R help - Dec 2012 - Get an expected value for Order Statistics by applying double integrals

Hi  R users,



 I’d like to get an expected value for  a minimum value from order
statistics of sample size, say, 5 for standard normal distribution, and the
formula would be 5* Integral (from -inf  to Inf) x*f(x)* (1-F(x))^4,where
f(x) and F(x) are a standard normal density  and a cumulative distribution
function respectively. After searching R posts, I applied the following
double integral formula which gave an error message saying “the integral is
probably divergent.”



 integrate(function(y){



+ sapply(y,function(y){



+
integrate(function(x)5*y*(1/sqrt(2*pi))*exp(-y^2/2)*(1-(1/sqrt(2*pi))*exp(-x^2/2))^4,-Inf,y)$value})



+ },-Inf,Inf)



I’m  not sure whether the syntax  is correct or not. Appreciate your help.



Kyong

	[[alternative HTML version deleted]]

Kyong-

It isn't clear to me why are you using a double integral.  You have the pdf
and your goal is to calculate the expected value of a univariate random
variable (the first order statistic).

Shouldn't the expected value be  E[X_(1)] = int x * 5 * f(x) *
(1-F(x))^4 dx?  A single integral?

Here are two different numerical techniques to calculate the expected value.

-tgs

fff <- function(x) 5 * x * dnorm(x) * (1 - pnorm(x))^4
integrate(fff,-Inf,Inf)

M <- 100000
N <- 5
D <- matrix(rnorm(M*N),nrow=M)
mean(apply(D,1,min))


On Thu, Dec 6, 2012 at 1:14 PM, kyong park <kpypark@gmail.com> wrote:
> Hi  R users,
>
>
>
>  I’d like to get an expected value for  a minimum value from order
> statistics of sample size, say, 5 for standard normal distribution, and the
> formula would be 5* Integral (from -inf  to Inf) x*f(x)* (1-F(x))^4,where
> f(x) and F(x) are a standard normal density  and a cumulative distribution
> function respectively. After searching R posts, I applied the following
> double integral formula which gave an error message saying “the integral is
> probably divergent.”
>
>
>
>  integrate(function(y){
>
>
>
> + sapply(y,function(y){
>
>
>
> +
>
>
integrate(function(x)5*y*(1/sqrt(2*pi))*exp(-y^2/2)*(1-(1/sqrt(2*pi))*exp(-x^2/2))^4,-Inf,y)$value})
>
>
>
> + },-Inf,Inf)
>
>
>
> I’m  not sure whether the syntax  is correct or not. Appreciate your help.
>
>
>
> Kyong
>
>         [[alternative HTML version deleted]]
>
>
> ______________________________________________
> R-help@r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>
>
	[[alternative HTML version deleted]]