SharpeR (version 1.3.0)

plambdap: The lambda-prime distribution.

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 \(P[X \le x]\), otherwise, \(P[X > x]\).

log.p

logical; if TRUE, probabilities p are given as \(\mbox{log}(p)\).

p

vector of probabilities.

n

number of observations. If 'length(n) > 1', the length is taken to be the number required.

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.

Details

Let \(t\) be distributed as a non-central t with \(\nu\) degrees of freedom and non-centrality parameter \(\delta\). We can view this as $$t = \frac{Z + \delta}{\sqrt{V/\nu}}.$$ where \(Z\) is a standard normal, \(\delta\) is the non-centrality parameter, \(V\) is a chi-square RV with \(\nu\) degrees of freedom, independent of \(Z\). We can rewrite this as $$\delta = t\sqrt{V/\nu} + Z.$$

Thus a 'lambda-prime' random variable with parameters \(t\) and \(\nu\) is one expressable as a sum $$t\sqrt{V/\nu} + Z$$ for Chi-square \(V\) with \(\nu\) d.f., independent from standard normal \(Z\)

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

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

# NOT RUN {
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)

# }