Solved – Compute cdf and quantile of a specific distribution

cumulative distribution functiondensity functionquantilesr

I want to calculate a quantile of a specific distribution. Therefore I need the cdf. My distribution is a standardized Student's-t distribution, this can be written as
\begin{align*}
f(l|\nu) =(\pi (\nu-2))^{-\frac{1}{2}}\Gamma \left(\frac{\nu}{2} \right)^{-1} \Gamma \left(\frac{\nu+1}{2} \right) \left(1+\frac{l^2}{\nu-2} \right)^{-\frac{1+\nu}{2}}
\end{align*}

This can be implemented in R with:

probabilityfunction<-function(x)(pinumber*(param-2))^(-1/2)*gamma(param
/2)^(-1)*gamma((param+1)/2)*(1+l^2/(param-2))^(-(1+param)/2)

Where pinumber is the value for pi and param is the $\nu$. Lets say $\nu=5$. Then I can get the probability by just inserting a certain value for my l. But I want to have the cumulative density, since I later want to compute the quantile. I thought about something like

cumsum(probabilityfunction(5))

to give me the cumulative value up to 5.

But obviously this does not work. How can I get the cumulative probability and later on the quantile?

EDIT: OK, I found a first improvement:

integrate(probabilityfunction,-Inf,2)

would be a good starting point, but how to do the other thing?

Best Answer

CDF and quantile functions of this (extremely restrictive) distribution are straightforward to obtain in R, using that this is a Student-t distribution with a particular scale parameter:

CDF

pt(x/sqrt((nu-2)/nu),df=nu)

Quantile

qt(p,df=nu)*sqrt((nu-2)/nu)

This looks like a duplicate of your previous question: quantile transformation with t distribution (sorry, you have to use the monotonic transformation ;) )

Related Solutions

Solved – Optim result highly dependent on starting value

In general, when using optim for one dimensional problem, you should set the method argument to Brent otherwise optim is indeed unreliable.

There are several problems with your code as written:

pinumber you can use pi instead (it's a reserved constant in R)
par is a reserved keyword in R so it's better to use another name.
loglikstandardizedt assumes a vector standresidalvewma which is not declared in the call to loglikstandardizedt

Kurtosis – Kurtosis of a Standardized Student’s-t Distribution

Sample moments typically converge slowly to the true moments. This is the reason why you are observing such discrepancies between the two methods. For instance, run the following code several times

# Simulated data
dat <- rt(2000,df=8)

# Sample kurtosis
kurtosis(dat)-3

# Theoretical kurtosis
6/(8-4)

# MLE kurtosis
LL <- function(par){
if(par>0) return(-sum(dt(dat,df=par,log=T)))
 else return(Inf)
 }

 parameter <-optim(8, fn=LL, method="Brent",lower=6,upper=11)$par

6/(parameter-4)

In many cases the two estimators (sample kurtosis and MLE) differ. You got one of those samples where they differ.

Moreover (and maybe more importantly), the sample kurtosis converges to the true kurtosis while the MLE kurtosis converges to the kurtosis of the distribution that better fits the true distribution according to this criterion.

I agree with @whuber that the fit of your proposal distribution is pretty bad. You are unnecesarilly restricting the distribution (a Student-t would provide a much better fit for almost the same computational cost). Check

 library(MASS)
 fitdistr(standresidsapewma,"t")

Best Answer

Related Solutions

Solved – Optim result highly dependent on starting value

Kurtosis – Kurtosis of a Standardized Student’s-t Distribution

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