plambdap: The lambda-prime distribution.

In shabbychef/SharpeR: Statistical Significance of the Sharpe Ratio

Description

Distribution function and quantile function for LeCoutre's lambda-prime distribution with df degrees of freedom and the observed t-statistic, tstat.

Usage

plambdap(q, df, tstat, lower.tail = TRUE, log.p = FALSE)

qlambdap(p, df, tstat, lower.tail = TRUE, log.p = FALSE)

rlambdap(n, df, tstat)

Arguments

q	vector of quantiles.
df	the degrees of freedom of the t-statistic.
tstat	the observed (non-central) t-statistic.
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
log.p	logical; if TRUE, probabilities p are given as log(p).
p	vector of probabilities.
n	number of observations. If 'length(n) > 1', the length is taken to be the number required.

Details

Let t be distributed as a non-central t with v degrees of freedom and non-centrality parameter ncp. We can view this as

t = (Z + ncp)/sqrt(V/v)

where Z is a standard normal, ncp is the non-centrality parameter, V is a chi-square RV with v degrees of freedom, independent of Z. We can rewrite this as

ncp = t sqrt(V/v) + Z

Thus a 'lambda-prime' random variable with parameters t and v is one expressable as a sum

t sqrt(V/v) + Z

for Chi-square V with v d.f., independent from standard normal Z

See ‘The Sharpe Ratio: Statistics and Applications’, section 2.4.

Value

dlambdap gives the density, plambdap gives the distribution function, qlambdap gives the quantile function, and rlambdap generates random deviates.

Invalid arguments will result in return value NaN with a warning.

Note

plambdap should be an increasing function of the argument q, and decreasing in tstat. qlambdap should be increasing in p

Author(s)

Steven E. Pav [e-mail]

References

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

Lecoutre, Bruno. "Another look at confidence intervals for the noncentral t distribution." Journal of Modern Applied Statistical Methods 6, no. 1 (2007): 107–116. https://eris62.eu/telechargements/Lecoutre_Another_look-JMSAM2007_6(1).pdf

Lecoutre, Bruno. "Two useful distributions for Bayesian predictive procedures under normal models." Journal of Statistical Planning and Inference 79 (1999): 93–105.

See Also

t-distribution functions, dt,pt,qt,rt

Other sr: as.sr(), confint.sr(), dsr(), is.sr(), power.sr_test(), predint(), print.sr(), reannualize(), se(), sr_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, summary.sr

Examples

rvs <- rnorm(128)
pvs <- plambdap(rvs, 253*6, 0.5)
plot(ecdf(pvs))
pvs <- plambdap(rvs, 253*6, 1)
plot(ecdf(pvs))
pvs <- plambdap(rvs, 253*6, -0.5)
plot(ecdf(pvs))
# test vectorization:
qv <- qlambdap(0.1,128,2)
qv <- qlambdap(c(0.1),128,2)
qv <- qlambdap(c(0.2),128,2)
qv <- qlambdap(c(0.2),253,2)
qv <- qlambdap(c(0.1,0.2),128,2)
qv <- qlambdap(c(0.1,0.2),c(128,253),2)
qv <- qlambdap(c(0.1,0.2),c(128,253),c(2,4))
qv <- qlambdap(c(0.1,0.2),c(128,253),c(2,4,8,16))
# random generation
rv <- rlambdap(1000,252,2)

shabbychef/SharpeR documentation built on Aug. 21, 2021, 8:50 a.m.