Description

Density, distribution function, quantile function and random generation for the lambda prime distribution.

Usage

 ```1 2 3 4 5 6 7``` ```dlambdap(x, df, t, log = FALSE, order.max=6) plambdap(q, df, t, lower.tail = TRUE, log.p = FALSE, order.max=6) qlambdap(p, df, t, lower.tail = TRUE, log.p = FALSE, order.max=6) rlambdap(n, df, t) ```

Arguments

 `x, q` vector of quantiles. `df` the degrees of freedom in the chi square. This is not recycled against the `x,q,p,n`. `t` the scaling parameter on the chi. This is not recycled against the `x,q,p,n`. `log` logical; if TRUE, densities f are given as log(f). `order.max` the order to use in the approximate density, distribution, and quantile computations, via the Gram-Charlier, Edeworth, or Cornish-Fisher expansion. `p` vector of probabilities. `n` number of observations. `log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

Details

Suppose y ~ x^2(v), and Z is a standard normal.

T = Z + t sqrt(y/v)

takes a lambda prime distribution with parameters v, t. A lambda prime random variable can be viewed as a confidence level on a non-central t because

t = (Z' + T)/sqrt(y/v)

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

The PDF, CDF, and quantile function are approximated, via the Edgeworth or Cornish Fisher approximations, which may not be terribly accurate in the tails of the distribution. You are warned.

The distribution parameters are not recycled with respect to the `x, p, q` or `n` parameters, for, respectively, the density, distribution, quantile and generation functions. This is for simplicity of implementation and performance. It is, however, in contrast to the usual R idiom for dpqr functions.

Author(s)

Steven E. Pav [email protected]

References

Lecoutre, Bruno. "Another look at confidence intervals for the noncentral t distribution." Journal of Modern Applied Statistical Methods 6, no. 1 (2007): 107–116. http://www.univ-rouen.fr/LMRS/Persopage/Lecoutre/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.

t distribution functions, `dt, pt, qt, rt`, K prime distribution functions, `dkprime, pkprime, qkprime, rkprime`, upsilon distribution functions, `dupsilon, pupsilon, qupsilon, rupsilon`,
 ```1 2 3 4``` ```rv <- rlambdap(100, 50, t=0.01) d1 <- dlambdap(1, 50, t=0.01) pv <- plambdap(rv, 50, t=0.01) qv <- qlambdap(ppoints(length(rv)), 50, t=1) ```