Maximum likelihood of Compound Poisson Distributions - r

I'm trying to compute a maximum likelihood of the compound Poisson-Gamma distribution in R. The distribution is defined by $ \sum_{j=1}^{N} Y_j $ where $Y_n$ is i.i.d sequence independent $gamma(k,\theta)$ values and $N$ is a Poisson distribution with parameter $\beta$. I'm trying to estimate the parameters $\theta$ and $\beta$ without luck.


If you wanted to do something similar, but for a negative binomial distribution, then you can use the the function negbin.mle from the package Rfast
y <- rpois(100, 2)
Rfast::negbin.mle(y)
Output
$iters
[1] 5
$loglik
[1] -162.855
$param
success probability number of failures mean
0.9963271 480.1317031 1.7700000
Also if you run the command:
Rfast::negbin.mle
You can see what the function is computing.
You can also check the functions manual with:
?Rfast::negbin.mle
Edit:
Unfortunately I haven't found something that perfectly fits your answer.
As Ben states, this answer is for a Poisson with Gamma-distributed mean.

Related

Using GAMLSS, the difference between fitDist() and gamlss()

When using the GAMLSS package in R, there are many different ways to fit a distribution to a set of data. My data is a single vector of values, and I am fitting a distribution over these values.
My question is this: what is the main difference between using fitDist() and gamlss() since they give similar but different answers for parameter values, and different worm plots?
Also, using the function confint() works for gamlss() fitted objects but not for objects fitted with fitDist(). Is there any way to produce confidence intervals for parameters fitted with the fitDist() function? Is there an accuracy difference between the two procedures? Thanks!

m1 <- fitDist()
fits many distributions and chooses the best according to a
generalized Akaike information criterion, GAIC(k), wit penalty k for each
fitted parameter in the distribution, where k is specified by the user,
e.g. k=2 for AIC,
k = log(n) for BIC,
k=4 for a Chi-squared test (rounded from 3.84, the 5% critical value of a Chi-squared distribution with 1 degree of fereedom), which is my preference.
m1$fits
gives the full results from the best to worst distribution according to GAIC(k).

nls error during the parameter estimation of power law with exponential cutoff distribution in R

I want to fit mydata with several known distributions, power law with exponential cutoff distribution is one of the candidates.
fitdistr function in package fitdistrplus is one of good methods to use for the parameter estimation using MLE, or MME, or QME.
But power law with exponential cutoff is not the base probability function according to CRAN Task View: Probability Distributions , so I try the nls function.
The pdf of power law with exponential cutoff is f(x;α,λ)=C*x^(−α)*exp(−λ*x)
First, I generate some random values to replace my real data:
data <- rlnorm(1000,0.6,1.23)
h <- hist(data,breaks=1000,plot=FALSE)
x <- h$mids
y <- h$density
Then, I use nls function to conduct parameter estimation:
nls(y~c*x^(-a)*exp(-b*x),start=list(a=1,b=1,c=1))
But it does not work and always throws one of these two errors:
Error in numericDeriv(form[[3L]], names(ind), env) : Missing value or an infinity produced when evaluating the model
Or: singular gradient matrix at initial parameter estimates
Before posting, I have read almost all the previous posts and google, there are several reasons for the errors:
bad start values for the nls. I tried a lot, but it does not work.
some negative values or values less than 1 or values equal to Inf may be generated. I tried to do the data cleaning, also, it does not work.
What should I do now? Or are there some other better methods to do the parameter estimation of power law with exponential cutoff? I need your help, thank you!

Inverse Moments of a Non-Central Chi-Square Distribution

I want to compute inverse moments and truncated inverse moments of a non-central chi-square distribution in R. How can I do that in R?
Suppose X follows the non-central chi-square distribution with degrees of freedom "k" and non-centrality parameter "t". My problem is to numerically compute the following expectations for various values of "t" so I can simulate the risk of James-Stein type estimators.
(i) E[X^(-1)] and E[X^(-2)]
(ii) E[X^(-1)I(A)] where I(A) is an indicator function of set A
(iii) E[1-c{X^(-2)}I(A)] where c is a constant.

In general, you can numerically compute the expected value of a random variable by drawing a large number of samples and then averaging them. For instance, you could estimate the expected values of X^(-1) and X^(-2) with something like:
mean(rchisq(1000000, df=3, ncp=10)^-1)
# [1] 0.1152163
mean(rchisq(1000000, df=3, ncp=10)^-2)
# [1] 0.1371877

Paolella's book, Intermediate Probability gives the moments of the non-central chi-square to various powers. See equation (10.10). You can find R code for these in the sadists package.

How to fit frequency distributions in R?

Is there a function that could be used to fit a frequency distribution in R? I'm aware of fitdistr but as far as I can tell it only works for data vectors (random samples). Also, I know that converting between the two formats is trivial but frequencies are so large that memory is a concern.
For example, fitdistr may be used the following way:
x<-rpois(100, lambda=10)
fitdistr(x,"poisson")
Is there a function that would do the same fitting on a frequency table? Something along the lines:
freqt <- as.data.frame(table(x))
fitfreqtable(freqt$x, weights=freqt$Freq, "poisson")
Thanks!

There's no built-in function that I know of for fitting a distribution to a frequency table. Note that, in theory, a continuous distribution is inappropriate for a table, since the data is discrete. Of course, for large enough N and a fine enough grid, this can be ignored.
You can build your own model-fitting function using optim or any other optimizer, if you know the density that you're interested in. I did this here for a gamma distribution (which was a bad assumption for that particular dataset, but never mind that).
Code reproduced below.
negll <- function(par, x, y)
{
shape <- par[1]
rate <- par[2]
mu <- dgamma(x, shape, rate) * sum(y)
-2 * sum(dpois(y, mu, log=TRUE))
}
optim(c(1, 1), negll, x=seq_along(g$count), y=g$count, method="L-BFGS-B", lower=c(.001, .001))
$par
[1] 0.73034879 0.00698288
$value
[1] 62983.18
$counts
function gradient
32 32
$convergence
[1] 0
$message
[1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"

For fitting a Poisson distribution, you only need the mean of your sample. Then the mean equals the lambda, which is the only parameter of the Poisson distribution. Example:
set.seed(1111)
sample<-rpois(n=10000,l=10)
mean(sample)
[1] 10.0191
which is almost equal to the lambda value put for creating the sample (l=10). The small difference (0.0191) is due to the randomness of the Poisson distribution random value generator. As you increase n the difference will get smaller.
Alternatively, you can fit the distribution using an optimization method:
library(fitdistrplus)
fitdist(sample,"pois")
set.seed(1111)
Fitting of the distribution ' pois ' by maximum likelihood
Parameters:
estimate Std. Error
lambda 10.0191 0.03165296
but it's only a waste of time.
For theoritical information on fitting frequency data, you can see my answer here.

The function fixtmixturegrouped from the package ForestFit does the job for other distribution models using frequency-by-group data.
It can fit simple or mixture distribution models based on "gamma", "log-normal", "skew-normal", and "weibull".
For a Poisson distribution, the population mean is the only parameter that is needed. Applying a simple summary function on your data would suffice (as suggested by ntzortzis)

Given a set of random numbers drawn from a continuous univariate distribution, find the distribution

Given a set of real numbers drawn from a unknown continuous univariate distribution (let's say is is one of beta, Cauchy, chi-square, exponential, F, gamma, Laplace, log-normal, normal, Pareto, Student's t, uniform and 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)
.. is there some easy way in R to programmatically and automatically find the most likely distribution and the estimated distribution parameters?
Please note that the distribution identification code will be part of an automated process, so manual intervention in the identification won't be possible.

My first approach would be to generate qq plots of the given data against the possible distributions.
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)
For more info see link
Another possibility is based on the fitdistr function in the MASS package. Here is the different distributions ordered by their log-likelihood
> 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

Another similar approach is using the fitdistrplus package
library(fitdistrplus)
Loop through the distributions of interest and generate 'fitdist' objects. Use either "mle" for maximum likelihood estimation or "mme" for matching moment estimation, as the fitting method.
f1<-fitdist(x,"norm",method="mle")
Use bootstrap re-sampling in order to simulate uncertainty in the parameters of the selected model
b_best<-bootdist(f_best)
print(f_best)
plot(f_best)
summary(f_best)
The fitdist method allows for using custom distributions or distributions from other packages, provided that the corresponding density function dname, the corresponding distribution function pname and the corresponding quantile function qname have been defined (or even just the density function).
So if you wanted to test the log-likelihood for the inverse normal distribution:
library(ig)
fitdist(x,"igt",method="mle",start=list(mu=mean(x),lambda=1))$loglik
You may also find Fitting distributions with R helpful.

(Answer edited to add additional explanation)
You can't really find "the" distribution; the actual distribution from which data are drawn can nearly always* be guaranteed not to be in any "laundry list" provided by any such software. At best you can find "a" distribution (more likely several), one that is an adequate description. Even if you find a great fit there are always an infinity of distributions that are arbitrarily close by. Real data tends to be drawn from heterogeneous mixtures of distributions that themselves don't necessarily have simple functional form.
* an example where you might hope to is where you know the data were actually generated from exactly one distribution on a list, but such situations are extremely rare.
I don't think just comparing likelihoods is necessarily going to make sense, since some distributions have more parameters than others. AIC might make more sense, except that ...
Attempting to identify a "best fitting" distribution from a list of candidates will tend to produce overfitting, and unless the effect of such model selection is accounted for properly will lead to overconfidence (a model that looks great but doesn't actually fit the data not in your sample). There are such possibilities in R (the package fitdistrplus comes to mind), but as a common practice I would advise against the idea. If you must do it, use holdout samples or cross-validation to obtain models with better generalization error.

I find it hard to imagine a realistic situation where this would be useful. Why not use a non-parametric tool like a kernel density estimate?

You could try using the Kolmogorov-Smirnov tests (ks.test in R).
If you have time-to-event data, here's software that does a Bayesian chi squared test against a list of common distributions to report the best fit.

As others have pointed out, this might be framed as a model selection question. It is a wrong approach to use the distribution that fits the data best without taking into account the complexity of the distribution. This is because the more complicated distribution will generally have better fit, but it will likely overfit the data.
You can use the Akaike Information Criteria (AIC) to take into account the complexity of the distribution. This is still unsatisfactory as you're only considering a limited number of distributions, but is still better than just using the log likelihood.
I use just a few distributions, but you can check the documentation to find others that could be relevant
Using the fitdistrplus you can run:
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
According to our plot and the AIC, it makes sense to use a normal. You can automatize this by just picking the distribution with the minimum AIC. You can check the estimated parameters with
> 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

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