inference: Inference on noncentrality parameter of F-like statistic

In SharpeR: Statistical Significance of the Sharpe Ratio

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. This is \delta_0 of Equations (6.67) and (6.68) of ‘The Sharpe Ratio: Statistics and Applications’.

• the Maximum Likelihood Estimator, which may be zero, but not negative.

• the estimator of Kubokawa, Roberts, and Shaleh (KRS), which is a shrinkage estimator. This is \delta_2 of Equations (6.67) and (6.68) of ‘The Sharpe Ratio: Statistics and Applications’.

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

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Kubokawa, T., C. P. Robert, and A. K. Saleh. "Estimation of noncentrality parameters." Canadian Journal of Statistics 21, no. 1 (1993): 45-57. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.2307/3315657")}

Spruill, M. C. "Computation of the maximum likelihood estimate of a noncentrality parameter." Journal of multivariate analysis 18, no. 2 (1986): 216-224. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/0047-259X(86)90070-9")}

See Also

F-distribution functions, df.

Other sropt Hotelling: asnr_confint(), 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')

SharpeR documentation built on April 3, 2025, 7:36 p.m.