# Apwm2BpwmRC: Conversion between A- and B-Type Probability-Weighted Moments...

### Description

This function converts “A”-type probability-weighted moments (PWMs, β^A_r) to the “B”-type β^B_r. The β^A_r are the ordinary PWMs for the m left noncensored or observed values. The β^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 β^A_r are known along with the parameters (para) of a right-tail censored distribution inclusive of the censoring fraction ζ=m/n. The value ζ is the right-tail censor fraction or the probability \mathrm{Pr}\lbrace \rbrace that x is less than the quantile at ζ nonexceedance probability (\mathrm{Pr}\lbrace x < X(ζ) \rbrace). The relation is

β^B_{r-1} = r^{-1}\lbraceζ^r r β^A_{r-1} + (1-ζ^r)X(ζ)\rbrace \mbox{,}

where 1 ≤ r ≤ n and n is the number of moments, and X(ζ) is the value of the quantile function at nonexceedance probability ζ. Finally, the RC in the function name is to denote Right-tail Censoring.

### Usage

 1 Apwm2BpwmRC(Apwm,para) 

### Arguments

 Apwm A vector of A-type PWMs: β^A_r. para The parameters of the distribution from a function such as pargpaRC in which the β^A_r are contained in a list element titled betas and the right-tail censoring fraction ζ is contained in an element titled zeta.

### Value

An R list is returned.

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.

Bpwm2ApwmRC, pwmRC

### Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 # 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 

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