Description Usage Arguments Value Note Author(s) References See Also Examples

This function computes the Mean Residual Quantile Function for quantile function *x(F)* (`par2qua`

, `qlmomco`

). The function is defined by Nair et al. (2013, p. 51) as

*M(u) = \frac{1}{1-u}\int_u^1 [x(p) - x(u)]\; \mathrm{d}p\mbox{,}*

where *M(u)* is the mean residual quantile for nonexceedance probability *u* and *x(u)* is a constant for *x(F = u)*. The variance of *M(u)* is provided in `rmvarlmomco`

.

The integration instead of from *0 \rightarrow 1* for the usual quantile function is *u \rightarrow 1*. Note that *x(u)* is a constant, so

*M(u) = \frac{1}{1-u}\int_u^1 x(p)\; \mathrm{d}p - x(u)\mbox{,}*

is equivalent and the basis for the implementation in `rmlmomco`

. Assuming that *x(F)* is a life distribution, the *M(u)* is interpreted (see Nair et al. [2013, p. 51]) as the average remaining life beyond the *100(1-F)\%* of the distribution. Alternatively, *M(u)* is the mean residual life conditioned that survival to lifetime *x(F)* has occurred.

If *u = 0*, then *M(0)* is the expectation of the life distribution or in otherwords *M(0) = λ_1* of the parent quantile function. If *u = 1*, then *M(u) = 0* (death has occurred)—there is zero residual life remaining. The implementation intercepts an intermediate *∞* and returns 0 for *u = 1*.

The *M(u)* is referred to as a quantile function but this quantity is not to be interpreted as a type of probability distribution. The second example produces a *M(u)* that is not monotonic increasing with *u* and therefore it is immediately apparent that *M(u)* is not the quantile function of some probability distribution by itself. Nair et al. (2013) provide extensive details on quantile-based reliability analysis.

1 | ```
rmlmomco(f, para)
``` |

`f` |
Nonexceedance probability ( |

`para` |
The parameters from |

Mean residual value for *F*.

The Mean Residual Quantile Function is the first of many other functions and “curves” associated with lifetime/reliability analysis operations that at their root use the quantile function (QF, *x(F)*) of a distribution. Nair et al. (2013) (NSB) is the authoritative text on which the following functions in lmomco were based

Residual mean QF | M(u) | `rmlmomco` | NSB[p.51] |

Variance residual QF | V(u) | `rmvarlmomco` | NSB[p.54] |

α-percentile residual QF | P_α(u) | `ralmomco` | NSB[p.56] |

Reversed α-percentile residual QF | R_α(u) | `rralmomco` | NSB[p.69--70] |

Reversed residual mean QF | R(u) | `rrmlmomco` | NSB[p.57] |

Reversed variance residual QF | D(u) | `rrmvarlmomco` | NSB[p.58] |

Conditional mean QF | μ(u) | `cmlmomco` | NSB[p.68] |

Vitality function (see conditional mean) | |||

Total time on test transform QF | T(u) | `tttlmomco` | NSB[p.171--172, 176] |

Scaled total time on test transform QF | φ(u) | `stttlmomco` | NSB[p.173] |

Lorenz curve | L(u) | `lrzlmomco` | NSB[p.174] |

Bonferroni curve | B(u) | `bfrlmomco` | NSB[p.179] |

Leimkuhler curve | K(u) | `lkhlmomco` | NSB[p.181] |

Income gap ratio curve | G(u) | `riglmomco` | NSB[p.230] |

Mean life: μ \equiv μ(0) \equiv λ_1(u=0) \equiv λ_1 |
|||

L-moments of residual life | λ_r(u) | `reslife.lmoms` | NSB[p.202] |

L-moments of reversed residual life | {}_\mathrm{r}λ_r(u) | `rreslife.lmoms` | NSB[p.211] |

W.H. Asquith

Kupka, J., and Loo, S., 1989, The hazard and vitality measures of ageing: Journal of Applied Probability, v. 26, pp. 532–542.

Nair, N.U., Sankaran, P.G., and Balakrishnan, N., 2013, Quantile-based reliability analysis: Springer, New York.

`qlmomco`

, `cmlmomco`

, `rmvarlmomco`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
# It is easiest to think about residual life as starting at the origin, units in days.
A <- vec2par(c(0.0, 2649, 2.11), type="gov") # so set lower bounds = 0.0
qlmomco(0.5, A) # The median lifetime = 1261 days
rmlmomco(0.5, A) # The average remaining life given survival to the median = 861 days
# 2nd example with discussion points
F <- nonexceeds(f01=TRUE)
plot(F, qlmomco(F, A), type="l", # usual quantile plot as seen throughout lmomco
xlab="NONEXCEEDANCE PROBABILITY", ylab="LIFETIME, IN DAYS")
lines(F, rmlmomco(F, A), col=2, lwd=3) # mean residual life
L1 <- lmomgov(A)$lambdas[1] # mean lifetime at start/birth
lines(c(0,1), c(L1,L1), lty=2) # line "ML" (mean life)
# Notice how ML intersects M(F|F=0) and again later in "time" (about F = 1/4) showing
# that this Govindarajulu as a peak mean resiudal life that is **greater** than the
# expected lifetime at start. The M(F) then tapers off to zero at infinity time (F=1).
# M(F) is non-monotonic for this example---not a proper probability dist.
``` |

```
[1] 1261.016
[1] 860.9676
```

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