Probabilty density function and Cumulative density function of three parameter log-normal distribution in R

Question

first off all i wanted to say that i am not so familar with statistics would like to apologize in advance if I have misrepresented or interpreted facts.

I have continous data x and i wanted to choose a distribution that fits to the data.

I used fitdistrplus in R and from the Cullen and Frey graph it seems that my data fits close to a logistic distribution.

Furthermore i used the software Easy Fit with the Result of the Summary of the Goodness of Fit that the data fits close to a lognormal(3P) distribution (Kolmogorv Smirnov Test and Chi Squared Test were ranked on the 1st place for the Lognormal(3P)).

With the given parameters for the 3-parameter lognormal distribution (σ = 0.03676, μ = 4.1258, γ = -61.826) from Easy Fit i wanted to plot the probabilty density function and the cumulative density function in R.

I tried to the function dlnorm3(x,shape=1,scale=1,thres=0,log=FALSE) and plnorm3(q,shape=1,scale=1,thres=0,lower.tail=TRUE,log.p=FALSE) from the package ‘FAdist’.

I interpreted the 3-parameter (σ = 0.03676, μ = 4.1258, γ = -61.826) as (shape=sdlog= σ = 0.03676, scale=meanlog= μ = 4.1258,threshold=γ = -61.826).

In R i got the following probabilty density function plot with the 3-parameter (σ = 0.03676, μ = 4.1258, γ = -61.826)

dat1 <- read.xlsx2('Tab.xlsx',"Sheet1")
dat1$x <- as.numeric(dat1$x)
str(dat1)
hist(dat1$x)

x= dat1$x

# > x
# [1]  2  1 -1 -7  1 -2 -2 -1  0  0  2 -1 -1 -2 -2  3  1 -1  3 -6 -1  1 -2  2 -2  4  0 -1  5 -2  3  0  0 -4  1  4 -2 -1 -1 -2
# [41] -4  0  2  1  2  0  3  0  3  1  1 -4  1 -2  2 -2 -1  4  2  0 -2 -2 -2  0  1  2  2  2  0 -3  0  1 -3  4  3  0  3  4 -2  0
# [81] -6 -2  1  3 -4 -1 -2 -2  1  0  3  2 -3 -3 -4  1  0 -1 -1  1  0  0  3  1  2  0 -1  0  1  1  0  2 -3  2 -1  2 -1 -4 -1 -2
# [121]  1  1  3  0  4  0 -1  2  1 -1 -3 -2 -3  0  0 -1  0  0 -2 -6 -2 -2  3 -1  5  4 -2  1 -1 -1 -1 -4  0 -2  0  1 -1 -1 -1  4
# [161]  1  0  5 -4 -1  0 -1  1  1  1  0  2  3 -2 -4 -2  1 -1  0 -1  3  1  1 -1 -1 -1  6 -1 -4 -3  0 -1 -2 -6 -1  1 -1  0  3 -1
# [201] -1 -2  0  2 -3 -1  1 -2  2  3  0  4  1 -3  1 -2  1 -4  0 -1  3  0  1  0  2 -3  2  1  3  2 -3  4  1  0  2  0 -1  1  0  2
# [241]  3  3  3  4  1  2 -3  1 -6 -3  6  3  2 -3  3 -1  0  0  1  0  4  3  1  5  1 -3  1  0 -2 -1  0  9  1  4 -1 -2 -1  2 -4  0
# [281] -1 -4 -2 -2  0  1 -1  0  3 -2 -4  2 -2  0  4 -2 -4  2 -2 -1 -8  0  6 -1 -3  2 -2 -1  2 -1  0 -2  2 -3  4 -2  1  2  1  2
# [321] -1  1 -3 -1  0  5  5 -2 -1  1  0 -1  1  1  3  3  2 -1  3 -2 -1  4  1  6 -1  1  7  2 -1  1  0  2  2  1  1  4 -2 -1  2  0
# [361]  0 -2  3  1  0  0  3  1  3  0  0  2 -3  0 -3  5 -1  1  2 -1  0  0  2 -2  1 -3  0  2 -7  1 -3  3 -1  3  3 -1 -1  6 -1  3
# [401]  0 -1 -3  0  2  2 -5  1 -1  0 -1  2  2 -1  0  0  1 -1  2  0 -1  0 -1 -1 -1  1  0 -1 -3  5  0  4 -4 -2 -1 10  3  0  1 -1
# [441]  2  0  3 -2 -6  0  1  1 -1  1  1  1  1  2 -1  0  0 -1  1  1  2  0  0  0  1  1  1  1  0 -3  0  4  0  0  1  0  5 -1  3 -2
# [481]  0 -2 -2 -1 -1  0  0  2  1  3 -2 -3  1 -1 -2  0  5 -1  4 -1 -2  0  0  1 -1  1 -2  2  0 -1 -1  0 -2 -3 -1 -1  2 -3 -1  2
# [521]  1  2 -3 -2  0  0  0 -2 -3 -1 -1 -1 -2  0 -1 -1  0  0 -1  0  3  0  1 -1 -1 -2  1  0 -1  2 -4  2  1 -1  2  3  0 -1  0  0
# [561]  0  0  3  3 -1  0 -2 -4  3  3  1  0 -3 -4  0  0  2  4  4  0 -3 -1  0  0  2  0 -2  0  3  0  1  2  0  0  3  0  1  0 -4  0
# [601]  2  3  1  1  0  2  1  4  3  3 -2 -2 -1 -1  1  1  1 -1 -1  0  3  3 -1  0  2  4  1  1  1 -1  1  0 -5  1  1  1  0 -2 -2 -1
# [641]  0  0 -2  1  0  2  2 -6  6 -3  0 -6  3  1  0  2  1 -2 -5  1  0  3 -1  2  0  2  1  1  2 -4  2 -1  2  0 -2  1 -1 -2  0 -2
# [681] -1  0  5  1  0 -2 -3  0

library(fitdistrplus)
library(logspline)

## Step (1) Plot 
descdist( data = x , discrete = FALSE)
descdist(data = x, discrete = FALSE, boot=1000)

# Cullen and Frey graph: Theoretical distribution "logistic"

## Step (2) Fit

fitdist(x,"logis")
max(x)

logis_ = fitdist(x, "logis")
normal_ = fitdist(x, "norm")

plot(logis_)
plot(normal_)

## Step (3) Estimate parameters

print(logis_)
print(normal_)

summary(logis_)
summary(normal_)

##################################################################################################################
#Plot the probabilty density function of the 3 parameter log-normal distribution with parameters from software Easy Fit
# 3-parameter from Software Easy Fit (σ = 0.03676, μ = 4.1258, γ = -61.826)

x1 <- seq(from=-61, to=-60 ,by=0.1)
y <- dlnorm3(x1, 0.03676, 4.1258,-61.826)
# Plot the log-normal distribution
plot(x1,y,type = "l", xlab = "x", ylab = "Density", main = "Log-Normal Distribution")

The plot looks a bit strange to me and i am not sure if I am on the right path.My aim is to use the cumulative density function to calculate the probabilty for different data x. But I am not sure if the 3-parameter lognormal distribution is the best fit for my data.

Thanks for your help in advance.

---

Edit

Data in dput format.

x <- c(2L, 1L, -1L, -7L, 1L, -2L, -2L, -1L, 0L, 0L, 2L, -1L, -1L, 
-2L, -2L, 3L, 1L, -1L, 3L, -6L, -1L, 1L, -2L, 2L, -2L, 4L, 0L, 
-1L, 5L, -2L, 3L, 0L, 0L, -4L, 1L, 4L, -2L, -1L, -1L, -2L, -4L, 
0L, 2L, 1L, 2L, 0L, 3L, 0L, 3L, 1L, 1L, -4L, 1L, -2L, 2L, -2L, 
-1L, 4L, 2L, 0L, -2L, -2L, -2L, 0L, 1L, 2L, 2L, 2L, 0L, -3L, 
0L, 1L, -3L, 4L, 3L, 0L, 3L, 4L, -2L, 0L, -6L, -2L, 1L, 3L, -4L, 
-1L, -2L, -2L, 1L, 0L, 3L, 2L, -3L, -3L, -4L, 1L, 0L, -1L, -1L, 
1L, 0L, 0L, 3L, 1L, 2L, 0L, -1L, 0L, 1L, 1L, 0L, 2L, -3L, 2L, 
-1L, 2L, -1L, -4L, -1L, -2L, 1L, 1L, 3L, 0L, 4L, 0L, -1L, 2L, 
1L, -1L, -3L, -2L, -3L, 0L, 0L, -1L, 0L, 0L, -2L, -6L, -2L, -2L, 
3L, -1L, 5L, 4L, -2L, 1L, -1L, -1L, -1L, -4L, 0L, -2L, 0L, 1L, 
-1L, -1L, -1L, 4L, 1L, 0L, 5L, -4L, -1L, 0L, -1L, 1L, 1L, 1L, 
0L, 2L, 3L, -2L, -4L, -2L, 1L, -1L, 0L, -1L, 3L, 1L, 1L, -1L, 
-1L, -1L, 6L, -1L, -4L, -3L, 0L, -1L, -2L, -6L, -1L, 1L, -1L, 
0L, 3L, -1L, -1L, -2L, 0L, 2L, -3L, -1L, 1L, -2L, 2L, 3L, 0L, 
4L, 1L, -3L, 1L, -2L, 1L, -4L, 0L, -1L, 3L, 0L, 1L, 0L, 2L, -3L, 
2L, 1L, 3L, 2L, -3L, 4L, 1L, 0L, 2L, 0L, -1L, 1L, 0L, 2L, 3L, 
3L, 3L, 4L, 1L, 2L, -3L, 1L, -6L, -3L, 6L, 3L, 2L, -3L, 3L, -1L, 
0L, 0L, 1L, 0L, 4L, 3L, 1L, 5L, 1L, -3L, 1L, 0L, -2L, -1L, 0L, 
9L, 1L, 4L, -1L, -2L, -1L, 2L, -4L, 0L, -1L, -4L, -2L, -2L, 0L, 
1L, -1L, 0L, 3L, -2L, -4L, 2L, -2L, 0L, 4L, -2L, -4L, 2L, -2L, 
-1L, -8L, 0L, 6L, -1L, -3L, 2L, -2L, -1L, 2L, -1L, 0L, -2L, 2L, 
-3L, 4L, -2L, 1L, 2L, 1L, 2L, -1L, 1L, -3L, -1L, 0L, 5L, 5L, 
-2L, -1L, 1L, 0L, -1L, 1L, 1L, 3L, 3L, 2L, -1L, 3L, -2L, -1L, 
4L, 1L, 6L, -1L, 1L, 7L, 2L, -1L, 1L, 0L, 2L, 2L, 1L, 1L, 4L, 
-2L, -1L, 2L, 0L, 0L, -2L, 3L, 1L, 0L, 0L, 3L, 1L, 3L, 0L, 0L, 
2L, -3L, 0L, -3L, 5L, -1L, 1L, 2L, -1L, 0L, 0L, 2L, -2L, 1L, 
-3L, 0L, 2L, -7L, 1L, -3L, 3L, -1L, 3L, 3L, -1L, -1L, 6L, -1L, 
3L, 0L, -1L, -3L, 0L, 2L, 2L, -5L, 1L, -1L, 0L, -1L, 2L, 2L, 
-1L, 0L, 0L, 1L, -1L, 2L, 0L, -1L, 0L, -1L, -1L, -1L, 1L, 0L, 
-1L, -3L, 5L, 0L, 4L, -4L, -2L, -1L, 10L, 3L, 0L, 1L, -1L, 2L, 
0L, 3L, -2L, -6L, 0L, 1L, 1L, -1L, 1L, 1L, 1L, 1L, 2L, -1L, 0L, 
0L, -1L, 1L, 1L, 2L, 0L, 0L, 0L, 1L, 1L, 1L, 1L, 0L, -3L, 0L, 
4L, 0L, 0L, 1L, 0L, 5L, -1L, 3L, -2L, 0L, -2L, -2L, -1L, -1L, 
0L, 0L, 2L, 1L, 3L, -2L, -3L, 1L, -1L, -2L, 0L, 5L, -1L, 4L, 
-1L, -2L, 0L, 0L, 1L, -1L, 1L, -2L, 2L, 0L, -1L, -1L, 0L, -2L, 
-3L, -1L, -1L, 2L, -3L, -1L, 2L, 1L, 2L, -3L, -2L, 0L, 0L, 0L, 
-2L, -3L, -1L, -1L, -1L, -2L, 0L, -1L, -1L, 0L, 0L, -1L, 0L, 
3L, 0L, 1L, -1L, -1L, -2L, 1L, 0L, -1L, 2L, -4L, 2L, 1L, -1L, 
2L, 3L, 0L, -1L, 0L, 0L, 0L, 0L, 3L, 3L, -1L, 0L, -2L, -4L, 3L, 
3L, 1L, 0L, -3L, -4L, 0L, 0L, 2L, 4L, 4L, 0L, -3L, -1L, 0L, 0L, 
2L, 0L, -2L, 0L, 3L, 0L, 1L, 2L, 0L, 0L, 3L, 0L, 1L, 0L, -4L, 
0L, 2L, 3L, 1L, 1L, 0L, 2L, 1L, 4L, 3L, 3L, -2L, -2L, -1L, -1L, 
1L, 1L, 1L, -1L, -1L, 0L, 3L, 3L, -1L, 0L, 2L, 4L, 1L, 1L, 1L, 
-1L, 1L, 0L, -5L, 1L, 1L, 1L, 0L, -2L, -2L, -1L, 0L, 0L, -2L, 
1L, 0L, 2L, 2L, -6L, 6L, -3L, 0L, -6L, 3L, 1L, 0L, 2L, 1L, -2L, 
-5L, 1L, 0L, 3L, -1L, 2L, 0L, 2L, 1L, 1L, 2L, -4L, 2L, -1L, 2L, 
0L, -2L, 1L, -1L, -2L, 0L, -2L, -1L, 0L, 5L, 1L, 0L, -2L, -3L, 0L)