Nieklasyczne metody statystyki

Nieparametryczna estymacja dystrybuanty

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] 0.7393409

   plot(f)

1.4 Przykład 2.

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

## $data
##   [1]  0.70224851 -2.76203446  5.74955598  0.27635959  1.54488277  5.95432521
##   [7]  5.34107725  0.39992366  3.56058110  3.67300299  4.29595430 -4.22530462
##  [13]  4.00207291  0.26709601  3.80772703  2.22932227 -1.21373116  4.24376521
##  [19] -1.07615231 -4.08604222  0.06220763  6.38807887  3.82729219  2.61265120
##  [25]  4.18469578  3.43482012 -0.22876756 -3.07441141  1.95376487  4.75396993
##  [31]  5.77111030  1.22959441  5.70601591  1.01754486  6.14899348 -0.19430168
##  [37]  2.76745160 -3.42717022 -0.35873376  4.73951021  2.01472326  3.24820939
##  [43]  1.71606253 -0.66060619  1.14513018  0.31794848  2.53340604 -3.08721827
##  [49]  2.02318543  2.64760863  3.13999173  3.42431445 -0.44512246  5.49292680
##  [55]  7.66826735  0.40282612  0.71002201 -0.06072497  2.41770841 -0.55489360
##  [61]  3.21281128  7.76056303  1.75737521 -2.20792855 -1.48719418  5.22174018
##  [67]  3.11250444  0.86143668  4.38222817  5.35708135  2.06191589  7.11487153
##  [73]  2.18712600  2.97045581  1.60661947 -4.41302011  6.72902689 -2.74435402
##  [79] -1.91576099  7.68131136  5.80176853  6.59577692  0.99899983  3.86476672
##  [85]  0.44499598  0.50681167  4.44140740  3.05461142  2.52344584  3.74073943
##  [91] -1.32725817  7.68387865  5.54446366 -2.08466736 -3.18675969  1.05985489
##  [97]  6.00801973  4.08049775  3.28690207 -4.76853368  3.62527977  0.99311858
## [103] -1.78834974  3.96932436  1.30501570  1.72552253  1.03638032 -0.28747062
## [109]  2.51169640 -1.71293680 -1.71189557  1.74962231  2.43658031  5.06515461
## [115]  6.56094158  1.08638243 -1.60180148 -0.70068743  3.43054595  5.66761679
## [121]  3.08057648  2.98572668  2.88102513  4.78081863 -2.61454012 -0.92341469
## [127]  3.47348153  3.34415459  4.65153254  3.23683107  0.50490108  2.83495038
## [133] -0.35557640  2.46886709  4.18017356 -0.95199810  0.47773940  5.80185010
## [139]  1.77999974  3.47114550  4.20264114 -0.22246419 -0.24547475  2.31012659
## [145]  7.08048242  2.84894689 -2.72590325  0.73941524 -0.17493913  2.60660355
## [151]  4.59616234  0.81559258 -3.52569815  0.32910778 10.01197079 -0.43499782
## [157]  3.94678587  2.44462227 -0.15751232 -3.75438513  2.04888250  2.52413536
## [163]  0.38389614  3.34202802  6.80226676  5.17459908  1.94892327  4.41241498
## [169]  2.88599195  2.06184109  2.49846607  2.43193981  3.71478238  0.16215750
## [175]  2.48773737 -1.13982571 -0.24327754 -0.80300475  0.95566112  4.42754907
## [181]  4.79451100  2.46965100  1.43653545  5.85666553 -1.43625268  5.09179417
## [187] -0.31997806  1.59076388  6.45670114  3.21830036  4.17120198  1.22453919
## [193]  5.61655234 -1.59066510  2.89705104  4.23592466 -0.27443105  2.69128758
## [199] -1.62895348  7.69069059
## 
## $xgrid
##   [1]  0.70224851 -2.76203446  5.74955598  0.27635959  1.54488277  5.95432521
##   [7]  5.34107725  0.39992366  3.56058110  3.67300299  4.29595430 -4.22530462
##  [13]  4.00207291  0.26709601  3.80772703  2.22932227 -1.21373116  4.24376521
##  [19] -1.07615231 -4.08604222  0.06220763  6.38807887  3.82729219  2.61265120
##  [25]  4.18469578  3.43482012 -0.22876756 -3.07441141  1.95376487  4.75396993
##  [31]  5.77111030  1.22959441  5.70601591  1.01754486  6.14899348 -0.19430168
##  [37]  2.76745160 -3.42717022 -0.35873376  4.73951021  2.01472326  3.24820939
##  [43]  1.71606253 -0.66060619  1.14513018  0.31794848  2.53340604 -3.08721827
##  [49]  2.02318543  2.64760863  3.13999173  3.42431445 -0.44512246  5.49292680
##  [55]  7.66826735  0.40282612  0.71002201 -0.06072497  2.41770841 -0.55489360
##  [61]  3.21281128  7.76056303  1.75737521 -2.20792855 -1.48719418  5.22174018
##  [67]  3.11250444  0.86143668  4.38222817  5.35708135  2.06191589  7.11487153
##  [73]  2.18712600  2.97045581  1.60661947 -4.41302011  6.72902689 -2.74435402
##  [79] -1.91576099  7.68131136  5.80176853  6.59577692  0.99899983  3.86476672
##  [85]  0.44499598  0.50681167  4.44140740  3.05461142  2.52344584  3.74073943
##  [91] -1.32725817  7.68387865  5.54446366 -2.08466736 -3.18675969  1.05985489
##  [97]  6.00801973  4.08049775  3.28690207 -4.76853368  3.62527977  0.99311858
## [103] -1.78834974  3.96932436  1.30501570  1.72552253  1.03638032 -0.28747062
## [109]  2.51169640 -1.71293680 -1.71189557  1.74962231  2.43658031  5.06515461
## [115]  6.56094158  1.08638243 -1.60180148 -0.70068743  3.43054595  5.66761679
## [121]  3.08057648  2.98572668  2.88102513  4.78081863 -2.61454012 -0.92341469
## [127]  3.47348153  3.34415459  4.65153254  3.23683107  0.50490108  2.83495038
## [133] -0.35557640  2.46886709  4.18017356 -0.95199810  0.47773940  5.80185010
## [139]  1.77999974  3.47114550  4.20264114 -0.22246419 -0.24547475  2.31012659
## [145]  7.08048242  2.84894689 -2.72590325  0.73941524 -0.17493913  2.60660355
## [151]  4.59616234  0.81559258 -3.52569815  0.32910778 10.01197079 -0.43499782
## [157]  3.94678587  2.44462227 -0.15751232 -3.75438513  2.04888250  2.52413536
## [163]  0.38389614  3.34202802  6.80226676  5.17459908  1.94892327  4.41241498
## [169]  2.88599195  2.06184109  2.49846607  2.43193981  3.71478238  0.16215750
## [175]  2.48773737 -1.13982571 -0.24327754 -0.80300475  0.95566112  4.42754907
## [181]  4.79451100  2.46965100  1.43653545  5.85666553 -1.43625268  5.09179417
## [187] -0.31997806  1.59076388  6.45670114  3.21830036  4.17120198  1.22453919
## [193]  5.61655234 -1.59066510  2.89705104  4.23592466 -0.27443105  2.69128758
## [199] -1.62895348  7.69069059
## 
## $fhat
##   [1] 0.107928281 0.038248356 0.066876269 0.103098480 0.118774705 0.062209978
##   [7] 0.077038958 0.104363640 0.127007862 0.124366044 0.106743578 0.019425619
##  [13] 0.115574573 0.103008975 0.120914943 0.129242187 0.071930638 0.108348969
##  [19] 0.075782211 0.021134415 0.100823638 0.052771253 0.120395872 0.133411871
##  [25] 0.110158243 0.129582485 0.096553454 0.033956402 0.125009483 0.092956790
##  [31] 0.066378265 0.114430271 0.067884289 0.111772599 0.057916197 0.097126724
##  [37] 0.134392645 0.029191228 0.094144273 0.093354909 0.125996721 0.132389279
##  [43] 0.121286271 0.087247195 0.113354696 0.103510398 0.132718264 0.033780571
##  [49] 0.126134590 0.133686731 0.133481663 0.129774579 0.092321357 0.073055380
##  [55] 0.024156654 0.104394903 0.108023855 0.099180074 0.131563433 0.089825897
##  [61] 0.132785329 0.022291265 0.121910884 0.047091737 0.064413522 0.080218515
##  [67] 0.133702586 0.109884370 0.104062858 0.076614472 0.126760585 0.036742913
##  [73] 0.128652989 0.134454326 0.119667116 0.017029610 0.045535722 0.038491782
##  [79] 0.053536653 0.023886646 0.065671813 0.048409616 0.111547958 0.119390210
##  [85] 0.104858682 0.105569677 0.102215805 0.134090431 0.132625787 0.122662032
##  [91] 0.068798877 0.023833720 0.071758753 0.049721641 0.032419945 0.112291066
##  [97] 0.061014705 0.113302668 0.131911716 0.012405115 0.125518438 0.111477095
## [103] 0.056564628 0.116498683 0.115433875 0.121428432 0.112002377 0.095516914
## [109] 0.132515268 0.058442792 0.058469291 0.121792846 0.131765934 0.084415806
## [115] 0.049150185 0.112618827 0.061339213 0.086224373 0.129661099 0.068780939
## [121] 0.133930430 0.134405591 0.134596989 0.092220175 0.040278220 0.080107344
## [127] 0.128843860 0.131107889 0.095832623 0.132520708 0.105547313 0.134570397
## [133] 0.094207941 0.132098815 0.110296338 0.079295452 0.105231684 0.065669938
## [139] 0.122258173 0.128889808 0.109609630 0.096660037 0.096266910 0.130305393
## [145] 0.037549413 0.134585934 0.038745506 0.108385039 0.097440147 0.133362176
## [151] 0.097463638 0.109324442 0.027898180 0.103623662 0.002079047 0.092542404
## [157] 0.117129793 0.131850454 0.097718230 0.025019348 0.126551069 0.132632226
## [163] 0.104192550 0.131139930 0.043904927 0.081479113 0.124931779 0.103120066
## [169] 0.134595659 0.126759386 0.132389168 0.131716680 0.123323917 0.101971759
## [175] 0.132285415 0.073990686 0.096304970 0.083488761 0.111027363 0.102647758
## [181] 0.091844482 0.132106683 0.117245271 0.064414964 0.065802899 0.083696166
## [187] 0.094907636 0.119436287 0.051334486 0.132726453 0.110570063 0.114364769
## [193] 0.069996959 0.061634970 0.134589786 0.108589734 0.095754275 0.133992401
## [199] 0.060621278 0.023693647
## 
## $Fhat
##   [1] 0.320274458 0.058913657 0.887882746 0.275378732 0.415647548 0.901096432
##   [7] 0.858547468 0.288194896 0.676685644 0.690818019 0.763014805 0.017167349
##  [13] 0.730330005 0.274424086 0.707343427 0.500508789 0.140896977 0.757402035
##  [19] 0.151057203 0.019992406 0.253531840 0.926016068 0.709704076 0.550919076
##  [25] 0.750948435 0.660545981 0.224784239 0.047636510 0.465460239 0.808679939
##  [31] 0.889318851 0.378893290 0.884949023 0.354912825 0.912787086 0.228121985
##  [37] 0.571655648 0.036503969 0.212387547 0.807332933 0.473110634 0.636082617
##  [43] 0.436192682 0.184972496 0.369273664 0.279674990 0.540373776 0.047202760
##  [49] 0.474177423 0.555587679 0.621693485 0.659183618 0.204332220 0.869941321
##  [55] 0.975445573 0.288497852 0.321113810 0.241235993 0.525083506 0.194333408
##  [61] 0.631389164 0.977588196 0.441216211 0.082365605 0.122256872 0.849164349
##  [67] 0.618021345 0.337611325 0.772108616 0.859777010 0.479074830 0.958673189
##  [73] 0.495067608 0.598967707 0.423007768 0.013744617 0.942784845 0.059592057
##  [79] 0.097048288 0.975758910 0.891343054 0.936524812 0.352842085 0.714197061
##  [85] 0.292909889 0.299413622 0.778212298 0.610269134 0.539052334 0.699184850
##  [91] 0.132908708 0.975820166 0.873672822 0.088330456 0.043908074 0.359652867
##  [97] 0.904404633 0.739305471 0.641196037 0.008515462 0.684855207 0.352186252
## [103] 0.104060582 0.726529944 0.387561331 0.437340723 0.357020275 0.219146276
## [109] 0.537494703 0.108396625 0.108457492 0.440271505 0.527568277 0.836275298
## [115] 0.934825531 0.362636018 0.115051517 0.181495926 0.659991954 0.882325147
## [121] 0.613748772 0.601020578 0.586935168 0.811165814 0.064704583 0.162961846
## [127] 0.665541698 0.648725877 0.799011686 0.634575498 0.299211943 0.580733988
## [133] 0.212684895 0.531827995 0.750449962 0.160683710 0.296349398 0.891348411
## [139] 0.443978308 0.665240661 0.752920343 0.225393188 0.223173487 0.510995685
## [145] 0.957395745 0.582617620 0.060304601 0.324294287 0.230005647 0.550112398
## [151] 0.793660503 0.332586588 0.033691657 0.280830724 0.997500000 0.205268062
## [157] 0.723897121 0.528628275 0.231706147 0.027645198 0.477424074 0.539143785
## [163] 0.286523580 0.648447032 0.946060366 0.845353056 0.464855181 0.775235712
## [169] 0.587603683 0.479065348 0.535742316 0.526956932 0.695992294 0.263668074
## [175] 0.534322506 0.146288985 0.223385049 0.172812148 0.348019028 0.776792765
## [181] 0.812425956 0.531931552 0.402861873 0.894913698 0.125573552 0.838514501
## [187] 0.216051096 0.421112206 0.929587902 0.632117873 0.749459202 0.378314985
## [193] 0.878781951 0.115736260 0.589092161 0.756551575 0.220393328 0.561433835
## [199] 0.113395793 0.975982043
## 
## $bw
## [1] 2.254658

# with specified bandwidth
kde(x, h = 4, kernel = "quar", plot = TRUE)

## $data
##   [1]  0.70224851 -2.76203446  5.74955598  0.27635959  1.54488277  5.95432521
##   [7]  5.34107725  0.39992366  3.56058110  3.67300299  4.29595430 -4.22530462
##  [13]  4.00207291  0.26709601  3.80772703  2.22932227 -1.21373116  4.24376521
##  [19] -1.07615231 -4.08604222  0.06220763  6.38807887  3.82729219  2.61265120
##  [25]  4.18469578  3.43482012 -0.22876756 -3.07441141  1.95376487  4.75396993
##  [31]  5.77111030  1.22959441  5.70601591  1.01754486  6.14899348 -0.19430168
##  [37]  2.76745160 -3.42717022 -0.35873376  4.73951021  2.01472326  3.24820939
##  [43]  1.71606253 -0.66060619  1.14513018  0.31794848  2.53340604 -3.08721827
##  [49]  2.02318543  2.64760863  3.13999173  3.42431445 -0.44512246  5.49292680
##  [55]  7.66826735  0.40282612  0.71002201 -0.06072497  2.41770841 -0.55489360
##  [61]  3.21281128  7.76056303  1.75737521 -2.20792855 -1.48719418  5.22174018
##  [67]  3.11250444  0.86143668  4.38222817  5.35708135  2.06191589  7.11487153
##  [73]  2.18712600  2.97045581  1.60661947 -4.41302011  6.72902689 -2.74435402
##  [79] -1.91576099  7.68131136  5.80176853  6.59577692  0.99899983  3.86476672
##  [85]  0.44499598  0.50681167  4.44140740  3.05461142  2.52344584  3.74073943
##  [91] -1.32725817  7.68387865  5.54446366 -2.08466736 -3.18675969  1.05985489
##  [97]  6.00801973  4.08049775  3.28690207 -4.76853368  3.62527977  0.99311858
## [103] -1.78834974  3.96932436  1.30501570  1.72552253  1.03638032 -0.28747062
## [109]  2.51169640 -1.71293680 -1.71189557  1.74962231  2.43658031  5.06515461
## [115]  6.56094158  1.08638243 -1.60180148 -0.70068743  3.43054595  5.66761679
## [121]  3.08057648  2.98572668  2.88102513  4.78081863 -2.61454012 -0.92341469
## [127]  3.47348153  3.34415459  4.65153254  3.23683107  0.50490108  2.83495038
## [133] -0.35557640  2.46886709  4.18017356 -0.95199810  0.47773940  5.80185010
## [139]  1.77999974  3.47114550  4.20264114 -0.22246419 -0.24547475  2.31012659
## [145]  7.08048242  2.84894689 -2.72590325  0.73941524 -0.17493913  2.60660355
## [151]  4.59616234  0.81559258 -3.52569815  0.32910778 10.01197079 -0.43499782
## [157]  3.94678587  2.44462227 -0.15751232 -3.75438513  2.04888250  2.52413536
## [163]  0.38389614  3.34202802  6.80226676  5.17459908  1.94892327  4.41241498
## [169]  2.88599195  2.06184109  2.49846607  2.43193981  3.71478238  0.16215750
## [175]  2.48773737 -1.13982571 -0.24327754 -0.80300475  0.95566112  4.42754907
## [181]  4.79451100  2.46965100  1.43653545  5.85666553 -1.43625268  5.09179417
## [187] -0.31997806  1.59076388  6.45670114  3.21830036  4.17120198  1.22453919
## [193]  5.61655234 -1.59066510  2.89705104  4.23592466 -0.27443105  2.69128758
## [199] -1.62895348  7.69069059
## 
## $xgrid
##   [1]  0.70224851 -2.76203446  5.74955598  0.27635959  1.54488277  5.95432521
##   [7]  5.34107725  0.39992366  3.56058110  3.67300299  4.29595430 -4.22530462
##  [13]  4.00207291  0.26709601  3.80772703  2.22932227 -1.21373116  4.24376521
##  [19] -1.07615231 -4.08604222  0.06220763  6.38807887  3.82729219  2.61265120
##  [25]  4.18469578  3.43482012 -0.22876756 -3.07441141  1.95376487  4.75396993
##  [31]  5.77111030  1.22959441  5.70601591  1.01754486  6.14899348 -0.19430168
##  [37]  2.76745160 -3.42717022 -0.35873376  4.73951021  2.01472326  3.24820939
##  [43]  1.71606253 -0.66060619  1.14513018  0.31794848  2.53340604 -3.08721827
##  [49]  2.02318543  2.64760863  3.13999173  3.42431445 -0.44512246  5.49292680
##  [55]  7.66826735  0.40282612  0.71002201 -0.06072497  2.41770841 -0.55489360
##  [61]  3.21281128  7.76056303  1.75737521 -2.20792855 -1.48719418  5.22174018
##  [67]  3.11250444  0.86143668  4.38222817  5.35708135  2.06191589  7.11487153
##  [73]  2.18712600  2.97045581  1.60661947 -4.41302011  6.72902689 -2.74435402
##  [79] -1.91576099  7.68131136  5.80176853  6.59577692  0.99899983  3.86476672
##  [85]  0.44499598  0.50681167  4.44140740  3.05461142  2.52344584  3.74073943
##  [91] -1.32725817  7.68387865  5.54446366 -2.08466736 -3.18675969  1.05985489
##  [97]  6.00801973  4.08049775  3.28690207 -4.76853368  3.62527977  0.99311858
## [103] -1.78834974  3.96932436  1.30501570  1.72552253  1.03638032 -0.28747062
## [109]  2.51169640 -1.71293680 -1.71189557  1.74962231  2.43658031  5.06515461
## [115]  6.56094158  1.08638243 -1.60180148 -0.70068743  3.43054595  5.66761679
## [121]  3.08057648  2.98572668  2.88102513  4.78081863 -2.61454012 -0.92341469
## [127]  3.47348153  3.34415459  4.65153254  3.23683107  0.50490108  2.83495038
## [133] -0.35557640  2.46886709  4.18017356 -0.95199810  0.47773940  5.80185010
## [139]  1.77999974  3.47114550  4.20264114 -0.22246419 -0.24547475  2.31012659
## [145]  7.08048242  2.84894689 -2.72590325  0.73941524 -0.17493913  2.60660355
## [151]  4.59616234  0.81559258 -3.52569815  0.32910778 10.01197079 -0.43499782
## [157]  3.94678587  2.44462227 -0.15751232 -3.75438513  2.04888250  2.52413536
## [163]  0.38389614  3.34202802  6.80226676  5.17459908  1.94892327  4.41241498
## [169]  2.88599195  2.06184109  2.49846607  2.43193981  3.71478238  0.16215750
## [175]  2.48773737 -1.13982571 -0.24327754 -0.80300475  0.95566112  4.42754907
## [181]  4.79451100  2.46965100  1.43653545  5.85666553 -1.43625268  5.09179417
## [187] -0.31997806  1.59076388  6.45670114  3.21830036  4.17120198  1.22453919
## [193]  5.61655234 -1.59066510  2.89705104  4.23592466 -0.27443105  2.69128758
## [199] -1.62895348  7.69069059
## 
## $fhat
##   [1] 0.106548107 0.042506884 0.069689560 0.099397511 0.116903034 0.064612222
##   [7] 0.079774296 0.101549399 0.112687736 0.111315235 0.101883002 0.020627702
##  [13] 0.106726799 0.099234062 0.109545870 0.119987145 0.071340193 0.102787789
##  [19] 0.073919139 0.022247241 0.095545901 0.054242916 0.109277379 0.119755771
##  [25] 0.103790307 0.114102811 0.090081231 0.037046006 0.119266659 0.093070093
##  [31] 0.069154278 0.113823831 0.070771069 0.111179087 0.059867957 0.090738308
##  [37] 0.119208953 0.031322669 0.087591600 0.093372054 0.119483653 0.115953195
##  [43] 0.118098714 0.081802886 0.112820653 0.100127833 0.119935852 0.036829587
##  [49] 0.119511184 0.119654634 0.116884277 0.114215082 0.085929786 0.076056710
##  [55] 0.029224811 0.101599250 0.106670025 0.093263493 0.120073085 0.083823362
##  [61] 0.116269446 0.027743053 0.118339733 0.052826163 0.066274757 0.082631226
##  [67] 0.117103814 0.108970517 0.100346300 0.079386092 0.119629413 0.038998112
##  [73] 0.119920538 0.118117585 0.117372213 0.018600885 0.046775490 0.042827180
##  [79] 0.058318592 0.029012706 0.068392660 0.049632651 0.110927679 0.108753582
##  [85] 0.102319980 0.103365793 0.099261053 0.117541721 0.119953488 0.110441650
##  [91] 0.069223017 0.028971061 0.074782314 0.055149141 0.035167783 0.111740478
##  [97] 0.063292868 0.105497477 0.115594738 0.015041998 0.111910231 0.110847285
## [103] 0.060706170 0.107225198 0.114661887 0.118155460 0.111431126 0.088958730
## [109] 0.119972843 0.062114078 0.062133474 0.118295839 0.120060908 0.086264911
## [115] 0.050391595 0.112083693 0.064174269 0.081037983 0.114148599 0.071725658
## [121] 0.117349567 0.118018421 0.118649810 0.092505579 0.045209939 0.076803629
## [127] 0.113681646 0.115040355 0.095178993 0.116056088 0.103333692 0.118892181
## [133] 0.087652268 0.120030632 0.103866054 0.076262450 0.102875674 0.068390633
## [139] 0.118464450 0.113707456 0.103488280 0.090201569 0.089762067 0.120068466
## [145] 0.039658972 0.118820784 0.043162653 0.107127457 0.091106401 0.119771956
## [151] 0.096289176 0.108288201 0.029829911 0.100322835 0.004809815 0.086124543
## [157] 0.107562794 0.120054474 0.091437123 0.026562091 0.119591057 0.119952303
## [163] 0.101273528 0.115061466 0.045238736 0.083739840 0.119247994 0.099796147
## [169] 0.118622399 0.119629197 0.119992815 0.120064282 0.110779967 0.097364515
## [175] 0.120007620 0.072724013 0.089804056 0.079089540 0.110327925 0.099517650
## [181] 0.092215851 0.120029753 0.115972847 0.067028886 0.067209069 0.085656972
## [187] 0.088335676 0.117255868 0.052692746 0.116221153 0.104015883 0.113765651
## [193] 0.072994717 0.064379314 0.118560461 0.102922208 0.089208337 0.119510118
## [199] 0.063673070 0.028860728
## 
## $Fhat
##   [1] 0.33092751 0.07154637 0.86894498 0.28704422 0.42553533 0.88269477
##   [7] 0.83841047 0.29945970 0.66370771 0.67630008 0.74286921 0.02656956
##  [13] 0.71220047 0.28612420 0.69117917 0.50685470 0.15970057 0.73752834
##  [19] 0.16969273 0.02955350 0.26616752 0.90844447 0.69331983 0.55285574
##  [25] 0.73142700 0.64944566 0.23915603 0.05913104 0.47387518 0.78757673
##  [31] 0.87044132 0.38913300 0.86588715 0.36526970 0.89480922 0.24227209
##  [37] 0.57135496 0.04708745 0.22761012 0.78622878 0.48115226 0.62797541
##  [43] 0.44565377 0.20204372 0.37956086 0.29119326 0.54335808 0.05865798
##  [49] 0.48216347 0.55704037 0.61537590 0.64824634 0.22011496 0.85024278
##  [55] 0.96097887 0.29975451 0.33175624 0.25456165 0.52947242 0.21079808
##  [61] 0.62386526 0.96360747 0.45053779 0.09794256 0.14088695 0.82871927
##  [67] 0.61216003 0.34808352 0.75159314 0.83968408 0.48679452 0.94217270
##  [73] 0.50179304 0.59545117 0.43276724 0.02288961 0.92565025 0.07230074
##  [79] 0.11418009 0.96135869 0.87254980 0.91922786 0.36321021 0.69740518
##  [85] 0.30405416 0.31041153 0.75749958 0.60536770 0.54216341 0.68381076
##  [91] 0.15172186 0.96143312 0.85412968 0.10459713 0.05507493 0.36998563
##  [97] 0.88612864 0.72052262 0.63245506 0.01691995 0.67097348 0.36255805
## [103] 0.12176270 0.70869713 0.39774972 0.44677126 0.36736619 0.23390092
## [109] 0.54075390 0.12639385 0.12645854 0.44962048 0.53173832 0.81549407
## [115] 0.91748569 0.37295439 0.13341160 0.19878029 0.64895786 0.86315128
## [121] 0.60841719 0.59725417 0.58486342 0.79006797 0.07801441 0.18120299
## [127] 0.65384893 0.63905742 0.77793426 0.62665547 0.31021408 0.57939099
## [133] 0.22788678 0.53561424 0.73095747 0.17901542 0.30741357 0.87255538
## [139] 0.45321659 0.65358334 0.73328685 0.23972422 0.23765369 0.51655389
## [145] 0.94082024 0.58105457 0.07309402 0.33489835 0.24403258 0.55213145
## [151] 0.77263334 0.34310346 0.04407531 0.29231170 0.99551923 0.22098596
## [157] 0.70627663 0.53270382 0.24562315 0.03763194 0.48523559 0.54224612
## [163] 0.29783432 0.63881276 0.92901964 0.82479777 0.47329778 0.75461399
## [169] 0.58545266 0.48678557 0.53916649 0.53118117 0.68093962 0.27580852
## [175] 0.53787904 0.16502412 0.23785096 0.19058843 0.35841568 0.75612221
## [181] 0.79133261 0.53570833 0.41291834 0.87626691 0.14428686 0.81778404
## [187] 0.23101922 0.43090716 0.91211336 0.62450334 0.73002495 0.38855775
## [193] 0.85945624 0.13412742 0.58676418 0.73672190 0.23506253 0.56226367
## [199] 0.13167595 0.96163009
## 
## $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")

dane <- diagnoza[, c("gp64", "plec")]

ggplot(dane, aes(x = gp64, color = plec, fill = plec)) +
  geom_density(alpha = 0.3) +
  labs(
    title = "Rozkład gęstości dochodu netto według płci",
    x = "Dochód miesięczny netto",
    y = "Gęstość",
    color = "Płeć",  
    fill = "Płeć"     
  ) +
  theme(
    text = element_text(color = "#555555"),
    plot.title = element_text(size = 10),
    axis.title = element_text(size = 10, color = '#555555'),
    axis.title.y = element_text(vjust = .5, angle = 90),
    axis.title.x = element_text(hjust = .5),
    legend.position = "bottom",
    legend.title = element_text(size = 9)
  )

## Warning: Removed 18773 rows containing non-finite outside the scale range
## (`stat_density()`).

wykres_dystrybuanta <- ggplot(dane, aes(x = gp64, color = plec)) +
  stat_ecdf(geom = "step") +
  labs(
    title = "Skumulowana gęstość dochodu netto według płci",
    x = "Dochód miesięczny netto",
    y = "Skumulowana gęstość",
    color = "Płeć"  # Zmienia nazwę legendy dla color
  ) +
  theme(
    text = element_text(color = "#555555"),
    plot.title = element_text(size = 10),
    axis.title = element_text(size = 10, color = '#555555'),
    axis.title.y = element_text(vjust = .5, angle = 90),
    axis.title.x = element_text(hjust = .5),
    legend.position = "bottom",
    legend.title = element_text(size = 9)
  )

wykres_dystrybuanta

## Warning: Removed 18773 rows containing non-finite outside the scale range
## (`stat_ecdf()`).

wykres1 <- ggplot(data = diagnoza, aes(x = gp64, fill = plec)) +
  geom_density(alpha = 0.5) +
  labs(
    title = "Rozkład gęstości dochodu miesięcznego netto według województwa i płci",
    x = "Dochód miesięczny netto (gp64)",
    y = "Gęstość",
    fill = "Płeć"
  ) +
  facet_wrap(~ wojewodztwo) +  
  theme(
    text = element_text(color = "#555555"),
    plot.title = element_text(size = 10),
    axis.title = element_text(size = 10, color = '#555555'),
    axis.title.y = element_text(vjust = .5, angle = 90),
    axis.title.x = element_text(hjust = .5),
    legend.position = "bottom",
    legend.title = element_text(size = 9)
  )

wykres1

## Warning: Removed 18773 rows containing non-finite outside the scale range
## (`stat_density()`).

wykres2 <- ggplot(data = diagnoza, aes(x = gp64, color = plec)) +
  stat_ecdf(geom = "step") +
  labs(
    title = "Skumulowana gęstość dochodu miesięcznego netto (gp64) według województwa i płci",
    x = "Dochód miesięczny netto (gp64)",
    y = "Skumulowana gęstość",
    color = "Płeć"
  ) +
  facet_wrap(~ wojewodztwo) + 
  theme(
    text = element_text(color = "#555555"),
    plot.title = element_text(size = 10),
    axis.title = element_text(size = 10, color = '#555555'),
    axis.title.y = element_text(vjust = .5, angle = 90),
    axis.title.x = element_text(hjust = .5),
    legend.position = "bottom",
    legend.title = element_text(size = 9)
  )

wykres2

## Warning: Removed 18773 rows containing non-finite outside the scale range
## (`stat_ecdf()`).