inference {SharpeR}

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"))

## S3 method for class 'sropt'
inference(z.s, type = c("KRS", "MLE", "unbiased"))

## S3 method for class 'del_sropt'
inference(z.s, type = c("KRS", "MLE", "unbiased"))

Arguments

z.s

an object of type sropt, or del_sropt

type

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

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!

Value

an estimate of the non-centrality parameter, which is the maximal population Sharpe ratio.

Author(s)

Steven E. Pav [e-mail]

References

Kubokawa, T., C. P. Robert, and A. K. Saleh. "Estimation of noncentrality parameters." Canadian Journal of Statistics 21, no. 1 (1993): 45-57. https://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. https://www.sciencedirect.com/science/article/pii/0047259X86900709

See Also

F-distribution functions, df.

Other sropt Hotelling: sric()

Examples

# generate some sropts
nfac <- 3
nyr <- 5
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 <- 3
df2 <- 512
ope <- 253
zeta.s <- 1.25
rvs <- rsropt(128, 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')

# for del_sropt:
nfac <- 5
nyr <- 10
ope <- 253
set.seed(as.integer(charToRaw("fix seed")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0.0005,sd=0.0125),ncol=nfac)
# hedge out the first one:
G <- matrix(diag(nfac)[1,],nrow=1)
asro <- as.del_sropt(Returns,G,drag=0,ope=ope)
est1 <- inference(asro,type='unbiased')  
est2 <- inference(asro,type='KRS')  
est3 <- inference(asro,type='MLE')