Учитывая набор случайных чисел, взятых из непрерывного одномерного распределения, найдем распределение

Учитывая набор действительных чисел, взятых из неизвестного непрерывного одномерного распределения (скажем, один из бета, коши, хи-квадрат, экспоненциальный, F, гамма, лаплас, лог-нормальный, нормальный, парето, студент t, равномерное и Weibull).

x <- c(7.7495976,12.1007857,5.8663491,9.9137894,11.3822335,7.4406175,8.6997212,9.4456074,11.8370711,6.4251469,9.3597039,8.7625700,10.3171063,8.0983110,11.7564283,11.7583461,7.3760516,14.5713098,14.3289690,12.8436795,7.1834376,12.2530520,8.9362235,11.8964391,5.4378782,7.8083060,0.1356370,14.9341847,6.8625143,9.0285873,10.2251998,10.3348486,7.7518365,2.8757024,9.2676577,10.6879259,11.7623207,14.0745924,9.3478318,7.6788852,9.7491924,14.9409955,11.0297640,8.5541261,8.6129808,9.2192320,12.3507414,8.9156903,11.6892831,10.2571897,11.1673235,10.5883741,8.2396129,7.3505839,3.4437525,8.3660082,10.5779227,8.5382177,13.6647484,9.0712034,4.1090454,13.4238382,16.1965937,14.2539891,14.6498816,6.9662381,12.3282141,10.9628268,10.8859495,11.6742822,12.0469869,9.1764119,4.2324549,12.6665295,10.7467579,6.4153703,10.3090806,12.0267082,9.2375369,13.8011813,13.0457227,14.0147179,6.9224316,7.1164269,10.7577799,8.0965571,13.3371566,14.6997535,8.8248384,8.0634834,10.2226001,8.5112199,8.1701147,8.1970784,10.5432878,5.9603389,6.6287037,13.3417943,3.1122822,10.4241008,11.4281520,9.4647825,10.5480176,14.2357819,9.4220778,9.7012755,10.9251006,5.3073151,10.8228672,12.0936384,8.5146227,8.4115865,7.7244591,7.2801474,7.3412563,4.5385940,7.8822841,12.7327836,11.5509252,13.0300876,10.0458138,11.3862972,11.3644867,12.6585391,5.8567192,9.8764841,7.6447620,8.7806429,9.2089114,9.1961781,7.2400724,14.7575303,8.6874476,4.6276043,14.0592724,10.3519708,8.2222625,8.7710501,8.5724602,11.4279232,9.6734741,12.1972490,10.1250074,4.8571327,8.0019245,9.8036286,17.7386541,10.8935339,4.7258581,14.2681556,7.4236474,9.4520797,9.2066764,7.7805317,0.4938756,13.0306624,8.0225287,11.1801478,8.7481126,16.5873192,6.0404763,9.5674318,10.8915023,13.2473727,5.5877557,1.4474869,10.9504070,10.8879749,10.7765684,9.1501230,11.0798794,10.0961631,9.5913525,14.0855129,7.3918195,16.6303158,9.1436327,11.9848346,11.4691572,16.0934172,13.1431040,8.2455786,10.7388841,13.7107201,9.6223990,7.6363513,9.5731838,7.0150930,14.1341888,7.5834625,13.8362695,12.9790060,10.4156690,6.4108920,6.3731019,6.3302824,8.4924571,11.2175143,11.6346609,6.0958761,12.8728176,10.2689647,9.7923411,11.3962741,7.3723701,8.1169299,9.7926014,8.7266379,10.7350973,12.7639103,7.4425159,15.9422109,9.9073852,6.2421614,5.2925668,9.9822059,13.9768971,9.3481404,6.8102106,12.6482884,9.8595946,12.8946675,6.3519119,9.2698768,4.9538608,13.8062408,14.7438135,8.5583994,12.4232260,9.4205371,13.6507205,11.7807767,10.9747222,15.9299602,10.0202244,11.9209419,12.8159324,7.0107459,7.8076222,8.0086965,14.7694984,6.4810687,6.6833260,3.9660939,16.2414479,9.3474497,10.2626126,11.7672786,10.1245905,2.3416774,9.2548226,12.3498943,9.1731074,8.6703280,3.8079927,12.0858349,11.1027140,11.9034505,11.1981903,9.5554276,11.5333311,4.1374535,7.9397446,10.6732513,5.4928081,5.9026714,7.1902350,7.3516027,9.5251792,12.8827838,8.6051567,9.9074448,4.7244414,9.4681156,17.4316786,15.0770196,7.4215510,7.2839984,8.2040354,11.2938556,12.2308244,17.2933409,5.7154747,9.9383524,7.9912142,10.2087560,13.0489301,10.2092634,11.4029668,10.3103281,10.2810316,8.9487624,14.2699307,12.8538251,10.7545354,18.0638133,7.2115769,7.4020585,7.9737234,13.1687588,13.7186238,9.6881618,4.2991770,11.4829896,8.0113006,10.0285544,8.3325591,8.8476239,9.3618137,11.0913308,10.2702207,12.0215701,11.8083744,8.1575837,10.0413629,11.7291752,13.8315537,12.4823312,13.3289096,8.5874403,9.8624401,7.0444818,13.9701389,10.0250634,14.3841966,17.4074390,13.1290358,8.3764673,7.8796107,6.4597773,12.4989708,11.3617236,5.0730931,13.5990536,9.4800716,11.1247161,12.6283343,12.5711367,10.8075848,13.2183856,12.4566869,17.0046899,9.9132293,13.8912393,10.4806343,6.7550983,18.4982020,4.6835563,4.6068688,8.4304188,7.8747286,9.4440702,12.1033704,10.7397568,12.4483258,12.0952273,9.4609549,16.1755646,13.2110564,12.5244792,14.5511670,14.9365263,6.6852081,14.6988321,9.8833093,11.1549852,14.4090081,6.2565184,8.3488705,10.8509966,7.6795679,13.5814813,10.1733942,12.1773482,4.7032686,9.9248308,17.7067155,8.2378404,12.8208154,12.7675305,9.0907063,9.5720411,4.5536981,5.2252539,10.7393508,8.1761239,7.8011878,10.8517959,12.8793471,10.1738281,9.0522516,9.7020267,8.5743543,7.1063673,9.4366173,7.5154902,9.2420952,13.7275687,8.2097051,12.4686117,8.6426135,10.6854081,14.8617929,14.2631291,11.1449327,8.4807248,5.9399190,6.7772300,7.2566033,10.3215210,9.2483564,10.8592844,13.8227188,5.8955118,6.8936159,11.4641992,8.6535466,14.1301887,10.2194653,9.3929177,11.8592296,9.3153675,10.8574024,9.5293558,14.1394531,7.1224090,5.6785198,13.1351723,7.1031658,7.6344684,8.6918016,6.8426780,8.6902514,9.9025967,6.1603559,6.3995948,6.7157089,14.9359341,13.1275476,11.2493476,10.7684760,8.5263731,5.1711855,10.2432689,6.7908688,9.2634794,5.6242460,7.7319788,13.7579540,10.5344149,11.2123002,9.5503450,11.3042249,6.6581916,13.0363709,9.0141363,6.8815546,8.6309000,9.4825677,6.9816465,9.4836443,8.5629547,12.5643187,13.2918150,4.9542483,3.8941388,12.0723769,14.6818075,6.2067566,8.6538934,11.4860264,9.6481396,12.7096758,7.8361298,12.0167492,9.2011051,6.7472607,13.5725275,15.0862343,12.5248807,10.8804527,12.7291198,7.7527975,7.8537703,10.5257599,11.2615216,5.2586963,9.3935784,4.8959811,14.9649019,9.7550081,9.0961317,3.0822901,10.4690830,11.4116176,11.8268286,9.6303294,12.6595176,10.3003485,10.6738841,7.1545388,13.1700952,8.8394611,11.7666496,5.3739818,12.5156287,10.5998309,7.9280247,11.3985509,9.3435626,9.1445783,7.5190392,10.5207065,5.5194295,14.4021779,7.9815022,7.3148241,5.0131517,12.1867856,3.4892615,14.7278153,10.0177503,9.0080577,6.2549383,11.5792232,10.0743671,4.6603495,9.1943305,10.0549778,13.3946923,11.0435648,11.9903902,7.5212459,6.9752799,9.7793759,3.0074422,9.9630136,8.2949444,14.4448033,8.8767257,10.4919437,12.8309614,11.9987884,9.4450733,7.1909711,7.7836130,12.0111407,7.8110426,8.8857522,7.2070115,6.1091037,15.5397454,12.4138856,11.0948175,10.3384724,4.0731303,11.9523302,11.7543732,8.6845056,11.3963952,9.1248950,9.8663549,14.4536098,10.5610537,9.6523570,9.9533877,10.1019772,12.0909679,12.1466894,9.8986813,14.2406526,10.1251599,13.5607593,8.3409267,7.3538062,9.2187909,8.3878572,9.6934979,6.8270478,6.9754722,14.7438670,6.2118150,4.3408116,11.4874280,12.9580969,9.5487183,10.2743684,11.2433385,14.4445854,10.3395096,5.7534609,10.5550234,10.9322053,10.2105928,11.3020951,12.9484069,6.5904212,8.4368601,11.3280691,8.6031823,7.6938566,11.3733151,12.3900593,11.7711757,11.2307516,13.4915701,10.7228153,7.3886924,8.4401787,10.2753493,8.4389663,12.1972728,10.4918743,10.6289742,10.5594228,6.7236908,11.2358099,8.5938861,12.3906280,14.4511787,7.4746119,15.8803774,2.5522927,9.6801286,8.5697501,10.8271935,13.5280438,10.6818935,13.5646711,3.5187030,10.4440143,9.8327296,9.7382627,14.1669606,6.9083257,3.8266181,13.6244062,11.0284378,9.5523319,8.9891586,9.9055215,8.3856238,8.7478998,6.6987620,14.7248918,9.2529918,10.2082195,4.9534370,9.2030317,5.2269606,8.0661516,13.1779369,5.2971835,15.0037013,7.2702621,6.9997505,9.6490126,13.9149660,10.7425870,9.7558964,12.5752855,10.5098261,20.2689637,9.8681830,7.8259004,9.4911900,9.6024895,7.6085691,12.0086596,6.6780724,8.2764670,8.9880572,15.9231426,5.9905542,13.5816388,8.9839322,9.5235545,10.1314783,13.1174616,8.1648447,12.5653484,12.4941364,10.5916275,12.7761500,9.8608664,8.1374522,10.6055768,6.5465219,11.7945966,7.0397647,4.4046833,12.4284773,0.4180241,12.0268339,10.0441325,5.3276329,8.4208769,8.5484829,9.8222639,9.4951750,9.3263556,13.7433301,10.1112279,12.3558939,10.8694158,9.7864777,5.5161601,7.0906274,14.5786803,12.9236138,8.9206195,7.0104273,5.8283839,7.6944516,6.2924265,10.0766522,10.3576597,8.5793193,11.2022858,4.9360148,6.5907700,13.0853471,9.5498965,10.8132248,7.3545704,9.3583861,10.5726301,6.8032692,9.5914570,6.1383186,7.0176580,16.8026498,6.7959168,9.2745414,7.7390857,12.5977623,8.6116698,13.6735060,10.8476068,9.6710713,10.1086791,9.6101003,11.2849373,14.3841286,10.0175111,5.9766042,9.2654916,12.3336237,11.0695365,9.4801954,6.6405542,11.7110714,9.2962742,4.5557592,7.9725970,10.3105591,9.1068024,8.1585631,14.9021906,9.2015137,15.0472571,9.1225965,13.9551835,15.1033478,10.6360240,12.0867865,15.6969704,9.5818060,8.1641150,8.2950194,8.6544478,7.9130456,8.8904450,13.9381998,8.9913977,14.0155779,6.2856039,10.7923301,8.8070441,11.2657258,10.7901363,9.1724396,6.6433443,9.5172255,12.3402514,2.7254577,12.4006210,13.2697124,10.0670987,15.3858112,8.2044828,10.7534955,7.9282064,10.9170642,12.8222748,18.2680638,9.0601854,13.2616197,7.0193571,12.2447467,5.3729936,14.8064727,10.5359554,10.4851627,11.8312380,13.3435483,10.5894537,5.0047413,7.5532502,11.9171854,12.1777692,7.6730359,5.5515027,12.3027227,10.1575062,14.8505769,9.6526219,11.2016182,10.7898901,13.6303578,12.8561220,13.3002161,9.0945849,4.9117132,8.0514791,8.3684288,4.7461608,6.3118847,14.3888758,15.8801467,11.6563489,7.9043481,6.1992280,10.4055679,6.4948166,11.8656277,3.8399970,9.5901581,8.6379262,7.4541442,7.1135626,7.9164363,9.6439593,15.6259631,7.3244170,8.4635798,12.0317526,17.1847365,12.5357554,6.0369018,12.9830581,11.2712555,12.3488084,9.3935706,8.1248854,11.4523131,9.6710694,9.5978474,15.1563587,7.5582530,10.8587757,13.5890062,10.1390991,8.1443215,16.1032757,6.5988579,9.6915113,7.6946942,10.5688193,7.9222074,6.0964578,7.0383112,11.5956154,6.6059072,13.5679685,15.1021379,10.2625096,10.2202339,15.7814051,16.3342713,6.1339245,0.9275113,15.8169582,11.0888355,7.8822788,15.2039942,9.6944328,11.7292036,11.6230714,8.4657438,7.6462181,7.1888162,8.1788400,13.7221572,12.4793501,10.4488461,8.9233659,8.9305724,7.4913262,12.5882791,10.6825315,10.8527571,12.1660301,12.4390247,13.8529219,8.5372836,11.2575812,6.4922496,9.5404721,10.7082122,11.2365487,10.2713802,14.8685632,10.7735798,10.6526134,4.8455022,8.3135583,10.8120056,7.2903999,7.0497880,4.9958942,5.9730174,9.8642732,11.5609671,10.1178216,6.6279774,9.2441754,9.9419299,13.4710469,6.0601435,8.2095239,7.9456672,12.7039825,7.4197810,9.5928275,8.2267352,2.8314614,11.5653497,6.0828073,11.3926117,10.5403929,14.9751607,11.7647580,8.2867261,10.0291522,7.7132033,6.3337642,14.6066222,11.3436587,11.2717791,10.8818323,8.0320657,6.7354041,9.1871676,13.4381778,7.4353197,8.9210043,10.2010750,11.9442048,11.0081195,4.3369520,13.2562675,15.9945674,8.7528248,14.4948086,14.3577443,6.7438382,9.1434984,15.4599419,13.1424011,7.0481925,7.4823108,10.5743730,6.4166006,11.8225244,8.9388744,10.3698150,10.3965596,13.5226492,16.0069239,6.1139247,11.0838351,9.1659242,7.9896031,10.7282936,14.2666492,13.6478802,10.6248561,15.3834373,11.5096033,14.5806570,10.7648690,5.3407430,7.7535042,7.1942866,9.8867927,12.7413156,10.8127809,8.1726772,8.3965665)

.. есть ли простой способ в R до программно и автоматически найти наиболее вероятное распределение и оцененные параметры распределения?

Обратите внимание, что идентификационный код распространения будет частью автоматизированного процесса, поэтому ручное вмешательство в идентификацию не будет возможным.

Ответы

x <- c(15.771062,14.741310,9.081269,11.276436,11.534672,17.980860,13.550017,13.853336,11.262280,11.049087,14.752701,4.481159,11.680758,11.451909,10.001488,11.106817,7.999088,10.591574,8.141551,12.401899,11.215275,13.358770,8.388508,11.875838,3.137448,8.675275,17.381322,12.362328,10.987731,7.600881,14.360674,5.443649,16.024247,11.247233,9.549301,9.709091,13.642511,10.892652,11.760685,11.717966,11.373979,10.543105,10.230631,9.918293,10.565087,8.891209,10.021141,9.152660,10.384917,8.739189,5.554605,8.575793,12.016232,10.862214,4.938752,14.046626,5.279255,11.907347,8.621476,7.933702,10.799049,8.567466,9.914821,7.483575,11.098477,8.033768,10.954300,8.031797,14.288100,9.813787,5.883826,7.829455,9.462013,9.176897,10.153627,4.922607,6.818439,9.480758,8.166601,12.017158,13.279630,14.464876,13.319124,12.331335,3.194438,9.866487,11.337083,8.958164,8.241395,4.289313,5.508243,4.737891,7.577698,9.626720,16.558392,10.309173,11.740863,8.761573,7.099866,10.032640)
> qqnorm(x)

> library(MASS)
> fitdistr(x, 't')$loglik
[1] -252.2659
Warning message:
In log(s) : NaNs produced
> fitdistr(x, 'normal')$loglik
[1] -252.2968
> fitdistr(x, 'logistic')$loglik
[1] -252.2996
> fitdistr(x, 'weibull')$loglik
[1] -252.3507
> fitdistr(x, 'gamma')$loglik
[1] -255.9099
> fitdistr(x, 'lognormal')$loglik
[1] -260.6328
> fitdistr(x, 'exponential')$loglik
[1] -331.8191
Warning messages:
1: In dgamma(x, shape, scale, log) : NaNs produced
2: In dgamma(x, shape, scale, log) : NaNs produced

library(fitdistrplus)

f1<-fitdist(x,"norm",method="mle")

b_best<-bootdist(f_best)
print(f_best)
plot(f_best)
summary(f_best)

library(ig)
fitdist(x,"igt",method="mle",start=list(mu=mean(x),lambda=1))$loglik

library(fitdistrplus)

distributions = c("norm", "lnorm", "exp",
          "cauchy", "gamma", "logis",
          "weibull")


# the x vector is defined as in the question

# Plot to see which distributions make sense. This should influence
# your choice of candidate distributions
descdist(x, discrete = FALSE, boot = 500)

distr_aic = list()
distr_fit = list()
for (distribution in distributions) {
    distr_fit[[distribution]] = fitdist(x, distribution)
    distr_aic[[distribution]] = distr_fit[[distribution]]$aic
}

> distr_aic
$norm
[1] 5032.269

$lnorm
[1] 5421.815

$exp
[1] 6602.334

$cauchy
[1] 5382.643

$gamma
[1] 5184.17

$logis
[1] 5047.796

$weibull
[1] 5058.336

> distr_fit[['norm']]
Fitting of the distribution ' norm ' by maximum likelihood 
Parameters:
     estimate Std. Error
mean 9.975849 0.09454476
sd   2.989768 0.06685321