There are two major forms (outside of the general plotting-position formula
pp) for estimation of the p_rth probability of the rth order statistic for a sample of size n: the mean is pp'_r = r/(n+1) (Weibull plotting position) and the Beta quantile function is pp_r(F) = IIB(F, r, n+1-r), where F represents the nonexceedance probability of the plotting position. IIB is the “inverse of the incomplete beta function” or the quantile function of the Beta distribution as provided in R by
qbeta(f, a, b). If F=0.5, then the median is returned but that is conveniently implemented in
pp.median. Readers might consult Gilchrist (2011, chapter 12) and Karian and Dudewicz (2011, p. 510).
A nonexceedance probability.
A vector of data. The ranks and the length of the vector are computed within the function.
vector is returned.
The function uses the R function
rank, which has specific settings to handle tied data. For implementation here, the
ties.method="first" method to
rank is used.
Gilchrist, W.G., 2000, Statistical modelling with quantile functions: Chapman and Hall/CRC, Boca Raton.
Karian, Z.A., and Dudewicz, E.J., 2011, Handbook of fitting statistical distributions with R: Boca Raton, FL, CRC Press.
X <- sort(rexp(10)) PPlo <- pp.f(0.25, X) PPhi <- pp.f(0.75, X) plot(c(PPlo,NA,PPhi), c(X,NA,X)) points(pp(X), X) # Weibull i/(n+1)
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