I have an Integral in terms of parameters a,b,p,q.
My probability densitiy function of data x is
GB2PdfCalc=function(x){
out=(abs(a)*x^(a*p-1))/(b^(a*p)*beta(p,q)*(1+(x/b)^(a))^(p+q))
return(out)
}
I want to an integral of this using GB2PdfCalc:
pdfIntegration<-function(x){
[...]
integral[i]=integrate(GB2PdfCalc,x[k],x[j])[1]
integral=as.vector(integral,mode="numeric")
return(integral)
}
pdfIntegration(x)
However, I want to maximize this Integral in terms of a,b,p,q. Eventually, I want to calculate a Maximum Likelihood function. Using mle(pdfIntegration) or optim() didn't help me here.
Anybody got ideas?
Related
I am writing a code for estimating the parameter of a GPD using weighted nonlinear least square(WNLS) method.
The WNLS method consist of 2 steps
step 1: $(\hat{\xi_1} , \hat{b_1}) = arg\ \min_{(\xi,b)} \sum_{i=1}^{n} [\log(1-F_n(x_i)) - log(1-G_{\xi,b}(x_i))]$,
here $F_n$ is the ECDF and $1-G_{\xi,b}$ is the generalized pareto distribution.
Can anyone let me know how to calculate EDF function $F_n$ for a data "X" in R?
Does ecdf(X)(X) will calculate the ECDF? If so then, what is the need for ecdf(X) other than plotting? Also it would be really helpful if someone share some example code which involves the calculation of ECDF for data.
The ecdf call creates a function. That is, you can apply ecdf(X) to other data, as your ecdf(X)(X) call does. However, you might want to apply ecdf(X) to something other than X itself. If you want to know the empirical quantile to which three numbers a, b, and c_ correspond, an easy way to do that is to call ecdf(X)(c(a, b, c_)).
I tried to run MLE codings for R, but the output is "initial value out of range"
library(maxLik)
# -log-likelihood function of the Weibull Burr X + fixed covariate distribution.
likelihood<-function(par){
x<-data2$x; yi<-data2$yi;
alpha<-par[1];beta<-par[2];theta<-par[3];tau0<-par[4];tau1<-par[5];
n<-nrow(data2)
a<-sum(((exp(tau0+tau1*yi))*x)^2)
b<-sum(log(x))
c<-sum(log(1-exp(-((exp(tau0+tau1*yi))*x)^2))^theta)
d<-sum(log(1-(1-exp(-(exp(tau0+tau1*yi))*x)^2)^theta))
e<-sum((((1-exp(-((exp(tau0+tau1*yi))*x)^2))^(theta*beta))/
((1-(1-exp(-((exp(-(tau0+tau1*yi)))*x)^2))^theta))^beta))
#n = 30
logL<-n*log(2*alpha*beta*((exp(tau0+tau1*yi))^2)*theta)- a +
b + (theta*beta-1)*c
-(beta-1)*d
-alpha*log(e)
return(logL)
}
max<-maxLik(likelihood,start=c(1.1,4.15,0.25,0.03,0.02))
summary(max)
How to fix this to estimate the parameters?
I would recommend calculating the log likelihood analytically first, where you pay attention to simplify it as much as possible, before putting it into your R function. This usually avoids running into numerical issues. For instance, taking the simple log(exp(x)) on the computer can lead to numerical issues for very large x, but log(exp(x))=x will not.
Secondly, have you tried plugging in your starting values in your function to see if a reasonable value is returned?
I'm trying to analyze repairable systems reliability using growth models.
I have already fitted a Crow-Amsaa model but I wonder if there is any package or any code for fitting a Generalized Renewal Process (Kijima Model I) or type II
in R and find it's parameters Beta, Lambda(or alpha) and q.
(or some other model for the mean cumulative function MCF)
The equation number 15 of this article gives an expression for the
Log-likelihood
I tried to create the function like this:
likelihood.G1=function(theta,x){
# x is a vector with the failure times, theta vector of parameters
a=theta[1] #Alpha
b=theta[2] #Beta
q=theta[3] #q
logl2=log(b/a) # First part of the equation
for (i in 1:length(x)){
logl2=logl2 +(b-1)*log(x[i]/(a*(1+q)^(i-1))) -(x[i]/(a*(1+q)^(i-1)))^b
}
return(-logl2) #Negavite of the log-likelihood
}
And then use some rutine for minimize the -Log(L)
theta=c(0.5,1.2,0.8) #Start parameters (lambda,beta,q)
nlm(likelihood.G1,theta, x=Data)
Or also
optim(theta,likelihood.G1,method="BFGS",x=Data)
However it seems to be some mistake, since the parameters it returns has no sense
Any ideas of what I'm doing wrong?
Thanks
Looking at equation (16) of the paper you reference and comparing it with your code it looks like you are missing one term in the for loop. It seems that each data point contributes to three terms of the log-likelihood but in your code (inside the loop) you only have two terms (not considering the updating term)
Specifically, your code does not include the 4th term in equation (16):
and neither it does the 7th term, and so on. This is at least one error in the code. An extra consideration would be that α and β are constrained to be greater than zero. I am not sure if the solver you are using is considering this constraint.
I am using R to run a simulation in which I use a likelihood ratio test to compare two nested item response models. One version of the LRT uses the joint likelihood function L(θ,ρ) and the other uses the marginal likelihood function L(ρ). I want to integrate L(θ,ρ) over f(θ) to obtain the marginal likelihood L(ρ). I have two conditions: in one, f(θ) is standard normal (μ=0,σ=1), and my understanding is that I can just pick a number of abscissa points, say 20 or 30, and use Gauss-Hermite quadrature to approximate this density. But in the other condition, f(θ) is a linearly transformed beta distribution (a=1.25,b=10), where the linear transformation B'=11.14*(B-0.11) is such that B' also has (approximately) μ=0,σ=1.
I am confused enough about how to implement quadrature for a beta distribution but then the linear transformation confuses me even more. My question is threefold: (1) can I use some variation of quadrature to approximate f(θ) when θ is distributed as this linearly transformed beta distribution, (2) how would I implement this in R, and (3) is this a ridiculous waste of time such that there is an obviously much faster and better method to accomplish this task? (I tried writing my own numerical approximation function but found that my implementation of it, being limited to the R language, was just too slow to suffice.)
Thanks!
First, I assume you can express your L(θ,ρ) and f(θ) in terms of actual code; otherwise you're kinda screwed. Given that assumption, you can use integrate to perform the necessary computations. Something like this should get you started; just plug in your expressions for L and f.
marglik <- function(rho) {
integrand <- function(theta, rho) L(theta, rho) * f(theta)
# set your lower/upper integration limits as appropriate
integrate(integrand, lower=-5, upper=5, rho=rho)
}
For this to work, your integrand has to be vectorized; ie, given a vector input for theta, it must return a vector of outputs. If your code doesn't fit the bill, you can use Vectorize on the integrand function before passing it to integrate:
integrand <- Vectorize(integrand, "theta")
Edit: not sure if you're also asking how to define f(θ) for the transformed beta distribution; that seems rather elementary for someone working with joint and marginal likelihoods. But if you are, then the density of B' = a*B + b, given f(B), is
f'(B') = f(B)/a = f((B' - b)/a) / a
So in your case, f(theta) is dbeta(theta/11.14 + 0.11, 1.25, 10) / 11.14
I have a probability density function (PDF)
(1-cos(x-theta))/(2*pi)
theta is the unknown parameter. How do I write a log likelihood function for this PDF? I am confused; the x will come from my data, but how do I handle the theta in the equation.
Thanks
You need to use an optimisation or maximisation function in R to compute the value of theta that maximises the log-likelihood. See help(nlmin) for starters.
The function you wrote is a likelihood function of theta given the known x:
ll(theta|x) = log((1-cos(x-theta))/(2*pi))
if you have many iid observations from this distribution, x1,x2,...xn just take the sum of the above:
ll(theta|x1,x2,...) = Sum[log((1-cos(xi-theta))/(2*pi))]
If f(x_i) = (1-cos(x_i-theta))/(2*pi) for observation i, then likelihood function L(Theta)=product(f(x_i)) and logL(theta)=sum(f(x_i)), of course assuming that x_i are independent.
I think log-likelihood only works for normal-distributions. The special property of the log-function is, that it cancels out the exp-function, but here's no exp-function.
Btw., your PDF is periodic and theta just manipulates the phase of that function. Where does this PDF come from? What should it describe?