Solved – quantile threshold of t distribution and normal distribution

normal distributionquantilesrt-distribution

I want to calculate the quantiles of a t distribution and a normal distribution fitted to my data (can be found here: http://uploadeasy.net/upload/nk0f.rar).

The t distribution is leptokurtic, therefore there will be an area, where the quantile of the t distribution is larger than the quantile of the normal distribution for the same probability level. Then there will be a threshold, at which they are the same and then an area, in which the quantile is smaller. Consider this picture:

Quantile

Ignore the "VaR" and read it as a quantile: At 0,05, the quantile of the leptokurtic density is smaller thant the quantile of the equivalent normal distribution. At 0,01 the quantile of the leptokurtic density is larger. I want to find this threshold. I am looking at the right side of the distribution. My code for calculating the quantiles refers to a special probability level. My questions are:

Are my calculations correct? Especially the plot?:
How can I get the threshold value?

I am wondering about the following: The threshold value is NOT at the intersection of both densities at the tail right?

My code gives the following picture:

DensityTail

I mean, is it correct, that the quantile of the t distribution is larger than the quantile of the normal distribution if a probability level is considered, which leads to quantiles before the intersection? Yes, I think so?! But the intersection is not the threshold right?! In this case, the threshold where the quantile of the t distribution is larger than the quantile of the normal distribution lies before the intersection point, does this always hold?

Best Answer

Hi your question was recently covered on r-bloggers in the context of value-at-risk. As far as I see your calculations are correct. And yes ... the crucial point is not where the densities cross, but where the cumulative distribution functions cross.

Recall that quantiles are not about densities - they are derived from the cdf.

Related Solutions

Quantiles from Normal Distributions – Combining Quantiles from Multiple Normal Distributions

Unfortunately, the standard normal (from which all others can be determined, since the normal is a location-scale family) quantile function does not admit a closed form (i.e. a 'pretty formula'). The closest thing to a closed form is that the standard normal quantile function is the function, $w$, that satisfies the differential equation

$$ \frac{d^2 w}{d p^2} = w \left(\frac{d w}{d p}\right)^2 $$

and the initial conditions $w(1/2) = 0$ and $w'(1/2) = \sqrt{2 \pi}$. In most computing environments there is a function that numerically calculates the normal quantile function. In R, you would type

qnorm(p, mean=mu, sd=sigma)

to get the $p$'th quantile of the $N(\mu, \sigma^2)$ distribution.

Edit: With a modified understanding of the problem, the data is generated from a mixture of normals, so that the density of the observed data is:

$$ p(x) = \sum_{i} w_{i} p_{i}(x) $$

where $\sum_{i} w_{i} = 1$ and each $p_{i}(x)$ is some normal density with mean $\mu_{i}$ and standard deviation $\sigma_{i}$. It follows that the CDF of the observed data is

$$ F(y) = \int_{-\infty}^{y} \sum_{i} w_{i} p_{i}(x) dx = \sum_{i} w_{i} \int_{-\infty}^{y} p_{i}(x) = \sum_{i} w_{i} F_{i}(y) $$

where $F_{i}(x)$ is the normal CDF with mean $\mu_{i}$ and standard deviation $\sigma_{i}$. Integration and summation can be interchanged because these integrals are finite. This CDF is continuous and easy enough to calculate on a computer, so the inverse CDF, $F^{-1}$, also known as the quantile function, can be calculated by doing a line search. I default to this option because no simple formula for the quantile function of a mixture of normals, as a function of the quantiles of the constituent distributions, comes to mind.

The following R code numerically calculates $F^{-1}$ using bisection for the line search. The function F_inv() is the quantile function, you need to supply the vector containing each $w_{i}, \mu_{i}, \sigma_{i}$ and the quantile to be solved for, $p$.

# evaluate the function at the point x, where the components 
# of the mixture have weights w, means stored in u, and std deviations
# stored in s - all must have the same length.
F = function(x,w,u,s) sum( w*pnorm(x,mean=u,sd=s) )

# provide an initial bracket for the quantile. default is c(-1000,1000). 
F_inv = function(p,w,u,s,br=c(-1000,1000))
{
   G = function(x) F(x,w,u,s) - p
   return( uniroot(G,br)$root ) 
}

#test 
# data is 50% N(0,1), 25% N(2,1), 20% N(5,1), 5% N(10,1)
X = c(rnorm(5000), rnorm(2500,mean=2,sd=1),rnorm(2000,mean=5,sd=1),rnorm(500,mean=10,sd=1))
quantile(X,.95)
    95% 
7.69205 
F_inv(.95,c(.5,.25,.2,.05),c(0,2,5,10),c(1,1,1,1))
[1] 7.745526

# data is 20% N(-5,1), 45% N(5,1), 30% N(10,1), 5% N(15,1)
X = c(rnorm(5000,mean=-5,sd=1), rnorm(2500,mean=5,sd=1),
      rnorm(2000,mean=10,sd=1), rnorm(500, mean=15,sd=1))
quantile(X,.95)
     95% 
12.69563 
F_inv(.95,c(.2,.45,.3,.05),c(-5,5,10,15),c(1,1,1,1))
[1] 12.81730

Solved – Ftting a mixture of two Gaussians

Just so this thread gets an answer (since we can't access your data any more, I don't think much more answering will be happening): what you are doing seems to be perfectly fine.

You could change your search for a quantile of a mixture to use KScorrect::qmixnorm(), as per Compute quantile function from a mixture of Normal distribution.

Best Answer

Related Solutions

Quantiles from Normal Distributions – Combining Quantiles from Multiple Normal Distributions

Solved – Ftting a mixture of two Gaussians

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