I got a sample data and i'm trying to obtain the parameters for two-parameter exponential function calculed based on maximum likelihood.
My sample:
sample = c(136.5,150,94.1,127.6,77.2,136.1,83.4,75.6,92.7,106.5,95.9,112.1,80.7,90.4,143.7,152.7,113.3,143.9,87.9,85.2,117.2,193,153.7,84.7,97.3,140.3,80,103.6,72.6,90.7,52.6,52.8)
My main goal is to use the cdf
or quantile
of exponential for maximum likelihood, just like that:
Example with GEV:
library(nsRFA)
parameters <- ML_estimation(sample, dist = "GEV")
p = c(0.1,0.066667,0.05,0.04,0.033333,0.02,0.01,0.005,0.002,0.001,0.0002,0.0001)
q = invF.GEV(1-p, parameters[1], parameters[2], parameters[3]); q
> 149.4 158.8 165.2 170 173.9 184.3 197.6 210 225.4 236.2 258.9 267.7
The two-parameter exponential function is an exponential function with a lower endpoint at xi
. Finding MLEs of distributions with such sharp boundary points is a bit of a special case: the MLE for the boundary is equal to the minimum value observed in the data set (see eg this CrossValidated question ). That makes the MLE of the two-parameter exponential equivalent to the MLE of the exponential distribution for x-xmin
.
So the MLE of xi
is
print(xi <- min(sample))
MASS::fitdistr
:(m0 <- MASS::fitdistr(sample-xi, "exponential"))
rate
0.018382353
(0.003249572)
bbmle
:m1 <- bbmle::mle2(y-xi~dexp(lambda),
data=data.frame(y=sample), start=list(lambda=1),
method="Brent", lower=0.001, upper=100)
broom::tidy(m1, conf.int=TRUE, conf.method="profile")
term estimate std.error statistic p.value conf.low conf.high
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 lambda 0.0184 0.00325 5.66 0.0000000154 0.0127 0.0255
1/mean(sample-xi)
## [1] 0.01838235
If you want a simple function that provides the shift and scale parameters (as apparently provided by your alternative software):
est_twoexp <- function(x) {
xi <- min(x)
c(xi = xi, scale = mean(sample-xi))
}
est_twoexp(sample)
## xi scale
## 52.6 54.4
est_twoexp <- function(x) {
xi <- min(x)
c(xi = xi, scale = mean(sample-xi))
}
est_twoexp(sample)
## xi scale
## 52.6 54.4
Plotting:
library(nsRFA)
ee <- ecdf(sample)
inv_ecdf <- approxfun(seq(0, 1, length=length(sample)),
sort(sample),
method="constant")
## homemade quantile function
q2exp <- function(p, xi, scale) {
xi + qexp(p, rate=1/scale)
}
curve(inv_ecdf(x), from=0, to=1, type="s", ylim=c(40,250))
with(as.list(est_twoexp(sample)),
curve( invF.exp (x, xi, scale), col=2, lwd=2, add=TRUE))
with(as.list(est_twoexp(sample)),
curve( q2exp(x, xi, scale), col=4, lwd=2, lty= 2, add=TRUE))
glm
with family=Gamma
doesn't work because it doesn't allow zero values (within the general family of Gamma distributions, x==0
only has a positive, finite density for the exponential distribution)
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