SharpeR (version 0.1305)

inference: Inference on noncentrality parameter of F-like statistic

Description

Estimates the non-centrality parameter associated with an observed statistic following an optimal Sharpe Ratio distribution.

Usage

inference(z.s,type=c("KRS","MLE","unbiased"))

Arguments

z.s

an object of type sropt.

type

the estimator type. one of c("KRS", "MLE", "unbiased")

Value

• an estimate of the non-centrality parameter.

Details

Let $F$ be an observed statistic distributed as a non-central F with $\nu_1$, $\nu_2$ degrees of freedom and non-centrality parameter $\delta^2$. Three methods are presented to estimate the non-centrality parameter from the statistic:

• an unbiased estimator, which, unfortunately, may be negative.
• the Maximum Likelihood Estimator, which may be zero, but not negative.
• the estimator of Kubokawa, Roberts, and Shaleh (KRS), which is a shrinkage estimator.

The sropt distribution is equivalent to an F distribution up to a square root and some rescalings.

The non-centrality parameter of the sropt distribution is the square root of that of the Hotelling, i.e. has units 'per square root time'. As such, the 'unbiased' type can be problematic!

References

Kubokawa, T., C. P. Robert, and A. K. Saleh. "Estimation of noncentrality parameters." Canadian Journal of Statistics 21, no. 1 (1993): 45-57. http://www.jstor.org/stable/3315657

Spruill, M. C. "Computation of the maximum likelihood estimate of a noncentrality parameter." Journal of multivariate analysis 18, no. 2 (1986): 216-224. http://www.sciencedirect.com/science/article/pii/0047259X86900709

See Also

F-distribution functions, df

Examples

# generate some sropts
nfac <- 5
nyr <- 10
ope <- 253
# simulations with no covariance structure.
# under the null:
set.seed(as.integer(charToRaw("determinstic")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
est1 <- inference(asro,type='unbiased')
est2 <- inference(asro,type='KRS')
est3 <- inference(asro,type='MLE')

# under the alternative:
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0.0005,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
est1 <- inference(asro,type='unbiased')
est2 <- inference(asro,type='KRS')
est3 <- inference(asro,type='MLE')

# sample many under the alternative, look at the estimator.
df1 <- 6
df2 <- 2000
ope <- 253
zeta.s <- 1.25
rvs <- rsropt(500, df1, df2, zeta.s, ope)
roll.own <- sropt(z.s=rvs,df1,df2,drag=0,ope=ope)
est1 <- inference(roll.own,type='unbiased')
est2 <- inference(roll.own,type='KRS')
est3 <- inference(roll.own,type='MLE')