## A von Mises variate…

Inspired from a mail that came along the previous random generation post the following question rised :

How to draw random variates from the Von Mises distribution?

First of all let’s check the pdf of the probability rule, it is , for .

Ok, I admit that Bessels functions can be a bit frightening, but there is a work around we can do. The solution is a Metropolis algorithm simulation. It is not necessary to know the normalizing constant, because it will cancel in the computation of the ratio. The following code is adapted from James Gentle’s notes on Mathematical Statistics .

n <- 1000 x <- rep(NA,n) a <-1 c <-3 yi <-3 j <-0 i<-2 while (i < n) { i<-i+1 yip1 <- yi + 2*a*runif(1)- 1 if (yip1 < pi & yip1 > - pi) { if (exp(c*(cos(yip1)-cos(yi))) > runif(1)) yi <- yip1 else yi <- x[i-1] x[i] <- yip1 } } hist(x,probability=TRUE,fg = gray(0.7), bty="7") lines(density(x,na.rm=TRUE),col="red",lwd=2)

## The distribution of rho…

There was a post here about obtaining non-standard p-values for testing the correlation coefficient. The R-library

SuppDists

deals with this problem efficiently.

library(SuppDists) plot(function(x)dPearson(x,N=23,rho=0.7),-1,1,ylim=c(0,10),ylab="density") plot(function(x)dPearson(x,N=23,rho=0),-1,1,add=TRUE,col="steelblue") plot(function(x)dPearson(x,N=23,rho=-.2),-1,1,add=TRUE,col="green") plot(function(x)dPearson(x,N=23,rho=.9),-1,1,add=TRUE,col="red");grid() legend("topleft", col=c("black","steelblue","red","green"),lty=1, legend=c("rho=0.7","rho=0","rho=-.2","rho=.9"))</pre>

This is how it looks like,

Now, let’s construct a table of critical values for some arbitrary or not significance levels.

```
q=c(.025,.05,.075,.1,.15,.2)
xtabs(qPearson(p=q, N=23, rho = 0, lower.tail = FALSE, log.p = FALSE) ~ q )
# q
# 0.025 0.05 0.075 0.1 0.15 0.2
# 0.4130710 0.3514298 0.3099236 0.2773518 0.2258566 0.1842217
```

We can calculate p-values as usual too…

```
1-pPearson(.41307,N=23,rho=0)
# [1] 0.0250003
```

## In search of a random gamma variate…

One of the most common exersices given to Statistical Computing,Simulation or relevant classes is the generation of random numbers from a gamma distribution. At first this might seem straightforward in terms of the lifesaving relation that exponential and gamma random variables share. So, it’s easy to get a gamma random variate using the fact that

.

The code to do this is the following

rexp1 <- function(lambda, n) { u <- runif(n) x <- -log(u)/lambda } rgamma1 <- function(k, lambda) { sum(rexp1(lambda, k)) }

This works unfortunately only for the case

Read more…

## A normal philosophy…

The following was sent by email to me. It originates to Youden.

THE NORMAL LAW OF ERROR STANDS OUT IN THE EXPERIENCE OF MANKIND AS ONE OF THE BROADEST GENERALIZATIONS OF NATURAL PHILOSOPHY . IT SERVES AS THE GUIDING INSTRUMENT IN RESEARCHES IN THE PHYSICAL AND SOCIAL SCIENCES AND IN MEDICINE AGRICULTURE AND ENGINEERING . IT IS AN INDISPENSABLE TOOL FOR THE ANALYSIS AND THE INTERPRETATION OF THE BASIC DATA OBTAINED BY OBSERVATION AND EXPERIMENT

*–W.J. Youden*

Youden is one of the truly inspiring statisticians to me.

## (Im)Perfect Detectors…

Detecting reliably an event is surely something to worry about in applied science. One of the main models used is the *perfect-imperfect detector*, obviously underlying a Poisson process… Two detectors are counting events generated by a source (eg a *photon* device). The first one detects efficiently (*perfect*) the events whether the other one lacks efficiency.

Then X~Poi(λ) and Y~Poi(λp), where p is the inefficiency ratio and estimation is (almost) trivial. Assume that m,r are the counts of X,Y respectively. Furthermore, let k be the total observations and n the observations of X.

The mle of λ is m/n as usual. What about p?

The likelihood is proportional to , so taking logarithms and differentiating with respect to p gives us that the mle is .

The real question is : what if ?

**Look it up… **S. S. Chitgopekar, “A note on the estimation of the Poisson parameter,” *International Journal of Mathematics and Mathematical Sciences*, vol. 8, no. 1, pp. 193-196, 1985. [pdf]