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List:       r-help
Subject:    Re: [R] Multiple Integrals
From:       David Winsemius <dwinsemius () comcast ! net>
Date:       2015-08-31 2:17:06
Message-ID: 5F866862-222A-48E7-93D6-4BCD055E9956 () comcast ! net
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On Aug 30, 2015, at 8:41 AM, Shant Ch via R-help wrote:

> Thank you very much to all for all your responses.
> 
> @Dr. Winsemius, E[f(X)] >=f(E(X)) if f is convex. Now we know |x| is convex \
> function, so clearly in this scenario if we compute the expectation of the \
> ((X1+X2+X3)/3-X4) and then take the absolute, then, we will get a lower bound of \
> the expectation I want to find.  
> On Saturday, August 29, 2015 7:24 PM, David Winsemius <dwinsemius@comcast.net> \
> wrote:

Using the adaptIntegrate function in package:cubature I seem to be getting \
convergence near  5.359359 as I extend the limits of integration. I don't think that \
`adaptIntegrate` can handle infinite range. Either than or the NaN it is returning is \
a signal of pathology that I don't understand.

> require(cubature)
> fx<-function(x){
+         dlnorm(x,meanlog=2.185,sdlog=0.562)
+       }
> I.4d <- function(x) {
+   x1 = x[1]; y1 <- x[3]
+   x2 = x[2]; y2 <- x[4];   abs(y1/3+y2/3+x1/3-x2)*fx(y1)*fx(y2)*fx(x1)*fx(x2)}
> 
> adaptIntegrate(I.4d, rep(0, 4), rep(1000, 4), maxEval=1000000)
$integral
[1] 5.359082

$error
[1] 0.001922979

$functionEvaluations
[1] 1000065

$returnCode
[1] 0

> I.4d <- function(x) {
+   x1 = x[1]; y1 <- x[3]
+   x2 = x[2]; y2 <- x[4];   abs(y1/3+y2/3+x1/3-x2)*fx(y1)*fx(y2)*fx(x1)*fx(x2)}
> 
> adaptIntegrate(I.4d, rep(0, 4), rep(100, 4), maxEval=1000000)
$integral
[1] 5.357679

$error
[1] 0.001820893

$functionEvaluations
[1] 1000065

$returnCode
[1] 0

> I.4d <- function(x) {
+   x1 = x[1]; y1 <- x[3]
+   x2 = x[2]; y2 <- x[4];   abs(y1/3+y2/3+x1/3-x2)*fx(y1)*fx(y2)*fx(x1)*fx(x2)}
> 
> adaptIntegrate(I.4d, rep(0, 4), rep(10000, 4), maxEval=1000000)
$integral
[1] 5.359359

$error
[1] 0.001871926

$functionEvaluations
[1] 1000065

$returnCode
[1] 0


Best;
David.

> 
> 
> 
> On Aug 29, 2015, at 11:35 AM, Shant Ch via R-help wrote:
> 
> > Hello Dr. Berry,
> > 
> > I know the theoretical side but note we are not talking about expectation of sums \
> > rather expectation of ABSOLUTE value of the function (X1/3+X2/3+X3/3-X4), i.e. \
> > E|X1/3+X2/3+X3/3-X4|  , I don't think this can be handled for log normal \
> > distribution by integrals by hand. 
> 
> To Shnant Ch;
> 
> I admit to puzzlement (being a humble country doctor). Can you explain why there \
> should be a difference between the absolute value of an expectation for a sum of \
> values from a function, in this case dlnorm,  that is positive definite versus an \
> expectation simply of the sum of such values? 
> -- 
> 
> David Winsemius
> Alameda, CA, USA
> 
> 
> 
> 	[[alternative HTML version deleted]]
> 
> ______________________________________________
> R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see
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> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

David Winsemius
Alameda, CA, USA

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