0. Datos.

x<-c(3.6,3.0,3.0,2.7,2.0,3.3,2.5,3.0,3.2,2.9,2.3,2.8,2.1,2.9,3.1,2.7,2.7,2.3,
3.3,2.8,3.5,2.0,2.6,2.9,2.5,3.6,3.1,3.4,2.9,2.3,1.6,3.3,3.1,2.8,2.7,2.7,
2.7,2.7,2.6,2.9,3.6,2.7,1.9,2.0,3.0,2.4,3.3,2.6,2.6,3.0,3.1,2.9,2.6,3.5,
2.8,2.1,3.2,2.3,2.2,2.8,2.6,3.1,2.9,2.5,2.8,2.8,3.6,2.6,3.1,2.3,1.9,2.6,
3.1,1.9,2.4,3.5,2.9,2.6,2.1,3.0,3.2,2.8,1.7,3.3,3.2,3.2,2.5,2.1,1.5,3.0,
2.6,3.5,2.3,2.8,2.7,3.0,2.2,2.9,3.5,2.8,2.7,2.4,2.3,3.8,1.8,3.2,3.6,3.1,
3.0,3.2,3.5,3.3,2.7,3.1,2.8,1.8,3.2,2.7,2.7,2.4,3.7,2.7,2.3,2.9,3.0,2.3,
2.8,3.0,2.9,3.5,2.0,3.0,1.8,2.9,3.1,1.6,1.7,3.3,2.7,2.3,3.8,3.4,2.9,2.0,
3.5,1.8,2.2,2.0,2.8,3.0,3.9,3.1,2.6,3.2,3.2,2.6,2.7,2.5,3.6,2.5,2.9,2.2,
2.5,3.0,3.0,2.4,3.4,3.5,2.8,2.0,3.5,3.4,3.4,3.3,2.7,2.1,2.2,3.4,2.7,3.2,
3.3,2.1,3.0,3.2,1.9,2.4,2.6,3.3,3.0,3.1,2.6,3.7,2.6,2.8,3.0,2.6,3.0,3.4,
3.1,3.1)
y<-c(6,6,2,2,4,6,4,2,3,2,3,5,4,4,1,3,3,5,2,2,5,3,2,2,5,5,0,4,5,3,2,3,1, 1,2,4,5,1,5,3,2,3,4,3,2,2,2,2,2,4)
z<-c(0.723,0.519,0.368,0.440,0.613,0.399,0.626,0.629,0.800,0.650,0.773, 0.699,0.845,0.844,0.281,0.590,0.667,0.639,0.614,0.838,0.650,0.756,0.246,0.562,0.683,0.844,0.811,0.341,0.699,0.749,0.562,0.572,0.641,0.316,0.886,0.544,0.621,0.299,0.755,0.818,0.760,0.396,0.395,0.745,0.809,0.597,0.195,0.802,0.648,0.637)

Gráfico de Cullen y Frey.

descdist(x)

## summary statistics
## ------
## min:  1.5   max:  3.9 
## median:  2.8 
## mean:  2.7965 
## estimated sd:  0.5063792 
## estimated skewness:  -0.3135622 
## estimated kurtosis:  2.622658

1. Distribución Normal.

norma<- fitdist(x, "norm",method=c("mle")) 
summary(norma)
## Fitting of the distribution ' norm ' by maximum likelihood 
## Parameters : 
##       estimate Std. Error
## mean 2.7965000 0.03571679
## sd   0.5051116 0.02525514
## Loglikelihood:  -147.1925   AIC:  298.3851   BIC:  304.9817 
## Correlation matrix:
##      mean sd
## mean    1  0
## sd      0  1
plot(norma)

2. Distribución Lognormal.

lonor<-fitdist(x, "lnorm",method="mle")
summary(lonor)
## Fitting of the distribution ' lnorm ' by maximum likelihood 
## Parameters : 
##          estimate  Std. Error
## meanlog 1.0106087 0.013660126
## sdlog   0.1931833 0.009658003
## Loglikelihood:  -157.0863   AIC:  318.1727   BIC:  324.7693 
## Correlation matrix:
##         meanlog sdlog
## meanlog       1     0
## sdlog         0     1
plot(lonor)

3. Distribución Gamma.

gama<-fitdist(x, "gamma",method="mle")
summary(gama)
## Fitting of the distribution ' gamma ' by maximum likelihood 
## Parameters : 
##       estimate Std. Error
## shape 28.31569   2.815052
## rate  10.12546   1.015592
## Loglikelihood:  -152.7337   AIC:  309.4673   BIC:  316.0639 
## Correlation matrix:
##          shape     rate
## shape 1.000000 0.991185
## rate  0.991185 1.000000
plot(gama)

4. Distribución Exponencial.

expo<-fitdist(x, "exp",method="mle")
summary(expo)
## Fitting of the distribution ' exp ' by maximum likelihood 
## Parameters : 
##       estimate Std. Error
## rate 0.3575898 0.02528522
## Loglikelihood:  -405.6737   AIC:  813.3475   BIC:  816.6458
plot(expo)

5. Distribución Weibull.

wei<-fitdist(x, "weibull",method="mle")
summary(wei)
## Fitting of the distribution ' weibull ' by maximum likelihood 
## Parameters : 
##       estimate Std. Error
## shape 6.459228 0.35802826
## scale 3.003726 0.03464716
## Loglikelihood:  -144.3051   AIC:  292.6102   BIC:  299.2068 
## Correlation matrix:
##           shape     scale
## shape 1.0000000 0.3150358
## scale 0.3150358 1.0000000
plot(wei)

6. Distribución logística.

logi<-fitdist(x, "logis",method="mle")
summary(logi)
## Fitting of the distribution ' logis ' by maximum likelihood 
## Parameters : 
##           estimate Std. Error
## location 2.8162287 0.03609619
## scale    0.2915427 0.01709282
## Loglikelihood:  -150.5521   AIC:  305.1042   BIC:  311.7008 
## Correlation matrix:
##             location       scale
## location  1.00000000 -0.03287957
## scale    -0.03287957  1.00000000
plot(logi)

7. Distribución Cauchy.

cau<-fitdist(x,"cauchy",method="mle")
summary(cau)
## Fitting of the distribution ' cauchy ' by maximum likelihood 
## Parameters : 
##           estimate Std. Error
## location 2.8507454 0.03475877
## scale    0.3054818 0.02760045
## Loglikelihood:  -184.5937   AIC:  373.1874   BIC:  379.784 
## Correlation matrix:
##             location       scale
## location  1.00000000 -0.03630391
## scale    -0.03630391  1.00000000
plot(cau)

8. Distribución Uniforme.

uni<-fitdist(x,"unif",method=c("mle"))
summary(uni)
## Fitting of the distribution ' unif ' by maximum likelihood 
## Parameters : 
##     estimate Std. Error
## min      1.5         NA
## max      3.9         NA
## Loglikelihood:  -175.0937   AIC:  354.1875   BIC:  360.7841 
## Correlation matrix:
## [1] NA
plot(uni)

9. Distribución Triangular.

tria<-fitdist(x,"triang", method="mge", start = list(min=1, mode=2,max=3))
## Warning in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg, : Some
## parameter names have no starting/fixed value but have a default value: mean.
## Warning in fitdist(x, "triang", method = "mge", start = list(min = 1, mode =
## 2, : maximum GOF estimation has a default 'gof' argument set to 'CvM'
summary(tria)
## Fitting of the distribution ' triang ' by maximum goodness-of-fit 
## Parameters : 
##      estimate
## min  1.528082
## mode 2.975057
## max  3.904715
## Loglikelihood:  -Inf   AIC:  Inf   BIC:  Inf
plot(tria)

10. Distribución Gumbel.

gum<-fitdist(x, "gumbel", start=list(a=1, b=1))
summary(gum)
## Fitting of the distribution ' gumbel ' by maximum likelihood 
## Parameters : 
##    estimate Std. Error
## a 2.5368439 0.03884157
## b 0.5170181 0.02633735
## Loglikelihood:  -168.5112   AIC:  341.0224   BIC:  347.619 
## Correlation matrix:
##           a         b
## a 1.0000000 0.3377982
## b 0.3377982 1.0000000
plot(gum)

11. Distribución Pareto.

memp <- function(x, order) mean(x^order)
pare<-fitdist(x,"pareto",method="mme",order=c(1, 2), memp="memp",
start=list(shape=10, scale=10))
summary(pare) 
## Fitting of the distribution ' pareto ' by matching moments 
## Parameters : 
##       estimate
## shape 171.2104
## scale 343.0643
## Loglikelihood:  -418.5859   AIC:  841.1717   BIC:  847.7684
plot(pare)

Bondad de ajuste.

gofstat(list(norma,lonor,gama,expo,wei,logi,cau,uni,tria,gum,pare), fitnames=c("Normal","Lognormal","Gamma","Exponencial","Weibull","Logistica","Cauchy","Uniforme","Triangular","Gumbel", "Pareto"))
## Goodness-of-fit statistics
##                                  Normal Lognormal     Gamma Exponencial
## Kolmogorov-Smirnov statistic 0.07862959 0.1177429 0.1045063   0.4496348
## Cramer-von Mises statistic   0.16888519 0.4726891 0.3470975  13.0714087
## Anderson-Darling statistic   1.04372638 2.8200607 2.0822445  61.7881324
##                                 Weibull  Logistica     Cauchy  Uniforme
## Kolmogorov-Smirnov statistic 0.05539806 0.06756679 0.09823015 0.1883333
## Cramer-von Mises statistic   0.08622222 0.13090674 0.32228902 2.0345833
## Anderson-Darling statistic   0.53814834 1.02424231 3.19523015       Inf
##                              Triangular    Gumbel    Pareto
## Kolmogorov-Smirnov statistic 0.06411800 0.1427094  0.566786
## Cramer-von Mises statistic   0.09115283 0.7506348 21.969231
## Anderson-Darling statistic          Inf 4.5633078 99.751608
## 
## Goodness-of-fit criteria
##                                  Normal Lognormal    Gamma Exponencial  Weibull
## Akaike's Information Criterion 298.3851  318.1727 309.4673    813.3475 292.6102
## Bayesian Information Criterion 304.9817  324.7693 316.0639    816.6458 299.2068
##                                Logistica   Cauchy Uniforme Triangular   Gumbel
## Akaike's Information Criterion  305.1042 373.1874 354.1875        Inf 341.0224
## Bayesian Information Criterion  311.7008 379.7840 360.7841        Inf 347.6190
##                                  Pareto
## Akaike's Information Criterion 841.1717
## Bayesian Information Criterion 847.7684

12. Distribución Beta.

bet <- fitdist(z, "beta")
summary(bet)
## Fitting of the distribution ' beta ' by maximum likelihood 
## Parameters : 
##        estimate Std. Error
## shape1 4.356784  0.8587149
## shape2 2.741216  0.5224440
## Loglikelihood:  18.70325   AIC:  -33.40651   BIC:  -29.58246 
## Correlation matrix:
##          shape1   shape2
## shape1 1.000000 0.863448
## shape2 0.863448 1.000000
plot(bet)

gofstat(bet)
## Goodness-of-fit statistics
##                              1-mle-beta
## Kolmogorov-Smirnov statistic  0.1145806
## Cramer-von Mises statistic    0.1290490
## Anderson-Darling statistic    0.7738691
## 
## Goodness-of-fit criteria
##                                1-mle-beta
## Akaike's Information Criterion  -33.40651
## Bayesian Information Criterion  -29.58246

13. Distribución Generalizada de Valores extremos.

gve<-fevd(x)
gve
## 
## fevd(x = x)
## 
## [1] "Estimation Method used: MLE"
## 
## 
##  Negative Log-Likelihood Value:  143.5896 
## 
## 
##  Estimated parameters:
##   location      scale      shape 
##  2.6436507  0.5296699 -0.3888485 
## 
##  Standard Error Estimates:
##   location      scale      shape 
## 0.04053423 0.02923480 0.03871168 
## 
##  Estimated parameter covariance matrix.
##               location         scale         shape
## location  0.0016430239 -0.0001819496 -0.0005669438
## scale    -0.0001819496  0.0008546733 -0.0007356433
## shape    -0.0005669438 -0.0007356433  0.0014985940
## 
##  AIC = 293.1791 
## 
##  BIC = 303.0741
plot(gve)

14. Distribución binomial.

bino<-fitdist(y,"binom",fix.arg=list(size=50), start=list(prob=0.3))
summary(bino)
## Fitting of the distribution ' binom ' by maximum likelihood 
## Parameters : 
##        estimate  Std. Error
## prob 0.06240494 0.004836805
## Fixed parameters:
##      value
## size    50
## Loglikelihood:  -90.49827   AIC:  182.9965   BIC:  184.9086
plot(bino)

15. Distribución Poisson.

poi<- fitdist(y,discrete=TRUE,"pois")
summary(poi)
## Fitting of the distribution ' pois ' by maximum likelihood 
## Parameters : 
##        estimate Std. Error
## lambda     3.12  0.2497999
## Loglikelihood:  -90.96812   AIC:  183.9362   BIC:  185.8483
plot(poi)

16. Distribución geométrica.

geo<-fitdist(y,"geom")
summary(geo)
## Fitting of the distribution ' geom ' by maximum likelihood 
## Parameters : 
##       estimate Std. Error
## prob 0.2427184 0.02987038
## Loglikelihood:  -114.1638   AIC:  230.3276   BIC:  232.2396
plot(geo)

17. Distribución binomial negativa.

binone<- fitdist(y,"nbinom")
summary(binone)
## Fitting of the distribution ' nbinom ' by maximum likelihood 
## Parameters : 
##          estimate Std. Error
## size 1.725944e+07 44.5879400
## mu   3.119957e+00  0.2497965
## Loglikelihood:  -90.96812   AIC:  185.9362   BIC:  189.7603 
## Correlation matrix:
##              size           mu
## size 1.000000e+00 3.956981e-08
## mu   3.956981e-08 1.000000e+00
plot(binone)

Bondad de ajuste.

gofstat(list(bino,poi,geo,binone),fitnames=c("Binomial","Poisson","Geométrica", "Binomial negativa"))
## Chi-squared statistic:  2.050607 2.145677 22.72921 2.145629 
## Degree of freedom of the Chi-squared distribution:  3 3 3 2 
## Chi-squared p-value:  0.5619697 0.5427274 4.598621e-05 0.3420444 
##    the p-value may be wrong with some theoretical counts < 5  
## Chi-squared table:
##      obscounts theo Binomial theo Poisson theo Geométrica
## <= 2        21     19.451377    19.842465       28.285885
## <= 3        10     11.523982    11.175932        5.270416
## <= 4         8      9.012476     8.717227        3.991189
## <= 5         8      5.518685     5.439550        3.022454
## > 5          3      4.493480     4.824826        9.430056
##      theo Binomial negativa
## <= 2              19.842928
## <= 3              11.175950
## <= 4               8.717120
## <= 5               5.439408
## > 5                4.824593
## 
## Goodness-of-fit criteria
##                                Binomial  Poisson Geométrica Binomial negativa
## Akaike's Information Criterion 182.9965 183.9362   230.3276          185.9362
## Bayesian Information Criterion 184.9086 185.8483   232.2396          189.7603

Paquetes utilizados:

library(fitdistrplus)

library(stats4)

library(MASS)

library(actuar)

library(mc2d)

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O.M.F.

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