1 Wprowadzenie

Cel: wyznaczenie obszaru ufności dla dystrybuanty nieznanego rozkładu, a nie tylko oszacowania parametrów, od jakich zależą jej wartości.

  • Brzegi tego obszaru są wykresami funkcji „przedziałami stałych” (funkcji schodkowych).

  • Jeżeli przy wyznaczaniu pasma ufności dla dystrybuanty otrzymamy lewy kraniec przedziału będący liczbą ujemną, to zastępujemy ją przez zero.

  • Jeżeli otrzymamy prawy kraniec przedziału większy od jedności, to przyjmujemy, że jest on równy jeden.

  • Określenie obszaru ufności dla dystrybuanty w przedstawiony sposób polega na wyznaczeniu przedziałowego oszacowania dla każdej wartości dystrybuanty.

1.1 Funkcja CDF

Funkcja w programie R odpowiedzialna za estymację to np. CDF z pakietu spatstat. CDF jest metodą ogólną, z metodą dla klasy “gęstość”.

Oblicza ona skumulowaną funkcję rozkładu, której gęstość prawdopodobieństwa została oszacowana i zapisana w obiekcie f. Obiekt f musi należeć do klasy “gęstość” i zazwyczaj zostałby uzyskany z wywołania funkcji gęstość.

1.2 Funkcja kde

Pakiet R o nazwie snpar zawiera kilka uzupełniających metod statystyki nieparametrycznej, w tym test kwantylowy, test trendu Coxa-Stuarta, test przebiegów, test normalnego wyniku, estymację jądra PDF i CDF, estymację regresji jądra i test jądra Kołmogorowa-Smirnowa.

Funkcja kde zawiera obliczanie zarówno nieparametrycznego estymatora jądra funkcji gęstości prawdopodobieństwa (PDF) jak i funkcji rozkładu skumulowanego (CDF).

1.3 Przykład 1.

b <- density(runif(10))
   f <- CDF(b)
   f(0.5)
## [1] 0.6478766
   plot(f)

1.4 Przykład 2.

x <- rnorm(200,2,3)
# with default bandwidth
kde(x, kernel = "quar", plot = TRUE)

## $data
##   [1]  3.90455452 -1.91369824  1.22820886  1.46819552  5.63166736  6.91427126
##   [7]  5.48163439 -5.45141272  2.93552853  2.89562635 -1.05495169  5.35192233
##  [13]  0.37498662  1.01865947 -0.32363796  1.00344704  4.45841879  3.38173957
##  [19]  0.74847748  2.24521582  6.48910178  2.36840092  0.22214504  0.44364591
##  [25]  0.79809201  2.28274280  5.89669784  2.44395968 -1.85442759  4.97098632
##  [31]  0.36141065  3.60594031  4.76435641 -1.12502282  6.83207972 -3.91121283
##  [37] -2.63339626  3.82211309  3.14628397 -4.04850000 -0.91690213  0.14294124
##  [43]  5.52111572  1.80868983  2.86302995 -0.26843978 -1.42190716  2.31005114
##  [49] -0.40113946  5.43128285  3.69678655  1.33001145  5.21504527 -0.62349035
##  [55]  2.88007901 -0.74323326  1.36277285  0.42727446  1.17366782 -0.79852075
##  [61]  3.21596881 -4.47105201 -6.39044682  2.91786721 -3.85662513 -1.51880869
##  [67]  2.49112010 -2.61244295  4.24057943  1.41519240  6.92991892  5.51183897
##  [73] -1.01155342  1.77378060  2.84827946  1.59314745  6.02538439  1.10323083
##  [79]  4.48485133 -5.67766821 -1.03879766 -1.35076231 -0.65098927  4.24135425
##  [85]  6.54374167  2.23885965  0.41253238 -2.40022792 -1.54927704 -2.37965840
##  [91]  3.95764191 -1.21614326  0.23756913  6.61623707  1.65515834  1.04927866
##  [97]  4.88941331  1.53290879  5.53738000  2.05567121 -0.92179853 -0.97409642
## [103]  0.17866489  6.77133792 -1.55825256  1.68765816  3.28061456  2.86499654
## [109]  2.19998139 11.73605476  1.52467892  4.02313678  5.47353160  1.13747245
## [115]  4.16654142  9.10820259 -5.46500865 -1.79440898  2.56425505 -1.39390186
## [121]  0.23007476 -3.99895092 -1.45734995  7.46515806  3.53722442  1.82547942
## [127]  4.88631894  6.04352045 -0.14341510  0.37015330  4.90338854 10.84330211
## [133] -0.97987273  3.67230310 -0.78142080 -0.11728579  4.70368117 -1.81378270
## [139]  0.33935213  4.28501857  0.48043073  5.35541005 -2.76109584  5.89429209
## [145] -1.59771850 -0.37566400  4.89317840  5.48245628  4.93842612  2.79243673
## [151] -1.45772315 -0.08004092  0.19345526 -2.96323224  4.86964199 -1.21694782
## [157]  2.99665453  3.11649258  4.96881130  3.80661872  0.88997840 -0.95237863
## [163]  3.08480613  0.81712068  2.86671994  2.64944198  4.50875638  5.35981965
## [169] -2.85980453 -1.64423863  2.51436127  1.49247599  6.10799414  0.33944416
## [175]  2.97106387  0.71106992 -0.29075462  1.37143274 -0.47236887  2.10704489
## [181]  1.40448112 -3.17719714  0.39949405  2.99928699 -3.16144574  1.21477131
## [187]  5.94048071  8.01600463  2.57942313 -0.81503937 -0.68497820  3.23387189
## [193]  5.82533012  1.62976075 -0.63655923  5.78145764 -0.05343094  6.21889911
## [199]  4.23001519  2.61247125
## 
## $xgrid
##   [1]  3.90455452 -1.91369824  1.22820886  1.46819552  5.63166736  6.91427126
##   [7]  5.48163439 -5.45141272  2.93552853  2.89562635 -1.05495169  5.35192233
##  [13]  0.37498662  1.01865947 -0.32363796  1.00344704  4.45841879  3.38173957
##  [19]  0.74847748  2.24521582  6.48910178  2.36840092  0.22214504  0.44364591
##  [25]  0.79809201  2.28274280  5.89669784  2.44395968 -1.85442759  4.97098632
##  [31]  0.36141065  3.60594031  4.76435641 -1.12502282  6.83207972 -3.91121283
##  [37] -2.63339626  3.82211309  3.14628397 -4.04850000 -0.91690213  0.14294124
##  [43]  5.52111572  1.80868983  2.86302995 -0.26843978 -1.42190716  2.31005114
##  [49] -0.40113946  5.43128285  3.69678655  1.33001145  5.21504527 -0.62349035
##  [55]  2.88007901 -0.74323326  1.36277285  0.42727446  1.17366782 -0.79852075
##  [61]  3.21596881 -4.47105201 -6.39044682  2.91786721 -3.85662513 -1.51880869
##  [67]  2.49112010 -2.61244295  4.24057943  1.41519240  6.92991892  5.51183897
##  [73] -1.01155342  1.77378060  2.84827946  1.59314745  6.02538439  1.10323083
##  [79]  4.48485133 -5.67766821 -1.03879766 -1.35076231 -0.65098927  4.24135425
##  [85]  6.54374167  2.23885965  0.41253238 -2.40022792 -1.54927704 -2.37965840
##  [91]  3.95764191 -1.21614326  0.23756913  6.61623707  1.65515834  1.04927866
##  [97]  4.88941331  1.53290879  5.53738000  2.05567121 -0.92179853 -0.97409642
## [103]  0.17866489  6.77133792 -1.55825256  1.68765816  3.28061456  2.86499654
## [109]  2.19998139 11.73605476  1.52467892  4.02313678  5.47353160  1.13747245
## [115]  4.16654142  9.10820259 -5.46500865 -1.79440898  2.56425505 -1.39390186
## [121]  0.23007476 -3.99895092 -1.45734995  7.46515806  3.53722442  1.82547942
## [127]  4.88631894  6.04352045 -0.14341510  0.37015330  4.90338854 10.84330211
## [133] -0.97987273  3.67230310 -0.78142080 -0.11728579  4.70368117 -1.81378270
## [139]  0.33935213  4.28501857  0.48043073  5.35541005 -2.76109584  5.89429209
## [145] -1.59771850 -0.37566400  4.89317840  5.48245628  4.93842612  2.79243673
## [151] -1.45772315 -0.08004092  0.19345526 -2.96323224  4.86964199 -1.21694782
## [157]  2.99665453  3.11649258  4.96881130  3.80661872  0.88997840 -0.95237863
## [163]  3.08480613  0.81712068  2.86671994  2.64944198  4.50875638  5.35981965
## [169] -2.85980453 -1.64423863  2.51436127  1.49247599  6.10799414  0.33944416
## [175]  2.97106387  0.71106992 -0.29075462  1.37143274 -0.47236887  2.10704489
## [181]  1.40448112 -3.17719714  0.39949405  2.99928699 -3.16144574  1.21477131
## [187]  5.94048071  8.01600463  2.57942313 -0.81503937 -0.68497820  3.23387189
## [193]  5.82533012  1.62976075 -0.63655923  5.78145764 -0.05343094  6.21889911
## [199]  4.23001519  2.61247125
## 
## $fhat
##   [1] 0.093569815 0.070832416 0.112251536 0.112104142 0.074130689 0.040155938
##   [7] 0.076611022 0.011505090 0.102133908 0.102525344 0.095832985 0.078494679
##  [13] 0.112835071 0.112218710 0.109535060 0.112211440 0.089095491 0.097632534
##  [19] 0.112212247 0.108845859 0.052981451 0.107703786 0.112945692 0.112693852
##  [25] 0.112181591 0.108517793 0.068784683 0.106945792 0.072741767 0.083469228
##  [31] 0.112860077 0.095655144 0.085849284 0.094153210 0.042632834 0.022473753
##  [37] 0.048778919 0.094108130 0.099969133 0.021181148 0.098951755 0.112822963
##  [43] 0.075993442 0.111454995 0.102840906 0.110170998 0.086152398 0.108267669
##  [49] 0.108516662 0.077365108 0.094972854 0.112204418 0.080382867 0.104873169
##  [55] 0.102675892 0.102569852 0.112183507 0.112728830 0.112262041 0.101452449
##  [61] 0.099261634 0.017902440 0.006589123 0.102308098 0.023061087 0.083270115
##  [67] 0.106471042 0.049403307 0.091143857 0.112145517 0.039686439 0.076140550
##  [73] 0.096838671 0.111569365 0.102984065 0.111985431 0.065723073 0.112253886
##  [79] 0.088824204 0.010315708 0.096210325 0.088195055 0.104358245 0.091137307
##  [85] 0.051349433 0.108899640 0.112759905 0.055836276 0.082342545 0.056461125
##  [91] 0.093223126 0.091852503 0.112955002 0.049157232 0.111880780 0.112232749
##  [97] 0.084439661 0.112050394 0.075731974 0.110241572 0.098845003 0.097687033
## [103] 0.112893566 0.044463790 0.082067637 0.111808455 0.098617978 0.102821861
## [109] 0.109217287 0.003283332 0.112057761 0.092781377 0.076734924 0.112261391
## [115] 0.091740324 0.004543208 0.011433093 0.074664862 0.105738618 0.086964541
## [121] 0.112951087 0.021622647 0.085110504 0.024938116 0.096216272 0.111396279
## [127] 0.084475675 0.065271149 0.111362993 0.112844140 0.084276151 0.003797452
## [133] 0.097557249 0.095151166 0.101801834 0.111571134 0.086508784 0.074045910
## [139] 0.112896982 0.090759201 0.112614169 0.078445738 0.045026525 0.068839172
## [145] 0.080851372 0.108866662 0.084395747 0.076598400 0.083860598 0.103527933
## [151] 0.085099453 0.111842474 0.112915533 0.039383668 0.084668859 0.091831518
## [157] 0.101514720 0.100275824 0.083495472 0.094211874 0.112165905 0.098171805
## [163] 0.100603576 0.112173909 0.102805174 0.104898261 0.088576733 0.078383828
## [169] 0.042220643 0.079405219 0.106237698 0.112085109 0.063629619 0.112896842
## [175] 0.101776601 0.112243737 0.109922875 0.112177537 0.107461059 0.109902958
## [181] 0.112153595 0.034056540 0.112786811 0.101487637 0.034416196 0.112255215
## [187] 0.067774510 0.013913295 0.105587648 0.101112051 0.103709076 0.099082451
## [193] 0.070356466 0.111929518 0.104629879 0.071273352 0.112015672 0.060687801
## [199] 0.091232510 0.105260481
## 
## $Fhat
##   [1] 0.727313618 0.125064705 0.446847080 0.473770656 0.874518809 0.949555530
##   [7] 0.863206174 0.014111662 0.632852786 0.628769586 0.197182488 0.853144673
##  [13] 0.351005663 0.423325072 0.272748093 0.421618008 0.778004704 0.677411215
##  [19] 0.393015167 0.560023571 0.929754690 0.573363300 0.333745891 0.358748090
##  [25] 0.398581702 0.564102129 0.893486561 0.581472686 0.129319592 0.822268711
##  [31] 0.349473646 0.699071293 0.804771191 0.190525814 0.946153337 0.039537258
##  [37] 0.082101295 0.719577520 0.654150924 0.036543206 0.210630109 0.324804208
##  [43] 0.866218762 0.511860027 0.625422485 0.278812122 0.163725753 0.567062176
##  [49] 0.264297524 0.859329550 0.707729760 0.458272422 0.842270484 0.240555174
##  [55] 0.627174418 0.228133728 0.461948062 0.356902841 0.440724423 0.222493623
##  [61] 0.661092520 0.028304473 0.005650343 0.631047427 0.040779866 0.155516255
##  [67] 0.586505090 0.083129912 0.758365695 0.467827692 0.950180202 0.865513106
##  [73] 0.201363388 0.507967193 0.623904476 0.487771533 0.902144183 0.432817148
##  [79] 0.780356145 0.011643531 0.198733630 0.169928119 0.237678329 0.758436313
##  [85] 0.932605092 0.559331557 0.355240754 0.094294297 0.152993259 0.095449251
##  [91] 0.732271832 0.182050237 0.335488053 0.936248372 0.494712855 0.426761333
##  [97] 0.815420059 0.481023593 0.867452621 0.539253615 0.210145862 0.205006616
## [103] 0.328835991 0.943508118 0.152255424 0.498347813 0.667488507 0.625624711
## [109] 0.555091494 0.996527275 0.480101403 0.738363079 0.862584908 0.436661049
## [115] 0.751595160 0.986706365 0.013955730 0.133743243 0.594265025 0.166149871
## [121] 0.334641541 0.037603544 0.160690690 0.967325031 0.692479205 0.513730821
## [127] 0.815158716 0.903332049 0.292664194 0.350460273 0.816598982 0.993325495
## [133] 0.204442720 0.705402322 0.224231446 0.295576781 0.799542225 0.132302700
## [139] 0.346983703 0.762407565 0.362892048 0.853418355 0.076112891 0.893321017
## [145] 0.149040511 0.267066524 0.815737900 0.863269134 0.819544557 0.618138349
## [151] 0.160658928 0.299737388 0.330505896 0.067588549 0.813748307 0.181976345
## [157] 0.639077016 0.651168143 0.822087135 0.718118570 0.408888520 0.207133436
## [163] 0.647985565 0.400716292 0.625801900 0.603236639 0.782476538 0.853764133
## [169] 0.071807648 0.145312886 0.588976889 0.476492368 0.907487698 0.346994092
## [175] 0.636475830 0.388817014 0.276356420 0.462919533 0.256604871 0.544908545
## [181] 0.466626427 0.059747649 0.353770377 0.639344214 0.060286912 0.445338668
## [187] 0.896476167 0.977785378 0.595867732 0.220820576 0.234142305 0.662868003
## [193] 0.888520904 0.491870717 0.239186186 0.885413938 0.302715856 0.914382719
## [199] 0.757402360 0.599351792
## 
## $bw
## [1] 2.519282
# with specified bandwidth
kde(x, h = 4, kernel = "quar", plot = TRUE)

## $data
##   [1]  3.90455452 -1.91369824  1.22820886  1.46819552  5.63166736  6.91427126
##   [7]  5.48163439 -5.45141272  2.93552853  2.89562635 -1.05495169  5.35192233
##  [13]  0.37498662  1.01865947 -0.32363796  1.00344704  4.45841879  3.38173957
##  [19]  0.74847748  2.24521582  6.48910178  2.36840092  0.22214504  0.44364591
##  [25]  0.79809201  2.28274280  5.89669784  2.44395968 -1.85442759  4.97098632
##  [31]  0.36141065  3.60594031  4.76435641 -1.12502282  6.83207972 -3.91121283
##  [37] -2.63339626  3.82211309  3.14628397 -4.04850000 -0.91690213  0.14294124
##  [43]  5.52111572  1.80868983  2.86302995 -0.26843978 -1.42190716  2.31005114
##  [49] -0.40113946  5.43128285  3.69678655  1.33001145  5.21504527 -0.62349035
##  [55]  2.88007901 -0.74323326  1.36277285  0.42727446  1.17366782 -0.79852075
##  [61]  3.21596881 -4.47105201 -6.39044682  2.91786721 -3.85662513 -1.51880869
##  [67]  2.49112010 -2.61244295  4.24057943  1.41519240  6.92991892  5.51183897
##  [73] -1.01155342  1.77378060  2.84827946  1.59314745  6.02538439  1.10323083
##  [79]  4.48485133 -5.67766821 -1.03879766 -1.35076231 -0.65098927  4.24135425
##  [85]  6.54374167  2.23885965  0.41253238 -2.40022792 -1.54927704 -2.37965840
##  [91]  3.95764191 -1.21614326  0.23756913  6.61623707  1.65515834  1.04927866
##  [97]  4.88941331  1.53290879  5.53738000  2.05567121 -0.92179853 -0.97409642
## [103]  0.17866489  6.77133792 -1.55825256  1.68765816  3.28061456  2.86499654
## [109]  2.19998139 11.73605476  1.52467892  4.02313678  5.47353160  1.13747245
## [115]  4.16654142  9.10820259 -5.46500865 -1.79440898  2.56425505 -1.39390186
## [121]  0.23007476 -3.99895092 -1.45734995  7.46515806  3.53722442  1.82547942
## [127]  4.88631894  6.04352045 -0.14341510  0.37015330  4.90338854 10.84330211
## [133] -0.97987273  3.67230310 -0.78142080 -0.11728579  4.70368117 -1.81378270
## [139]  0.33935213  4.28501857  0.48043073  5.35541005 -2.76109584  5.89429209
## [145] -1.59771850 -0.37566400  4.89317840  5.48245628  4.93842612  2.79243673
## [151] -1.45772315 -0.08004092  0.19345526 -2.96323224  4.86964199 -1.21694782
## [157]  2.99665453  3.11649258  4.96881130  3.80661872  0.88997840 -0.95237863
## [163]  3.08480613  0.81712068  2.86671994  2.64944198  4.50875638  5.35981965
## [169] -2.85980453 -1.64423863  2.51436127  1.49247599  6.10799414  0.33944416
## [175]  2.97106387  0.71106992 -0.29075462  1.37143274 -0.47236887  2.10704489
## [181]  1.40448112 -3.17719714  0.39949405  2.99928699 -3.16144574  1.21477131
## [187]  5.94048071  8.01600463  2.57942313 -0.81503937 -0.68497820  3.23387189
## [193]  5.82533012  1.62976075 -0.63655923  5.78145764 -0.05343094  6.21889911
## [199]  4.23001519  2.61247125
## 
## $xgrid
##   [1]  3.90455452 -1.91369824  1.22820886  1.46819552  5.63166736  6.91427126
##   [7]  5.48163439 -5.45141272  2.93552853  2.89562635 -1.05495169  5.35192233
##  [13]  0.37498662  1.01865947 -0.32363796  1.00344704  4.45841879  3.38173957
##  [19]  0.74847748  2.24521582  6.48910178  2.36840092  0.22214504  0.44364591
##  [25]  0.79809201  2.28274280  5.89669784  2.44395968 -1.85442759  4.97098632
##  [31]  0.36141065  3.60594031  4.76435641 -1.12502282  6.83207972 -3.91121283
##  [37] -2.63339626  3.82211309  3.14628397 -4.04850000 -0.91690213  0.14294124
##  [43]  5.52111572  1.80868983  2.86302995 -0.26843978 -1.42190716  2.31005114
##  [49] -0.40113946  5.43128285  3.69678655  1.33001145  5.21504527 -0.62349035
##  [55]  2.88007901 -0.74323326  1.36277285  0.42727446  1.17366782 -0.79852075
##  [61]  3.21596881 -4.47105201 -6.39044682  2.91786721 -3.85662513 -1.51880869
##  [67]  2.49112010 -2.61244295  4.24057943  1.41519240  6.92991892  5.51183897
##  [73] -1.01155342  1.77378060  2.84827946  1.59314745  6.02538439  1.10323083
##  [79]  4.48485133 -5.67766821 -1.03879766 -1.35076231 -0.65098927  4.24135425
##  [85]  6.54374167  2.23885965  0.41253238 -2.40022792 -1.54927704 -2.37965840
##  [91]  3.95764191 -1.21614326  0.23756913  6.61623707  1.65515834  1.04927866
##  [97]  4.88941331  1.53290879  5.53738000  2.05567121 -0.92179853 -0.97409642
## [103]  0.17866489  6.77133792 -1.55825256  1.68765816  3.28061456  2.86499654
## [109]  2.19998139 11.73605476  1.52467892  4.02313678  5.47353160  1.13747245
## [115]  4.16654142  9.10820259 -5.46500865 -1.79440898  2.56425505 -1.39390186
## [121]  0.23007476 -3.99895092 -1.45734995  7.46515806  3.53722442  1.82547942
## [127]  4.88631894  6.04352045 -0.14341510  0.37015330  4.90338854 10.84330211
## [133] -0.97987273  3.67230310 -0.78142080 -0.11728579  4.70368117 -1.81378270
## [139]  0.33935213  4.28501857  0.48043073  5.35541005 -2.76109584  5.89429209
## [145] -1.59771850 -0.37566400  4.89317840  5.48245628  4.93842612  2.79243673
## [151] -1.45772315 -0.08004092  0.19345526 -2.96323224  4.86964199 -1.21694782
## [157]  2.99665453  3.11649258  4.96881130  3.80661872  0.88997840 -0.95237863
## [163]  3.08480613  0.81712068  2.86671994  2.64944198  4.50875638  5.35981965
## [169] -2.85980453 -1.64423863  2.51436127  1.49247599  6.10799414  0.33944416
## [175]  2.97106387  0.71106992 -0.29075462  1.37143274 -0.47236887  2.10704489
## [181]  1.40448112 -3.17719714  0.39949405  2.99928699 -3.16144574  1.21477131
## [187]  5.94048071  8.01600463  2.57942313 -0.81503937 -0.68497820  3.23387189
## [193]  5.82533012  1.62976075 -0.63655923  5.78145764 -0.05343094  6.21889911
## [199]  4.23001519  2.61247125
## 
## $fhat
##   [1] 0.093035572 0.069714566 0.109292874 0.108871962 0.066927528 0.042139969
##   [7] 0.069672886 0.012482917 0.101309989 0.101573612 0.087120631 0.071985490
##  [13] 0.106639598 0.109256433 0.099028802 0.109240241 0.086253966 0.098073211
##  [19] 0.108670481 0.105401512 0.050240668 0.104734103 0.105389480 0.107122108
##  [25] 0.108831862 0.105199538 0.061879960 0.104317956 0.071011311 0.078488407
##  [31] 0.106538225 0.096085416 0.081785880 0.085846465 0.043669502 0.029399031
##  [37] 0.053903186 0.093924924 0.099872900 0.027250427 0.089569772 0.104642888
##  [43] 0.068958328 0.107622424 0.101786986 0.099788269 0.080115209 0.105051745
##  [49] 0.097921389 0.070576896 0.095208234 0.109174670 0.074375433 0.094498227
##  [55] 0.101675622 0.092534832 0.109116948 0.107011699 0.109318983 0.091605680
##  [61] 0.099366433 0.021382503 0.007411623 0.101426926 0.030279509 0.078139580
##  [67] 0.104053639 0.054351570 0.089074429 0.109006583 0.041850521 0.069126697
##  [73] 0.087898818 0.107777972 0.101882982 0.108481941 0.059355007 0.109314567
##  [79] 0.085892275 0.011120499 0.087411256 0.081535286 0.094053739 0.089064797
##  [85] 0.049180213 0.105435579 0.106909770 0.058948222 0.077509674 0.059399362
##  [91] 0.092444005 0.084146372 0.105526533 0.047780616 0.108259607 0.109283937
##  [97] 0.079814184 0.108680078 0.068662432 0.106400354 0.089484250 0.088564265
## [103] 0.104988440 0.044814345 0.077323362 0.108134987 0.098878163 0.101774162
## [109] 0.105642865 0.002629897 0.108705839 0.091692577 0.069818899 0.109321905
## [115] 0.089983819 0.009539098 0.012396633 0.072313837 0.103634481 0.080677343
## [121] 0.105460278 0.028013551 0.079397781 0.032098470 0.096721858 0.107545604
## [127] 0.079863885 0.058997445 0.101417094 0.106603729 0.079589152 0.003395673
## [133] 0.088461977 0.095448797 0.091894564 0.101741333 0.082714553 0.071894735
## [139] 0.106369414 0.088517353 0.107359488 0.071923952 0.051186710 0.061926832
## [145] 0.076499972 0.098290672 0.079753650 0.069658063 0.079021003 0.102242621
## [151] 0.079390191 0.102193639 0.105127274 0.046963347 0.080131010 0.084131106
## [157] 0.100903339 0.100084127 0.078524112 0.094088279 0.109064063 0.088947696
## [163] 0.100305097 0.108887289 0.101762920 0.103130610 0.085560666 0.071846109
## [169] 0.049112013 0.075521502 0.103921569 0.108803013 0.057725214 0.106370129
## [175] 0.101074095 0.108532243 0.099484088 0.109100233 0.096861880 0.106132186
## [181] 0.109030980 0.042610966 0.106817660 0.100885718 0.042926034 0.109301709
## [187] 0.061024335 0.022772288 0.103546143 0.091325274 0.093498798 0.099233178
## [193] 0.063262737 0.108353158 0.094287507 0.064104193 0.102509783 0.055536959
## [199] 0.089205464 0.103351650
## 
## $Fhat
##   [1] 0.72126534 0.14520792 0.44576207 0.47195022 0.86136937 0.93127509
##   [7] 0.85112107 0.01808029 0.62673522 0.62268746 0.21279523 0.84193310
##  [13] 0.35314864 0.42285729 0.28103642 0.42119535 0.77097610 0.67124620
##  [19] 0.39340175 0.55533124 0.91165065 0.56827415 0.33694218 0.36048730
##  [25] 0.39879749 0.55928286 0.87844369 0.57617205 0.14937842 0.81325769
##  [31] 0.35170159 0.69301636 0.79669548 0.20673509 0.92774882 0.04831290
##  [37] 0.10075675 0.71355846 0.64793738 0.04442548 0.22499212 0.32862403
##  [43] 0.85385776 0.50882465 0.61937305 0.28652370 0.18208612 0.56215366
##  [49] 0.27340414 0.84759014 0.70170584 0.45688314 0.83191577 0.25200566
##  [55] 0.62110747 0.24080702 0.46045893 0.35873445 0.43980029 0.23571662
##  [61] 0.65487947 0.03419210 0.00885357 0.62494492 0.04994170 0.17441820
##  [67] 0.58108552 0.10189089 0.75187502 0.46617600 0.93193222 0.85321727
##  [73] 0.21659303 0.50506490 0.61787093 0.48553047 0.88624465 0.43210010
##  [79] 0.77325124 0.01541270 0.21420492 0.18783658 0.24941317 0.75194403
##  [85] 0.91436680 0.55466118 0.35715763 0.11391100 0.17204700 0.11512818
##  [91] 0.72618872 0.19898974 0.33856878 0.91788136 0.49225074 0.42620307
##  [97] 0.80680089 0.47898961 0.85497691 0.53525784 0.22455376 0.21989793
## [103] 0.33236847 0.92506155 0.17135215 0.49576715 0.66128741 0.61957321
## [109] 0.55055799 0.99581810 0.47809508 0.73221885 0.85055593 0.43584337
## [115] 0.74524634 0.98364108 0.01791116 0.15367957 0.58868021 0.18433765
## [121] 0.33777817 0.04579456 0.17925931 0.95170993 0.68639177 0.51063094
## [127] 0.80655384 0.88731788 0.29910287 0.35263331 0.80791474 0.99313486
## [133] 0.21938665 0.69937186 0.23728555 0.30175708 0.79170483 0.15228264
## [139] 0.34935337 0.75582107 0.36443217 0.84218406 0.09404709 0.87829476
## [145] 0.16831674 0.27590345 0.80710128 0.85117832 0.81069341 0.61217143
## [151] 0.17922968 0.30555488 0.33392232 0.08412891 0.80521972 0.19892205
## [157] 0.63291549 0.64495888 0.81308694 0.71210189 0.40880903 0.22182552
## [163] 0.64178405 0.40086895 0.61974860 0.59748719 0.77530054 0.84250104
## [169] 0.08909710 0.16478068 0.58350231 0.47459285 0.89108065 0.34936316
## [175] 0.63033112 0.38933920 0.28430033 0.46140380 0.26646676 0.54071714
## [181] 0.46500827 0.07454856 0.35576430 0.63318109 0.07522222 0.44429338
## [187] 0.88113426 0.96676672 0.59025148 0.23420574 0.24622580 0.65665724
## [193] 0.87397801 0.48950001 0.25077205 0.87118404 0.30827847 0.89736127
## [199] 0.75093332 0.59367028
## 
## $bw
## [1] 4

1.5 Przeczytaj

Przeczytaj artykuł naukowy “Kernel-smoothed cumulative distribution function estimation with akdensity” autorstwa Philippe Van Kerm.

1.6 Zadanie

Posłużymy się zbiorem danych diagnozy społecznej.

Na jego podstawie Twoim zadaniem jest oszacowanie rozkładu “p64 Pana/Pani wlasny (osobisty) dochod miesieczny netto (na reke)” według województw/płci.

Postaraj się oszacować zarówno rozkład gęstości jak i skumulowanej gęstości (dystrybuanty).

data("diagnoza")
data("diagnozaDict")
data <- diagnoza %>%
  select(wojewodztwo, plec, gp64) %>%
  filter(!is.na(gp64), gp64 > 0)

2 Dochód miesięczny netto według województw

# wykres gęstości
ggplot(data.frame(data), aes(x = gp64, color = wojewodztwo)) +
  geom_density() +
  ggtitle("Rozkład gęstości dochodu") +
  xlab("Dochód") +
  ylab("Gęstość")+
  scale_x_continuous(limits = c(0, 7000))

# Wykres dystrybuanty
ggplot(data.frame(data), aes(x = gp64, color = wojewodztwo)) +
  stat_ecdf(geom = "step") +
  ggtitle("Rozkład skumulowanej gęstości dochodu") +
  xlab("Dochód") +
  ylab("Skumulowana gęstość")+
  scale_x_continuous(limits = c(0, 7000))

3 Dochód miesięczny netto według płci

# wykres gęstości
ggplot(data.frame(data), aes(x = gp64, color = plec)) +
  geom_density() +
  ggtitle("Rozkład gęstości dochodu") +
  xlab("Dochód") +
  ylab("Gęstość")+
  scale_x_continuous(limits = c(0, 7000))

# Wykres dystrybuanty
ggplot(data.frame(data), aes(x = gp64, color = plec)) +
  stat_ecdf(geom = "step") +
  ggtitle("Rozkład skumulowanej gęstości dochodu") +
  xlab("Dochód") +
  ylab("Skumulowana gęstość")+
  scale_x_continuous(limits = c(0, 7000))

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