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, \(\beta^A_r\)) to the “B”-type \(\beta^B_r\). The \(\beta^A_r\) are the ordinary PWMs for the \(m\) left noncensored or observed values. The \(\beta^B_r\) 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 \(\beta^A_r\) 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 \(\mathrm{Pr}\lbrace \rbrace\) that \(x\) is less than the quantile at \(\zeta\) nonexceedance probability (\(\mathrm{Pr}\lbrace x < X(\zeta) \rbrace\)). The relation is $$\beta^B_{r-1} = r^{-1}\lbrace\zeta^r r \beta^A_{r-1} + (1-\zeta^r)X(\zeta)\rbrace \mbox{,}$$ 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: \(\beta^A_r\).

para

The parameters of the distribution from a function such as pargpaRC in which the \(\beta^A_r\) 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