What’s the distribution of these data?

I got the data, and plot the distribution of the data, and use the qqnorm function, but is seems doesn’t follow a normal distribution, so which distribution should I use to discribe the data?

Empirical cumulative distribution function
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Answer

I suggest you give heavy-tail Lambert W x F or skewed Lambert W x F distributions a try (disclaimer: I am the author). In R they are implemented in the LambertW package.

They arise from a parametric, non-linear transformation of a random variable (RV) XF, to a heavy-tailed (skewed) version YLambert W×F. For F being Gaussian, heavy-tail Lambert W x F reduces to Tukey’s h distribution. (I will here outline the heavy-tail version, the skewed one is analogous.)

They have one parameter δ0 (γR for skewed Lambert W x F) that regulates the degree of tail heaviness (skewness). Optionally, you can also choose different left and right heavy tails to achieve heavy-tails and asymmetry. It transforms a standard Normal UN(0,1) to a Lambert W × Gaussian Z by
Z=Uexp(δ2U2)

If δ>0 Z has heavier tails than U; for δ=0, ZU.

If you don’t want to use the Gaussian as your baseline, you can create other Lambert W versions of your favorite distribution, e.g., t, uniform, gamma, exponential, beta, … However, for your dataset a double heavy-tail Lambert W x Gaussian (or a skew Lambert W x t) distribution seem to be a good starting point.

library(LambertW)
set.seed(10)

### Set parameters ####
# skew Lambert W x t distribution with 
# (location, scale, df) = (0,1,3) and positive skew parameter gamma = 0.1
theta.st <- list(beta = c(0, 1, 3), gamma = 0.1)
# double heavy-tail Lambert W x Gaussian
# with (mu, sigma) = (0,1) and left delta=0.2; right delta = 0.4 (-> heavier on the right)
theta.hh <- list(beta = c(0, 1), delta = c(0.2, 0.4))

### Draw random sample ####
# skewed Lambert W x t
yy <- rLambertW(n=1000, distname="t", theta = theta.st)

# double heavy-tail Lambert W x Gaussian (= Tukey's hh)
zz =<- rLambertW(n=1000, distname = "normal", theta = theta.hh)

### Plot ecdf and qq-plot ####
op <- par(no.readonly=TRUE)
par(mfrow=c(2,2), mar=c(3,3,2,1))
plot(ecdf(yy))
qqnorm(yy); qqline(yy)

plot(ecdf(zz))
qqnorm(zz); qqline(zz)
par(op)

ecdf and qqplot of skewed/heavy-tailed Lambert W x F distributions

In practice, of course, you have to estimate θ=(β,δ), where β is the parameter of your input distribution (e.g., β=(μ,σ) for a Gaussian, or β=(c,s,ν) for a t distribution; see paper for details):

### Parameter estimation ####
mod.Lst <- MLE_LambertW(yy, distname="t", type="s")
mod.Lhh <- MLE_LambertW(zz, distname="normal", type="hh")

layout(matrix(1:2, ncol = 2))
plot(mod.Lst)
plot(mod.Lhh)

enter image description here

Since this heavy-tail generation is based on a bijective transformations of RVs/data, you can remove heavy-tails from data and check if they are nice now, i.e., if they are Gaussian (and test it using Normality tests).

### Test goodness of fit ####
## test if 'symmetrized' data follows a Gaussian
xx <- get_input(mod.Lhh)
normfit(xx)

enter image description here

This worked pretty well for the simulated dataset. I suggest you give it a try and see if you can also Gaussianize() your data.

However, as @whuber pointed out, bimodality can be an issue here. So maybe you want to check in the transformed data (without the heavy-tails) what’s going on with this bimodality and thus give you insights on how to model your (original) data.

Attribution
Source : Link , Question Author : PepsiCo , Answer Author : Georg M. Goerg

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