Jawaban Tugas Mandiri 3 No 4
Dariani Matualage
2025-02-15
TUGAS MANDIRI 3 NO 4 STA1701
Melakukan pembangkitan data (X) dengan n = 1000 dengan 5 sebaran.
Data 1 dibangkitkan dengan menggunakan sebaran Poisson, selanjutkan akan digunakan untuk menentukan penduga bagi sebaran Poisson, Exponential, LogNormal, Pareto, dan Weibull. Data 2 dibangkitkan menggunakan sebaran Exponential, dan akan digunakan juga untuk menentukan penduga bagi sebaran-sebaran tersebut. Dengan langkah yang sama, saya akan menggunakan data 3 yang dibangkitkan menggunakan sebaran LogNormal, data 4 yang dibangkitkan menggunakan sebaran Pareto, dan data 5 menggunakan sebaran Weibull.
#menggunakan package EstimationTools
library(EstimationTools)## Loading required package: survival## Loading required package: DEoptim## Loading required package: parallel##
## DEoptim package
## Differential Evolution algorithm in R
## Authors: D. Ardia, K. Mullen, B. Peterson and J. Ulrich## Loading required package: BBmisc##
## Attaching package: 'BBmisc'## The following object is masked from 'package:base':
##
## isFALSE## Loading required package: GA## Loading required package: foreach## Loading required package: iterators## Package 'GA' version 3.2.4
## Type 'citation("GA")' for citing this R package in publications.##
## Attaching package: 'GA'## The following object is masked from 'package:utils':
##
## de## Loading required package: gaussquad## Loading required package: orthopolynom##
##
## ><<<<<<<<<<<<<<<<<<<<<<<< EstimationTools Version 4.0.0 >>>>>>>>>>>>>>>>>>>>>>>><
## Feel free to report bugs in https://github.com/Jaimemosg/EstimationTools/issuesset.seed(3041)
data_poisson <- rpois(1000, lambda = 5)
hist(data_poisson,breaks = 15,
main = "Histogram Data Poisson",
xlab = "Data",
ylab = "Frekuensi",
ylim = c(0,250),
xlim = c(0,15),
axes = FALSE)
axis(1)
axis(2, seq(0,200,by=50))
box()
Dengan Data 1, dilakukan pendugaan parameter sebaran Poisson
#Sebaran Poisson
teta_poison1 = maxlogL(x = data_poisson, dist = "dpois", start = 1, optimizer =
"nlminb")
summary(teta_poison1)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 4392.029 4396.937
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## lambda 5.01900 0.07084 70.84 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 1 yang diasumsikan mengikuti sebaran exponential
#Sebaran Exponential
teta_expo1 = maxlogL(x = data_poisson, dist = "dexp", start = 1, optimizer ="nlminb")
summary(teta_expo1)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 5228.461 5233.369
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## rate 0.1992 0.0063 31.62 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---Karena mengandung angka 0, tidak sesuai dengan sebaran LogNormal.
akan dilakukan pendugaan parameter Data 1 yang diasumsikan mengikuti sebaran Pareto
library(extremefit)Tidak memenuhi sebaran Poisson karena mengandung 0
#Weibull menggunakan data yang telah diubah
teta_weibull1 = maxlogL(x = data_poisson, dist = "dweibull", start = c(1, 1), optimizer = "nlminb")## Warning in nlminb(start = start, objective = ll, lower = lower, upper = upper,
## : NA/NaN function evaluationNilai AIC dan BIC dari Hasil pendugaan untuk masing-masing sebaran menggunakan Data 1:
Sebaran AIC BIC
Poisson 4392.029 4396.937
Exponential 5228.461 5233.369
LogNormal Tidak dapat dilakukan
Pareto Tidak dapat dilakukan
Weibull Tidak dapat dilakukan
Karena nilai AIC dan BIC lebih kecil pada Sebaran Poisson, maka data tersebut lebih mendekati ke sebaran Poisson.
—————————————-Data 2———————————————-
Membangkitkan data 2 dari distribusi Eksponential
set.seed(3042)
data_ekspo = rexp(n = 1000, rate = 0.5)
print(data_ekspo)## [1] 9.611970e-01 4.471103e-01 4.604340e-01 3.726131e+00 5.020449e-01
## [6] 4.059542e+00 3.258043e+00 1.717572e+00 2.625011e+00 2.356195e+00
## [11] 6.431947e+00 6.809789e-01 1.052680e+00 1.306791e+00 4.431809e-01
## [16] 3.006671e-01 2.093004e-01 1.821425e+00 4.067016e-01 9.002208e-01
## [21] 2.409847e+00 1.152958e+00 4.405070e+00 6.013685e+00 3.598336e-01
## [26] 7.488996e-01 6.936049e+00 1.283443e+00 1.019377e+00 1.255353e+00
## [31] 4.167802e+00 5.973976e+00 2.003292e+00 6.794869e-01 9.441672e+00
## [36] 7.657564e-01 4.085198e+00 2.886080e+00 6.684481e-01 7.336810e+00
## [41] 2.109726e+00 3.442046e+00 3.679936e+00 2.632842e+00 1.219303e-01
## [46] 1.663411e+00 2.649271e-01 5.038362e-02 9.082583e+00 4.498299e+00
## [51] 3.537351e+00 2.357351e-01 5.993668e-01 1.156797e-01 2.502731e+00
## [56] 1.911479e+00 3.395779e+00 1.723457e+00 1.484345e-01 2.833001e+00
## [61] 1.247897e+00 2.411863e+00 7.532320e+00 3.649915e+00 5.613422e-01
## [66] 5.931805e+00 4.441958e-01 1.859705e+00 1.835541e+00 9.635788e-01
## [71] 6.370734e-01 7.125027e+00 6.822156e-01 6.796949e-01 5.222728e+00
## [76] 7.297738e-02 1.729737e+00 1.854316e-01 8.600749e+00 3.547485e-01
## [81] 3.254861e+00 7.805003e-02 4.477338e-01 3.289487e+00 2.102687e+00
## [86] 9.854721e+00 9.189260e-01 3.206972e+00 7.197867e-02 2.070350e+00
## [91] 8.328154e-01 3.079852e+00 1.561393e+00 1.141460e+00 1.150306e+00
## [96] 2.335721e+00 2.271192e+00 1.854312e+00 2.104132e+00 2.964543e+00
## [101] 1.214253e+00 4.154129e+00 7.695499e-01 1.521411e+00 7.964493e-01
## [106] 3.465812e+00 1.157564e+01 1.087079e+00 7.288953e-01 1.490574e+00
## [111] 1.946109e+00 1.085482e+00 2.582566e-01 7.186599e+00 1.773282e+00
## [116] 1.990491e+00 1.683733e-01 5.547802e-01 5.313996e+00 2.485467e+00
## [121] 5.385935e-01 1.058632e-01 4.563908e+00 4.101445e-01 5.212618e+00
## [126] 2.982472e+00 5.875220e-01 1.345787e+00 7.751094e-01 1.924057e+00
## [131] 3.289180e+00 1.038526e-01 4.752340e+00 6.070730e+00 5.832408e+00
## [136] 1.401991e+00 1.685161e-01 8.154068e-01 9.579108e-02 5.400258e+00
## [141] 1.101521e+01 1.337855e-01 1.918254e+00 3.428566e-02 8.149213e-01
## [146] 2.168051e+00 3.153305e+00 3.863635e+00 3.143021e+00 7.761362e-01
## [151] 1.474254e-01 4.256605e+00 1.024116e+00 4.698896e-01 5.741298e-01
## [156] 4.890438e+00 2.023928e+00 2.272058e+00 2.338579e+00 4.341230e+00
## [161] 2.591778e+00 1.489544e-01 1.332830e+00 3.657053e+00 2.594268e+00
## [166] 1.211449e+00 6.409964e-01 1.247901e+00 1.862002e+00 1.725957e+00
## [171] 1.316120e+00 6.517733e+00 5.756148e-02 5.387814e-01 8.497171e-01
## [176] 5.127500e-01 2.579750e-01 1.248288e+00 4.915687e+00 4.454797e-02
## [181] 1.479806e+00 5.787287e-01 4.829843e-01 2.586168e+00 3.223165e+00
## [186] 2.908464e+00 3.398189e+00 9.935297e-01 1.229291e+00 3.532526e+00
## [191] 2.564283e-01 5.626882e-01 7.174407e-01 3.928704e-01 3.759045e+00
## [196] 2.539671e+00 4.312698e-02 8.498470e-01 4.133591e-01 3.075146e-01
## [201] 4.149653e+00 2.917949e+00 1.924808e+00 1.810470e-01 3.863122e+00
## [206] 3.684155e+00 2.514122e-01 2.530007e+00 9.082550e-02 1.719121e+00
## [211] 2.759130e+00 3.374878e-01 6.239648e-02 2.674026e+00 1.466444e+00
## [216] 1.677978e+00 1.508703e+00 1.053269e+00 6.555014e+00 2.571565e+00
## [221] 5.805751e-01 5.948571e-02 9.753923e+00 2.451121e+00 6.686358e+00
## [226] 4.245241e+00 1.955924e+00 1.777890e+00 5.540561e+00 4.712873e+00
## [231] 3.149612e+00 2.876519e-01 1.657114e+00 2.112319e+00 1.123763e-01
## [236] 1.412339e+00 9.881042e-01 5.870614e-01 2.968755e+00 1.505845e+00
## [241] 4.980610e-01 7.589844e-01 6.518932e+00 1.484278e+00 1.716844e-01
## [246] 1.555554e-01 4.737780e-01 2.417175e-01 3.246995e+00 2.867560e+00
## [251] 3.397288e-01 4.005500e+00 1.938423e+00 2.330013e+00 4.734560e+00
## [256] 9.764221e-01 3.682254e+00 1.072792e+00 1.130131e+00 3.515829e-01
## [261] 1.506823e+00 3.661895e-01 2.123656e+00 1.155775e+01 2.121190e-01
## [266] 4.086418e-01 2.957695e+00 4.827355e+00 1.498715e+00 3.547661e-01
## [271] 2.754774e+00 1.716355e+00 1.938047e+00 1.887418e+00 1.381571e+00
## [276] 3.506277e+00 3.412123e+00 1.205447e+00 3.394137e+00 4.780423e-01
## [281] 5.956790e-01 4.146266e-01 5.901109e-01 6.608509e+00 5.359496e+00
## [286] 2.105448e+00 4.878028e-01 1.400401e+00 1.467470e+00 1.701006e+00
## [291] 3.388727e+00 3.978190e+00 1.049869e+00 9.657178e-01 1.963494e+00
## [296] 4.330452e-01 1.472092e+00 2.745008e+00 5.609738e+00 7.185040e-01
## [301] 9.857578e-01 1.373776e+00 2.441091e+00 4.288679e-01 1.626105e+00
## [306] 2.102005e+00 2.654520e+00 8.636619e-01 1.765622e+00 1.810815e+00
## [311] 1.513302e-01 8.456358e-01 2.614092e+00 7.310127e+00 3.011631e+00
## [316] 1.017209e-01 7.671092e+00 1.230722e+00 4.513878e+00 6.476109e-01
## [321] 3.496771e+00 9.647795e+00 2.460651e+00 4.611289e+00 4.352832e-01
## [326] 2.258983e-01 6.886480e-01 2.810829e+00 1.368372e+00 2.794682e+00
## [331] 1.463760e+00 1.488594e-01 2.365200e+00 9.146679e-01 8.917006e-01
## [336] 1.592558e+00 3.736339e+00 3.279600e-01 1.277071e+01 2.131197e+00
## [341] 9.545109e-01 3.522106e+00 2.377902e+00 3.548352e+00 4.809552e-01
## [346] 1.660620e+00 2.664486e-01 7.258756e+00 1.680958e+00 2.441747e+00
## [351] 1.001908e+00 2.338714e-01 2.128662e+00 4.941977e+00 3.011858e+00
## [356] 3.409913e+00 2.418936e-01 2.552109e+00 1.038840e+00 7.073872e-01
## [361] 1.486499e+00 6.198555e-01 3.676526e-01 2.128920e+00 3.791816e+00
## [366] 3.526955e+00 3.514246e-01 2.609439e+00 4.855416e+00 7.119747e+00
## [371] 2.311972e+00 1.548966e+00 2.723346e-01 5.475496e-01 2.498844e-01
## [376] 8.646042e+00 4.045726e+00 9.811593e-01 3.247876e-01 1.889727e+00
## [381] 1.469307e+00 6.384328e-01 1.211450e-01 1.197338e+00 5.824199e-01
## [386] 2.655837e+00 2.838327e+00 5.162009e-03 1.975006e-01 2.912842e+00
## [391] 1.306590e-01 2.584623e+00 6.686978e+00 7.992000e-01 2.718606e+00
## [396] 2.098266e+00 2.026328e+00 1.936878e+00 4.660290e-01 8.162185e+00
## [401] 3.193057e+00 7.329015e-01 1.135677e+00 2.060659e-01 2.181416e+00
## [406] 1.820039e+00 6.565495e-01 3.431032e+00 1.371881e-01 6.276520e-01
## [411] 7.180961e-01 4.170289e+00 5.693694e+00 3.509268e+00 4.651265e-02
## [416] 3.366107e+00 7.735848e-01 4.436979e+00 1.518435e+01 1.057643e+00
## [421] 1.977713e+00 2.723951e+00 6.383143e-01 2.290590e+00 1.613518e+00
## [426] 1.109587e+00 1.685325e-01 1.486969e+01 4.622430e+00 4.723008e+00
## [431] 3.649784e+00 6.339417e-01 6.389522e-01 2.136206e+00 2.515289e+00
## [436] 4.361572e+00 1.362048e+00 5.421652e-01 6.383806e+00 5.210865e+00
## [441] 1.782455e+00 7.846795e-01 5.523509e+00 2.067096e+00 7.198323e-01
## [446] 1.304831e+00 2.579861e+00 1.042936e+00 4.870795e+00 2.845695e+00
## [451] 1.489017e-01 6.219602e-01 4.930093e+00 4.110458e-02 2.128259e+00
## [456] 2.278594e+00 5.434935e-01 2.998063e+00 2.569549e+00 2.721663e-03
## [461] 4.968883e+00 2.015354e-01 5.954926e+00 7.224900e-01 6.772106e-02
## [466] 1.646679e+00 1.185315e+00 9.980327e-01 1.096701e+00 2.809584e-01
## [471] 1.639314e+00 3.659544e-01 4.240139e-01 5.013653e+00 3.954738e-01
## [476] 5.491864e-01 3.887326e-01 2.671108e-01 4.919620e+00 3.310500e-01
## [481] 1.248634e-01 8.396312e-01 1.521832e+00 5.285259e+00 3.541781e+00
## [486] 9.492300e-01 1.692933e+00 1.895469e+00 1.456384e-01 2.955899e+00
## [491] 1.816637e+00 1.779446e+00 3.808267e+00 2.972394e+00 1.199134e-01
## [496] 7.524607e-01 5.068312e+00 9.457218e-01 9.944094e-01 2.302334e-01
## [501] 4.568875e-01 5.699159e-01 9.452606e+00 1.589484e-01 1.481522e+00
## [506] 1.916701e+00 5.013798e-01 1.971816e-01 2.541517e+00 9.534277e-01
## [511] 5.882941e-01 2.195838e+00 1.016204e+01 4.921269e+00 3.115772e+00
## [516] 1.006823e+00 1.093489e+00 8.104762e-01 2.843209e+00 3.863643e+00
## [521] 1.123818e+00 8.968681e-01 3.517551e+00 1.318319e-01 9.055408e-02
## [526] 5.832014e+00 8.122547e+00 3.457910e+00 5.744490e+00 4.253165e+00
## [531] 1.274986e-01 1.361427e+00 7.658620e+00 1.887234e+00 1.664303e-01
## [536] 1.168657e+00 8.819751e-02 2.454486e+00 1.980809e+00 8.938310e-01
## [541] 1.277765e+00 9.867036e-01 4.652131e+00 2.944300e+00 4.064351e+00
## [546] 2.710768e+00 2.436550e+00 1.474740e-01 2.761967e-01 3.116415e+00
## [551] 7.427746e+00 2.658794e-01 1.040388e-01 7.098966e+00 7.638305e-01
## [556] 4.711586e+00 1.190307e+00 1.483824e-01 9.108276e-01 3.186933e-01
## [561] 2.332543e+00 5.839498e+00 1.726278e+00 3.533648e-01 4.181242e+00
## [566] 6.453571e+00 1.530232e+00 5.262916e-01 1.540595e-01 8.711861e-01
## [571] 7.166208e+00 1.082215e+00 2.325396e+00 1.104211e-01 6.804121e-01
## [576] 6.701746e-02 3.106114e+00 9.044228e-01 4.850343e-01 7.578443e-01
## [581] 1.565444e+00 1.366300e+00 6.103048e-01 8.467531e-01 8.457895e-01
## [586] 1.199946e+01 1.588760e+00 3.125928e-01 1.299397e+00 1.229126e+00
## [591] 1.082425e+00 1.746557e+00 7.681141e-01 5.236555e-01 4.265022e+00
## [596] 2.117732e+00 2.336967e+00 6.502569e-01 1.901807e+00 4.691389e+00
## [601] 2.258471e+00 2.737306e+00 2.651204e+00 2.116715e-01 5.222989e+00
## [606] 1.530239e+00 1.008197e+00 1.408144e+00 4.763166e+00 9.339959e-01
## [611] 7.528431e-01 2.399314e+00 4.553228e-01 1.218614e+01 3.528119e+00
## [616] 1.220626e+00 3.490964e+00 1.590861e+00 6.547265e-01 2.148563e-01
## [621] 2.818947e+00 4.064481e-01 1.169858e+00 2.039229e+00 1.581605e+00
## [626] 3.416760e+00 5.398663e-04 1.309981e+00 2.298368e+00 5.592222e-01
## [631] 2.526774e+00 4.082028e+00 1.092500e+00 4.860325e-01 1.721327e+00
## [636] 9.642107e-01 3.015314e-01 1.770468e+00 3.958116e+00 1.403590e+00
## [641] 4.301982e+00 1.043269e+00 4.625218e-01 1.208533e+00 1.493208e+00
## [646] 8.745771e-01 1.477777e+00 1.351775e+00 5.360403e+00 2.533034e-01
## [651] 4.564402e+00 6.892161e+00 2.413406e+00 8.057006e-01 5.467910e+00
## [656] 1.381943e+00 1.180232e+00 7.394933e-01 1.003065e+00 4.282325e+00
## [661] 5.337605e-02 7.713851e-01 3.255420e+00 4.864365e+00 4.917762e-01
## [666] 1.105809e+00 1.336551e+00 5.473315e+00 4.104274e+00 7.080560e-01
## [671] 1.678578e-01 3.860947e-01 1.212909e+00 6.597365e-01 9.058001e-01
## [676] 3.283287e+00 2.823964e-01 4.654371e-01 3.074461e+00 2.548027e+00
## [681] 2.507828e+00 4.489694e-01 9.221051e-01 6.541084e-01 3.335893e+00
## [686] 2.967396e+00 2.337191e-01 5.442791e+00 3.258042e+00 3.294447e+00
## [691] 3.969854e+00 3.843368e+00 7.082488e+00 4.399079e-01 1.022770e+00
## [696] 3.802958e+00 5.378963e+00 2.246003e+00 6.114682e-01 6.945726e-02
## [701] 2.781306e+00 6.826681e+00 1.035862e+00 9.042342e-01 1.337201e-02
## [706] 3.495085e+00 2.515190e+00 6.740794e-03 1.453948e-01 3.756416e+00
## [711] 3.077126e-01 3.790136e+00 7.426329e-01 1.717744e+00 1.456672e+00
## [716] 2.667202e+00 2.332275e+00 1.865123e+00 1.495910e+00 2.627890e+00
## [721] 4.575125e-01 2.864209e+00 6.765619e-01 7.704293e-01 3.389684e+00
## [726] 3.811282e+00 1.532261e+00 2.683196e+00 3.040219e+00 2.802181e-01
## [731] 5.395858e+00 1.305491e+00 7.542011e-01 1.590148e+00 2.099007e-01
## [736] 4.338719e+00 1.139347e-01 1.704215e-01 1.720370e+00 8.156615e-01
## [741] 1.056458e+00 1.569566e+00 3.757679e+00 9.680486e-01 4.597147e+00
## [746] 1.658396e+00 2.283261e-01 8.660689e-01 7.788148e-01 8.722987e-01
## [751] 2.415878e-01 2.967962e-01 1.703379e+00 3.024805e+00 4.756541e+00
## [756] 5.803374e-01 1.679476e+00 5.731994e-01 3.098382e+00 1.084444e+01
## [761] 2.704418e+00 2.893601e+00 5.712950e+00 4.454776e+00 4.019265e-01
## [766] 1.067381e+00 3.249516e+00 3.960421e+00 3.544373e-01 1.994861e+00
## [771] 9.041294e-01 1.702278e+00 2.351727e+00 4.788464e-01 9.601633e-02
## [776] 1.054727e+00 5.004586e-01 6.686837e-01 3.641418e+00 1.281263e+00
## [781] 4.457303e-02 3.429660e+00 5.588619e-01 5.359428e-01 8.639530e-02
## [786] 1.610469e+00 7.562733e-02 9.755238e-01 2.466902e+00 9.689746e-02
## [791] 1.247320e+00 3.776801e+00 4.216875e+00 8.096550e-01 7.426203e-01
## [796] 1.626456e+00 1.527648e-01 1.191077e+00 2.680981e+00 2.941274e-01
## [801] 6.119938e+00 4.062062e+00 1.857122e+00 4.954357e-01 3.642811e+00
## [806] 1.429154e+00 2.049937e-01 3.985457e-01 5.549292e-01 1.847606e-01
## [811] 2.147234e+00 1.373867e+00 3.008490e+00 8.294675e+00 5.004794e-02
## [816] 2.434718e-01 1.385543e+00 2.537370e+00 1.059230e+00 2.007776e+00
## [821] 1.847250e-01 1.546335e+00 2.611046e-01 1.188682e-01 5.466776e+00
## [826] 1.443231e+00 2.069432e+00 1.707011e+00 4.452378e-01 5.236626e-01
## [831] 2.721634e+00 2.982102e-01 1.835568e-01 2.783480e+00 2.184528e+00
## [836] 2.598599e+00 1.961178e-01 2.012317e+00 7.252556e-03 1.407471e+00
## [841] 2.747799e-01 2.820011e+00 1.347933e+00 1.753757e+00 4.558588e-01
## [846] 3.803909e+00 1.259045e+00 2.270443e+00 4.117717e+00 4.846269e+00
## [851] 6.395828e-01 1.415458e-01 2.847792e-01 1.477357e+00 1.670564e+00
## [856] 6.378132e-01 1.373650e-01 1.034483e+00 9.787844e-01 1.036129e+01
## [861] 1.902367e+00 3.604585e+00 1.216823e+00 1.310464e+00 1.657971e+00
## [866] 4.066808e+00 2.828044e+00 2.092649e+00 5.639767e+00 2.977833e+00
## [871] 8.561882e-01 2.255627e-01 3.281084e-01 6.690428e-01 1.110818e-01
## [876] 2.437704e+00 1.853185e+00 2.149701e+00 4.253368e+00 1.611815e+00
## [881] 1.020402e-01 2.177996e-01 2.760202e+00 2.478908e+00 3.606985e+00
## [886] 4.149911e+00 8.183264e-01 8.077833e-01 1.153882e+00 2.435107e+00
## [891] 2.106925e+00 3.307739e-01 1.521299e+00 1.301622e+00 1.209881e+00
## [896] 6.235670e-03 1.033249e+00 1.369654e+00 1.956513e+00 1.198681e+00
## [901] 8.965734e-01 1.353844e+00 2.700599e+00 2.311106e-01 4.666926e-03
## [906] 3.781777e+00 1.551955e-01 1.507286e+00 1.013871e+00 3.271447e+00
## [911] 2.703113e+00 3.890960e+00 2.988522e+00 2.287355e+00 1.828982e-01
## [916] 1.944944e-01 3.055442e-01 1.830012e+00 2.155500e+00 6.282906e+00
## [921] 2.917882e+00 5.604697e+00 1.995216e-01 1.999955e+00 3.930700e+00
## [926] 2.869382e+00 6.397186e-01 3.442255e+00 1.775683e+00 4.883515e+00
## [931] 3.462320e-01 2.402043e-01 9.472352e-01 2.422385e-01 2.585498e+00
## [936] 1.044155e+00 4.077437e+00 2.257277e+00 3.048783e+00 1.770708e+00
## [941] 3.035593e+00 4.288062e+00 1.021559e+00 1.389056e+00 2.202666e+00
## [946] 9.458874e-01 2.728273e+00 2.738529e-02 5.603979e+00 7.185896e-01
## [951] 2.633818e-01 1.577949e+00 2.547023e+00 3.559599e+00 6.529550e+00
## [956] 6.017661e+00 9.431346e-01 7.755544e-01 4.397052e-01 1.151237e+00
## [961] 4.077276e+00 3.243424e-01 2.675390e+00 4.095496e+00 1.649816e+00
## [966] 6.531947e-01 3.557544e+00 3.452190e-02 5.468492e-02 7.373502e+00
## [971] 2.777584e+00 4.232994e+00 2.685818e+00 2.082003e-01 9.086358e-01
## [976] 9.205899e-01 2.007788e+00 9.769537e-01 4.958766e-02 7.347013e-01
## [981] 1.505164e+00 8.927287e-01 1.234999e+00 4.448032e-01 6.162297e-01
## [986] 1.063918e+00 2.912123e-01 9.450153e-01 1.410403e+00 1.022373e+00
## [991] 1.992624e+00 5.512001e+00 1.101825e+00 7.915929e-01 2.725137e+00
## [996] 6.185298e-01 3.510269e+00 3.206170e-01 2.670166e-01 5.794883e-01hist(data_ekspo,
main = "Histogram Data Eksponential",
xlab = "Data",
ylab = "Frekuensi",
ylim = c(0,400),
xlim = c(0,20))
Saya tidak menggunakan data 2 untuk menghitung parameter sebaran Poisson karena Jenis datanya berbeda. Sebaran Poisson untuk data kategori, sedangkan Eksponential data kontinu.
akan dilakukan pendugaan parameter Data 2 yang diasumsikan mengikuti sebaran exponential
teta_expo2 = maxlogL(x = data_ekspo, dist = "dexp", start = 1, optimizer =
"nlminb")summary(teta_expo2)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 3514.351 3519.258
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## rate 0.46946 0.01485 31.62 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 2 yang diasumsikan mengikuti sebaran LogNormal
teta_lognorm2 = maxlogL(x = data_ekspo, dist = "dlnorm", start = c(1,1), optimizer
= "nlminb")
summary(teta_lognorm2)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 3695.039 3704.854
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## meanlog 0.18307 0.04034 4.538 5.68e-06 ***
## sdlog 1.27572 0.02853 44.721 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 2 yang diasumsikan mengikuti sebaran Weibull
teta_weibull2 = maxlogL(x = data_ekspo, dist = "dweibull", start = c(1,1),
optimizer = "nlminb")
summary(teta_weibull2)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 3516.324 3526.139
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## shape 1.00407 0.02477 40.53 <2e-16 ***
## scale 2.13376 0.07076 30.15 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---Nilai AIC dan BIC dari Hasil pendugaan untuk masing-masing sebaran menggunakan Data 2:
Sebaran AIC BIC
Poisson beda jenis data
Exponential 3514.351 3519.258
LogNormal 3695.039 3704.854
Pareto Tidak dapat dilakukan
Weibull 3516.324 3526.139
Karena nilai AIC dan BIC lebih kecil pada Sebaran Exponential, maka data tersebut lebih mendekati ke sebaran Exponential.
—————————————-Data 3———————————————-
set.seed(3043)
data_lognormal = rlnorm(n = 1000, meanlog = 3, sdlog = 1)Membuat histogram Data 3
# Tampilkan histogram
hist(data_lognormal,
breaks = 50, # Jumlah bin
main = "Histogram Data Lognormal",
xlab = "Nilai",
ylab = "Frekuensi",
probability = TRUE) # Menampilkan dalam bentuk probabilitas
akan dilakukan pendugaan parameter Data 3 yang diasumsikan mengikuti sebaran LogNormal
teta_lognorm3 = maxlogL(x = data_lognormal, dist = "dlnorm", start = c(1,1), optimizer
= "nlminb")## Warning in dlnorm(x = c(3.14235030643907, 20.1270513005772, 6.59394620972228, :
## NaNs produced## Warning in nlminb(start = start, objective = ll, lower = lower, upper = upper,
## : NA/NaN function evaluationsummary(teta_lognorm3)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 8789.07 8798.886
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## meanlog 2.99251 0.03103 96.44 <2e-16 ***
## sdlog 0.98127 0.02194 44.72 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 3 yang diasumsikan mengikuti sebaran Weibull
teta_weibull3 = maxlogL(x = data_lognormal, dist = "dweibull", start = c(1,1),
optimizer = "nlminb")
summary(teta_weibull3)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 8975.197 8985.012
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## shape 1.00320 0.02233 44.93 <2e-16 ***
## scale 32.69465 1.09344 29.90 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---Nilai AIC dan BIC dari Hasil pendugaan untuk masing-masing sebaran menggunakan Data 3:
Sebaran AIC BIC
Poisson beda jenis data
Exponential tidak dapat dilakukan
LogNormal 8789.07 8798.886
Pareto Tidak dapat dilakukan
Weibull 8975.197 8985.012
Karena nilai AIC dan BIC lebih kecil pada Sebaran LogNormal, maka data tersebut lebih mendekati ke sebaran LogNormal.
—————————————-Data 4———————————————-
Data 4 akan dibangkitkan menggunakan sebaran Pareto
set.seed(3044)
library(VGAM)## Loading required package: stats4## Loading required package: splines##
## Attaching package: 'VGAM'## The following objects are masked from 'package:extremefit':
##
## dpareto, ppareto, qpareto, rpareto# Bangkitkan data Pareto dengan shape = 1 dan scale = 1
set.seed(3043)
data_pareto <- VGAM::rpareto(n = 1000, scale = 1, shape = 1)
# Tampilkan histogram
hist(data_pareto,
breaks = 50,
col = "skyblue",
border = "white",
main = "Histogram Data Pareto",
xlab = "Nilai Data",
ylab = "Frekuensi",
probability = TRUE)
akan dilakukan pendugaan parameter Data 4 yang diasumsikan mengikuti sebaran exponential
teta_expo4 = maxlogL(x = data_pareto, dist = "dexp", start = 1, optimizer =
"nlminb")summary(teta_expo4)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 5820.607 5825.515
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## rate 0.148184 0.004686 31.62 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 4 yang diasumsikan mengikuti sebaran LogNormal
teta_lognorm4 = maxlogL(x = data_pareto, dist = "dlnorm", start = c(1,1), optimizer
= "nlminb")summary(teta_lognorm4)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 4683.373 4693.189
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## meanlog 0.97188 0.03005 32.35 <2e-16 ***
## sdlog 0.95015 0.02125 44.72 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 4 yang diasumsikan mengikuti sebaran Pareto
library(actuar)##
## Attaching package: 'actuar'## The following objects are masked from 'package:VGAM':
##
## dgumbel, dlgamma, dpareto, pgumbel, plgamma, ppareto, qgumbel,
## qlgamma, qpareto, rgumbel, rlgamma, rpareto## The following objects are masked from 'package:extremefit':
##
## dburr, dpareto, pburr, ppareto, qburr, qpareto, rburr, rpareto## The following objects are masked from 'package:stats':
##
## sd, var## The following object is masked from 'package:grDevices':
##
## cmlibrary(maxLik)## Loading required package: miscTools##
## Please cite the 'maxLik' package as:
## Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.
##
## If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
## https://r-forge.r-project.org/projects/maxlik/# Estimasi parameter dengan maxlogL()
teta_pareto4 <- maxlogL(x = data_pareto,
dist = "dpareto", # Menggunakan fungsi dari actuar
start = c(1, 1), # Hanya ada dua parameter: shape dan scale
optimizer = "nlminb")
# Cetak hasil estimasi
summary(teta_pareto4)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 4956.169 4965.984
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## shape 2.3809 0.1898 12.544 <2e-16 ***
## scale 6.8454 0.7113 9.624 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 5 yang diasumsikan mengikuti sebaran Weibull
teta_weibull4 = maxlogL(x = data_pareto, dist = "dweibull", start = c(1,1),
optimizer = "nlminb")summary(teta_weibull4)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 5387.785 5397.601
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## shape 0.71763 0.01357 52.87 <2e-16 ***
## scale 4.53964 0.21301 21.31 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---Nilai AIC dan BIC dari Hasil pendugaan untuk masing-masing sebaran menggunakan Data 4:
Sebaran AIC BIC
Poisson beda jenis data
Exponential 5820.607 5825.515
LogNormal 4683.373 4693.189
Pareto 4956.169 4965.984
Weibull 5387.785 5397.601
Karena nilai AIC dan BIC lebih kecil pada Sebaran LogNormal, maka data tersebut lebih mendekati ke sebaran LogNormal.
—————————————-Data 5———————————————-
Data 4 akan dibangkitkan menggunakan sebaran Weibull
set.seed(3045)
data_weibull = rweibull(n = 1000, shape = 2, scale = 1)# Tampilkan histogram
hist(data_weibull,
breaks = 50,
main = "Histogram Data Weibull" ,
xlab = "Nilai Data",
ylab = "Frekuensi",
ylim = c(0,1.5),
xlim = c(0,3),
probability = TRUE)
akan dilakukan pendugaan parameter Data 5 yang diasumsikan mengikuti sebaran exponential
teta_expo5 = maxlogL(x = data_weibull, dist = "dexp", start = 1, optimizer =
"nlminb")summary(teta_expo5)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 1766.353 1771.261
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## rate 1.12505 0.03558 31.62 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 5 yang diasumsikan mengikuti sebaran LogNormal
teta_lognorm5 = maxlogL(x = data_weibull, dist = "dlnorm", start = c(1,1), optimizer = "nlminb")## Warning in dlnorm(x = c(0.827337726336412, 0.280420424396878,
## 0.961273458446978, : NaNs produced## Warning in nlminb(start = start, objective = ll, lower = lower, upper = upper,
## : NA/NaN function evaluation## Warning in dlnorm(x = c(0.827337726336412, 0.280420424396878,
## 0.961273458446978, : NaNs produced## Warning in nlminb(start = start, objective = ll, lower = lower, upper = upper,
## : NA/NaN function evaluationakan dilakukan pendugaan parameter Data 5 yang diasumsikan mengikuti sebaran Pareto
# Estimasi parameter dengan maxlogL()
teta_pareto5 <- maxlogL(x = data_weibull,
dist = "dpareto", # Menggunakan fungsi dari actuar
start = c(1, 1), # Hanya ada dua parameter: shape dan scale
optimizer = "nlminb")## Warning in sqrt(diag(solve(fit$hessian))): NaNs produced# Cetak hasil estimasi
summary(teta_pareto5)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 1769.434 1779.25
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## shape 191346 NaN NaN NaN
## scale 164616 0 -Inf <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---akan dilakukan pendugaan parameter Data 5 yang diasumsikan mengikuti sebaran Weibull
teta_weibull5 = maxlogL(x = data_weibull, dist = "dweibull", start = c(1,1),
optimizer = "nlminb")summary(teta_weibull5)## _______________________________________________________________
## Optimization routine: nlminb
## Standard Error calculation: Hessian from optim
## _______________________________________________________________
## AIC BIC
## 1175.103 1184.918
## _______________________________________________________________
## Estimate Std. Error Z value Pr(>|z|)
## shape 2.02113 0.04908 41.18 <2e-16 ***
## scale 1.00431 0.01657 60.61 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## _______________________________________________________________
## Note: p-values valid under asymptotic normality of estimators
## ---Nilai AIC dan BIC dari Hasil pendugaan untuk masing-masing sebaran menggunakan Data 5:
Sebaran AIC BIC
Poisson beda jenis data
Exponential 1766.353 1771.261
LogNormal tidak dapat dilakukan
Pareto 1769.434 1779.25
Weibull 1175.103 1184.918
Karena nilai AIC dan BIC lebih kecil pada Sebaran Weibull, maka data tersebut lebih mendekati ke sebaran Weibull