| plambdap {SharpeR} | R Documentation |
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
|
log.p |
logical; if TRUE, probabilities p are given as |
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 \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.
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 shabbychef@gmail.com
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)