lmomco (version 2.4.14)

Apwm2BpwmRC: Conversion between A- and B-Type Probability-Weighted Moments for Right-Tail Censoring of an Appropriate Distribution

Description

This function converts “A”-type probability-weighted moments (PWMs, ${\beta }_r^A$) to the “B”-type ${\beta }_r^B$. The ${\beta }_r^A$ are the ordinary PWMs for the $m$ left noncensored or observed values. The ${\beta }_r^B$ are more complex and use the $m$ observed values and the $m-n$ right-tailed censored values for which the censoring threshold is known. The “A”- and “B”-type PWMs are described in the documentation for pwmRC.

This function uses the defined relation between to two PWM types when the ${\beta }_r^A$ are known along with the parameters (para) of a right-tail censored distribution inclusive of the censoring fraction $\zeta =m/n$. The value $\zeta$ is the right-tail censor fraction or the probability $\mathrm{Pr}\{\}$ that $x$ is less than the quantile at $\zeta$ nonexceedance probability ($\mathrm{Pr}\{x<X(\zeta )\}$). The relation is $${\beta }_{r-1}^B=r^{-1}\{{\zeta }^{r}r{\beta }_{r-1}^A+(1-{\zeta }^r)X(\zeta )\}\text{,}$$ where $1\le r\le n$ and $n$ is the number of moments, and $X(\zeta )$ is the value of the quantile function at nonexceedance probability $\zeta$. Finally, the RC in the function name is to denote Right-tail Censoring.

Usage

Apwm2BpwmRC(Apwm,para)

Value

An R

list is returned.

Arguments

Apwm

A vector of A-type PWMs: ${\beta }_r^A$.

para

The parameters of the distribution from a function such as pargpaRC in which the ${\beta }_r^A$ are contained in a list element titled betas and the right-tail censoring fraction $\zeta$ is contained in an element titled zeta.

Author

W.H. Asquith

References

Hosking, J.R.M., 1995, The use of L-moments in the analysis of censored data, in Recent Advances in Life-Testing and Reliability, edited by N. Balakrishnan, chapter 29, CRC Press, Boca Raton, Fla., pp. 546--560.

See Also

Bpwm2ApwmRC, pwmRC

Examples

# Data listed in Hosking (1995, table 29.2, p. 551)
H <- c(3,4,5,6,6,7,8,8,9,9,9,10,10,11,11,11,13,13,13,13,13,
             17,19,19,25,29,33,42,42,51.9999,52,52,52)
      # 51.9999 was really 52, a real (noncensored) data point.
z <-  pwmRC(H,52)
# The B-type PMWs are used for the parameter estimation of the
# Reverse Gumbel distribution. The parameter estimator requires
# conversion of the PWMs to L-moments by pwm2lmom().
para <- parrevgum(pwm2lmom(z$Bbetas),z$zeta) # parameter object
Bbetas <- Apwm2BpwmRC(z$Abetas,para)
Abetas <- Bpwm2ApwmRC(Bbetas$betas,para)
# Assertion that both of the vectors of B-type PWMs should be the same.
str(Abetas)   # A-type PWMs of the distribution
str(z$Abetas) # A-type PWMs of the original data