomes total was dstrbuted among 76.7 % of poorer households (havng ncomes less or equal to 4,874.65 euros), the second half of ncomes total was dstrbuted among 3.9 % of rcher households (havng ncomes greater or equal to 4,874.65 euros). For eample n regon Banská Bystrca, the half of the ncomes total was dstrbuted among 74,88 % of poorer households (havng ncomes less or equal to 8,30.86 euros), the second half of ncomes total was dstrbuted among 5. % of rcher households (havng ncomes greater or equal to 8,30.86 euros). Regon number Table no. Regonal structure of medan whole gross household ncome n 04 Regon name Estmate of medan whole gross household ncome n 04 (Euros) Order of regon accordng to medan whole gross household ncome Bratslava 4,49.37. Trnava 3,969. 4. 3 Trenčín 4,368.47. 4 Ntra,379.67 7. 5 Žlna 4,054.85 3. 6 Banská Bystrca,746.4 8. 7 Prešov 3,595. 5. 8 Košce 3,8.6 6. Source: own Regon number Table no. Regonal structure of medal whole gross household ncome n 04 Regon name Estmate of medal whole gross household ncome n 04 (Euros) Order of regon accordng to medal whole gross household ncome Percentage of households havng ncomes less or equal to medal Bratslava 4,874.65. 76.7 Trnava 0,33.04 5. 7.44 3 Trenčín 0,555.89. 73.40 4 Ntra 9,364.40 6. 74.6 5 Žlna 0,49.4 3. 73.88 6 Banská Bystrca 8,30.86 7. 74.88 7 Prešov 0,489.5 4. 73.34 8 Košce 8,93.63 8. 7.90 Source: own The dsperson of ncomes n regons was characterzed by standard devaton. The populaton varance was estmated accordng to (8). Then the standard devaton estmate was calculated as square root of varance estmate. The estmate of standard devaton of the whole gross household ncome n 04 for the whole Slovak republc was,893.55 euros. The obtaned results for regons are n Table no. 3.
78 Terek, M. Table no. 3 Regonal structure of standard devaton of the whole gross household ncome n 04 Regon number Regon name Estmate of Standard devaton of the whole gross household ncome n 04 (Euros) Order of regon accordng to standard devaton of the whole gross household ncome Bratslava 7,74.84. Trnava 0,53.97 7. 3 Trenčín,808.75. 4 Ntra,40.05 3. 5 Žlna,06.9 4. 6 Banská Bystrca 0,933.77 5. 7 Prešov 0,65.63 6. 8 Košce 8,970.3 8. Source: own Fnally, the samplng weghts were used to construct a hstogram that estmates the populaton hstogram for the whole Slovak republc and for each regon separately. The wdths of classes n hstograms are 0,000 euros. The denstes of relatve frequences n hstograms were calculated accordng to (6). The area of rectangle n the hstogram s equal to relatve frequency. The number of Slovak households n 04 can be estmated by the sum of samplng weghts: w,850,84. When the number of households s known, the S number of households n the classes can be easly calculated. In Table no. 4 are denstes of relatve frequences and numbers of households n defned classes. Table no. 4 Dstrbuton estmatng dstrbuton of Slovak households accordng to the whole gross household ncome n 04 Income (euros) Densty of relatve frequency Number of households -0,000 0.0000368000 68,765 0,000-0,000 0.0000364000 673,063 0,000-30,000 0.000078000 39,995 30,000-40,000 0.0000056500 04,565 40,000-50,000 0.000009700 36,54 50,000-60,000 0.000000640,86 60,000-70,000 0.00000070 5,038 70,000-80,000 0.0000000747,38 80,000-90,000 0.0000000890,647 90,000-00,000 0.0000000968,79 00,000-0.000000730 3,95 Sum,850,84 Source: own The hstogram estmatng populaton hstogram for the Slovak republc s presented n Fgure no.. As can be seen n Fgure no., the greatest proporton of households n Slovak republc has ncomes less or equal to 0,000 euros, the households havng ncomes greater than 0,000 and less or equal to 0,000 euros are also frequent. The proportons of ncomes greater than 70,000 euros are not dscernble n the hstogram.
Scentfc Annals of Economcs and Busness, 07, Vol. 64, Issue, pp. 7-85 79 Fgure no. Hstogram of the Slovak households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households has ncomes less or equal to 0,000 euros also n Bratslava regon (n Fgure no. ), but n ths hstogram also the households havng ncomes greater than 70,000 euros are n dscernble proportons. The number of households n Bratslava regon s estmated to be equal to 45,997. Fgure no. Hstogram of the Bratslava regon households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households n the regon Trnava has ncomes greater than 0,000 and less or equal to 0,000 euros (see Fgure no. 3). That s dfference n comparson to the whole Slovak republc and also to Bratslava regon. The dscernable proportons of ncomes greater than 70,000 and less or equal than 80,000 and also greater than 90,000 and less or equal to 00,000 are present n that regon. The number of households n Trnava regon s estmated to be equal to 9,08.
80 Terek, M. Fgure no. 3 Hstogram of the Trnava regon households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households n the regon Trenčín has ncomes less or equal to 0,000 that s the most frequent are the poorest households but on the other hand there are also the households havng ncomes greater than 00,000 euros n dscernble proporton (see Fgure no. 4). The number of households n Trenčín regon s estmated to be equal to 07,585. Fgure no. 4 Hstogram of the Trenčín regon households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households n the regon Ntra has ncomes greater as 0,000 and less or equal to 0,000 euros (see Fgure no. 5). The dstrbuton s very smlar to dstrbuton of Trnava regon, only dfference s the dscernble proporton of the rchest households, havng ncomes greater than 00,000 euros. The number of households n Ntra regon s estmated to be equal to 44,85.
Scentfc Annals of Economcs and Busness, 07, Vol. 64, Issue, pp. 7-85 8 Fgure no. 5 Hstogram of the Ntra regon households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households n the regon Žlna has ncomes greater as 0,000 euros and less or equal to 0,000 euros (see Fgure no. 6). There s not the dscernable proporton of households havng ncomes greater than 80,000 euros. The number of households n Žlna regon s estmated to be equal to 8,788. Fgure no. 6 Hstogram of the Žlna regon households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households n the regon Banská Bystrca has ncomes less or equal to 0,000 euros (see Fgure no. 7). There s also the dscernable proporton of households havng ncomes more than 80,000 euros. The number of households n Banská Bystrca regon s estmated to be equal to 39,708.
8 Terek, M. Fgure no. 7 Hstogram of the Banská Bystrca regon households dstrbuton accordng to the whole gross ncome n 04 Fgure no. 8 Hstogram of the Prešov regon households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households n the regon Prešov has ncomes less or equal to 0,000 euros (see Fgure no. 8). There s not the dscernble proporton of households havng ncomes greater than 60,000 euros. On the other hand the proporton of households wth mddle ncomes, for eample greater than 0,000 and less or equal to 30,000 s about 0%, n Banská Bystrca regon only less than 5%. The number of households n Prešov regon s estmated to be equal to 37,454.
Scentfc Annals of Economcs and Busness, 07, Vol. 64, Issue, pp. 7-85 83 Fgure no. 9 Hstogram of the Košce regon households dstrbuton accordng to the whole gross ncome n 04 The greatest proporton of households n the regon Košce has ncomes greater than 0,000 euros and less or equal to 0,000 euros (see Fgure no. 9). There s not the dscernble proporton of households havng ncomes greater than 60,000 euros. The number of households n Košce regon s estmated to be equal to 65,467. 4. CONCLUSIONS All calculatons were realzed n Ecel 03. The obtaned orderng of regons accordng to medan whole gross household ncome s very nterestng. Obvously the great dfference among Bratslava regon wth the captal of Slovaka Bratslava and the rest of Slovaka s epected. The analyss results show that the dfference between frst Bratslava and second Trenčín regons s not very large. The medan household ncomes of the thrd Žlna, fourth Trnava and ffth Prešov are also very close. There are bgger dfferences among last three regons. The medan whole gross household ncome of Banská Bystrca s surprsngly low. The regons Bratslava, Trenčín, Žlna, Trnava, Prešov have the medan whole gross household ncome greater and the regons Košce, Ntra and Banská Bystrca less than n the whole Slovak republc. In the analyss based on medal whole gross household ncome, the results of regonal orderng are not very dfferent. The frst three places are occuped by the same regons as accordng to medan, the changes n the other places of orderng are only moderate. The regons Bratslava, Trenčín, Žlna, Prešov have the medal whole gross household ncome greater and the regons Trnava, Košce, Ntra and Banská Bystrca less than n the whole Slovak republc. The fndng that n all Slovak regons the half of the ncomes total s dstrbuted among 7.90 76.7% of poorer households and the second half of ncomes total s dstrbuted among the rest of rcher households s very nterestng. The dfferences among Slovak regons n ths ndcator are only moderate. The regons Bratslava, Ntra and Banská Bystrca have percentage of households havng ncomes less or equal to medal greater and the regons Trnava, Trenčín, Žlna, Prešov and Košce less than Slovak republc percentage.
84 Terek, M. The regonal orderng accordng to standard devaton follows appromately the orderng accordng to medan and medal. Only Bratslava regon has markedly greater standard devaton of ncomes 7,74.84 euros, what s natural because ths regon ncludes the captal of Slovaka wth a lot of central nsttutons. Others regons have not very dfferent standard devatons of ncomes. The dstrbuton of Slovak households accordng to the whole gross household ncome n 04 s nterestng on the level of the whole Slovak republc as well as on the level of regons. The obtaned nformaton can be very useful for eample for some marketng studes. The applcaton of correct methodology of estmaton s very mportant n the contet of the data from comple surveys analyses. It s clear that the estmates obtaned wth ad of fnte weghts whch allow the used sample desgn, nonresponse and potentally also coverage error better reflects the realty. Acknowledgements The paper was supported by grants from Grant Agency of VEGA no. /009/5 enttled Modern Approaches to Comple Statstcal Surveys Desgn and no. /0393/6 enttled European Unon n Post Crss Perod Macro and Mcroeconomc Aspects. References Atknson, A. B., and Salverda, W., 005. Top Incomes n the Netherlands and the Unted Kngdom over the 0th Century. Journal of the European Economc Assocaton, 3(4), 883-93. do: http://d.do.org/0.6/5447605443086 Barnett, V., and Lews, T., 994. Outlers n Statstcal Data. Hoboken: Wley and Sons. Chotkapanch, D., Grffths, W. E., and Rao, D. S. P., 007. Estmatng and Combnng Natonal Income Dstrbutons Usng Lmted Data. Journal of Busness & Economc Statstcs, 5(), 97-09. Cochran, W. G., 977. Samplng Technques. New York: J. Wley and Sons. Cowell, F. A., and Flachare, E., 007. Income dstrbuton and nequalty measurement: The problem of etreme values. Journal of Econometrcs, 4(), 044-07. do: http://d.do.org/0.06/j.jeconom.007.0.00 Dagnele, P., 998. Statstque Theorque et Applquee. Tom - Statstque Descrptve et Bases de l' Inference Statstque. Pars: DeBoeck and Larcer. Dowrck, S., and Akmal, M., 005. Contradctory Trends n Global Income Inequalty: A Tale of Two Bases. Revew of Income and Wealth, 5(), 0-9. do: http://d.do.org/0./j.475-499.005.005. EUROSTAT, 007. European Unon Statstcs on Income and Lvng Condtons. from http://ec. europa.eu/eurostat/web/mcrodata/european-unon-statstcs-on-ncome-and-lvng-condtons Ghosh, M., Nanga, N., and Km, D. H., 996. Estmaton of Medan Income of Four-Person Famles: A Bayesan Tme Seres Approach. Journal of the Amercan Statstcal Assocaton, 9(436), 43-43. do: http://d.do.org/0.307/9568 Halley, R. M., 004. Measures of Central Tendency, Locaton, and Dsperson n Salary Survey Research. Compensaton & Benefts Revew, 36(5), 39-5. do: http://d.do.org/0.77/088636870468598 Kloek, T., and van Djk, H. K., 978. Effcent estmaton of ncome dstrbuton parameters. Journal of Econometrcs, 8(), 6-74. do: http://d.do.org/0.06/0304-4076(78)90090-8 Levy, P. S., and Lemeshow, S., 008. Samplng of Populatonas. Methods and Applcatons (4th ed. ed.). Hoboken: Wley and Sons. do:http://d.do.org/0.00/9780470374597 Lohr, S. L., 00. Samplng: Desgn and Analyss (nd ed. ed.). Boston: Brooks/Cole.
Scentfc Annals of Economcs and Busness, 07, Vol. 64, Issue, pp. 7-85 85 Pegorsch, W. W., 05. Statstcal Data Analyss. Foundatons for Data Mnng, Informatcs, and Knowledge Dscovery. Chchester: Wley and Sons. Sala--Martn, X., 006. The World Dstrbuton of Income: Fallng Poverty and Convergence, Perod*. The Quarterly Journal of Economcs, (), 35-397. do: http://d.do.org/0.6/qjec.006...35 Terek, M., 06. Odľahlé dáta a charakterstky polohy v analýzach mezd a príjmov. Revue socálnoekonomckého rozvoja : vedecký recenzovaný on-lne časops, (), 4-6. Tosenovsky, J., and Noskevcova, D., 000. Statstcke metody pro zlepsovan jakost. Ostrava: Montane. Wang, X., and Woo, W. T., 0. The Sze and Dstrbuton of Hdden Household Income n Chna. Asan Economc Papers, 0(), -6. do: http://d.do.org/0.6/asep_a_00064 Wonnacott, T. H., and Wonnacott, R., 984. Statstcs for Busness and Economcs. New York: Wley and Sons. Notes More n detals, see n: European Unon Statstcs on Income and Lvng Condtons (EU-SILC), avalable at EUROSTAT (007). Doman can be defned as subpopulaton. 3 We shall defne an outler n a set of data to be an observaton (or subset of observatons) whch appears to be nconsstent wth the remander of that set of data (Barnett and Lews, 994, p. 7). 4 More n detals see n Halley (004, pp. 39-5). 5 More n detals see n Terek (06). 6 More n detals see for eample n Terek (06). 7 Alternatvely the term desgn weghts s used. 8 In fact, t s an emprcal dstrbuton of the observaton from the populaton. 9 More n detals see for eample n Tosenovsky and Noskevcova (000). 0 More n detals see for eample n Levy and Lemeshow (008). For more detals, see n Cochran (977, pp. 8-9). Copyrght Ths artcle s an open access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton-NonCommercal-NoDervatves 4.0 Internatonal Lcense.