# [R] Anscombe-Glynn, Bonett-Seier, D'Agostino

From: Lukasz Komsta <luke_at_novum.am.lublin.pl>
Date: Sat 30 Apr 2005 - 04:11:38 EST

Dear useRs,

I was searching CRAN for implementation of kurtosis and skewness tests, and found that there is some kind of lack on it.

So, I have written three functions:

1. Anscombe-Glynn test for kurtosis
2. Bonett-Seier test based on Geary's kurtosis (which is not widely known, but I was inspired by original paper describing it, found coincidentally in Elsevier database)
3. D'Agostino test for skewness

These three functions are not enough to make another small package, so I am waiting for ideas about implementing it in some existing package. If there is a need, I will contact maintainer and write manpages with appropriate examples and references.

Regards,

```--
Lukasz Komsta
Department of Medicinal Chemistry
Medical University of Lublin
6 Chodzki, 20-093 Lublin, Poland
Fax +48 81 7425165

Code:

agostino.test <- function (x, alternative=c("two.sided","less","greater"))
{
DNAME <- deparse(substitute(x))
x <- sort(x[complete.cases(x)])
n <- length(x)

s <- match.arg(alternative)
alter <- switch(s, two.sided=0, less=1, greater=2)

if ((n < 8 || n > 46340))
stop("sample size must be between 8 and 46340")

s3 <- (sum((x-mean(x))^3)/n)/(sum((x-mean(x))^2)/n)^(3/2)
y <- s3*sqrt((n+1)*(n+3)/(6*(n-2)))
b2 <- 3*(n*n+27*n-70)*(n+1)*(n+3)/((n-2)*(n+5)*(n+7)*(n+9))
w <- sqrt(-1+sqrt(2*(b2-1)));
d <- 1/sqrt(log10(w));
a <- sqrt(2/(w*w-1));
z <- d*log10(y/a+sqrt((y/a)^2+1));

pval <- pnorm(z, lower.tail = FALSE)

if (alter == 0) {
pval <- 2*pval
if (pval > 1) pval<-2-pval
alt <- "data have a skewness"

}
else if (alter == 1)
{
alt <- "data have positive skewness"

}
else
{
pval <- 1-pval
alt <- "data have negative skewness"

}

RVAL <- list(statistic = c(g1 = s3, z = z), p.value = pval,
alternative = alt, method = "D'Agostino skewness test",
data.name = DNAME)
class(RVAL) <- "htest"
return(RVAL)
}

bonett.test <- function (x, alternative=c("two.sided","less","greater"))
{
DNAME <- deparse(substitute(x))
x <- sort(x[complete.cases(x)])
n <- length(x)

s <- match.arg(alternative)
alter <- switch(s, two.sided=0, less=1, greater=2)

rho <- sqrt(sum((x-mean(x))^2)/n);
tau <- sum(abs(x-mean(x)))/n;
omega <- 13.29*(log(rho)-log(tau));
z <- sqrt(n+2)*(omega-3)/3.54;

pval <- pnorm(z, lower.tail = FALSE)

if (alter == 0) {
pval <- 2*pval
if (pval > 1) pval<-2-pval
alt <- "kurtosis is not equal to 3"

}
else if (alter == 1)
{
alt <- "kurtosis is greater than 3"

}
else
{
pval <- 1-pval
alt <- "kurtosis is lower than 3"

}

RVAL <- list(statistic = c(tau = tau, z = z), alternative = alt,
p.value = pval, method = "Bonett-Seier kurtosis test",
data.name = DNAME)
class(RVAL) <- "htest"
return(RVAL)
}

anscombe.test <- function (x, alternative=c("two.sided","less","greater"))
{
DNAME <- deparse(substitute(x))
x <- sort(x[complete.cases(x)])
n <- length(x)
s <- match.arg(alternative)
alter <- switch(s, two.sided=0, less=1, greater=2)

b <- n*sum( (x-mean(x))^4 )/(sum( (x-mean(x))^2 )^2);

eb2 <- 3*(n-1)/(n+1);
vb2 <- 24*n*(n-2)*(n-3)/ ((n+1)^2*(n+3)*(n+5));
m3 <- (6*(n^2-5*n+2)/((n+7)*(n+9)))*sqrt((6*(n+3)*(n+5))/(n*(n-2)*(n-3)));
a <- 6+(8/m3)*(2/m3+sqrt(1+4/m3));
xx <- (b-eb2)/sqrt(vb2);
z <- ( 1-2/(9*a)-( (1-2/a) / (1+xx*sqrt(2/(a-4))) )^(1/3))/ sqrt(2/(9*a));

pval <- pnorm(z, lower.tail = FALSE)

if (alter == 0) {
pval <- 2*pval
if (pval > 1) pval<-2-pval
alt <- "kurtosis is not equal to 3"

}
else if (alter == 1)
{
alt <- "kurtosis is greater than 3"

}
else
{
pval <- 1-pval
alt <- "kurtosis is lower than 3"

}

RVAL <- list(statistic = c(b2 = b, z = z), p.value = pval,
alternative = alt, method = "Anscombe-Glynn kurtosis test",
data.name = DNAME)
class(RVAL) <- "htest"
return(RVAL)
}

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