MM-, Tau-, CM-, and MTL- Estimators for Nonlinear Robust Regression

"MM":: Compute an MM-estimator for nonlinear robust (constrained) regression.
"tau":: Compute a Tau-estimator for nonlinear robust (constrained) regression.
"CM":: Compute a “Constrained M” (=: CM) estimator for nonlinear robust (constrained) regression.
"MTL":: Compute a “Maximum Trimmed Likelihood” (=: MTL) estimator for nonlinear robust (constrained) regression.

## You can *not* call the  nlrob(*, method = <M>) like this ==> see  help(nlrob)
## ------- ===== ------------------------------------------

nlrob.MM(formula, data, lower, upper,
   tol = 1e-06,
   psi = c("bisquare", "lqq", "optimal", "hampel"),
         init = c("S", "lts"),
   ctrl = nlrob.control("MM", psi = psi, init = init, fnscale = NULL,
           tuning.chi.scale = .psi.conv.cc(psi, .Mchi.tuning.defaults[[psi]]),
           tuning.psi.M     = .psi.conv.cc(psi, .Mpsi.tuning.defaults[[psi]]),
           optim.control = list(), optArgs = list(...)),
   ...)

nlrob.tau(formula, data, lower, upper,
    tol = 1e-06, psi = c("bisquare", "optimal"),
    ctrl = nlrob.control("tau", psi = psi, fnscale = NULL,
      tuning.chi.scale = NULL, tuning.chi.tau = NULL,
      optArgs = list(...)),
    ...)

nlrob.CM(formula, data, lower, upper,
   tol = 1e-06,
   psi = c("bisquare", "lqq", "welsh", "optimal", "hampel", "ggw"),
   ctrl = nlrob.control("CM", psi = psi, fnscale = NULL,
                        tuning.chi = NULL, optArgs = list(...)),
   ...)

nlrob.mtl(formula, data, lower, upper,
    tol = 1e-06,
    ctrl = nlrob.control("mtl", cutoff = 2.5, optArgs = list(...)),
    ...)

Arguments

formula

nonlinear regression formula, using both variable names from data and parameter names from either lower or upper.

data

data to be used, a data.frame

lower, upper

bounds aka “box constraints” for all the parameters, in the case "CM" and "mtl" these must include the error standard deviation as "sigma", see nlrob() about its names, etc.

Note that one of these two must be a properly “named”, e.g., names(lower) being a character vector of parameter names (used in formula above).

tol

numerical convergence tolerance.

psi, init

see nlrob.control.

ctrl

a list, typically the result of a call to nlrob.control.

tuning.psi.M

optim.control: ..

optArgs: a list of optional arguments for optimization, e.g., trace = TRUE, passed to to the optimizer, which currently must be JDEoptim(.).
...: alternative way to pass the optArgs above.

Value

an R object of class "nlrob.<meth>", basically a list with components

Details

Currently, all four methods use JDEoptim() from DEoptimR, which subsamples using sample(). From R version 3.6.0, sample depends on RNGkind(*, sample.kind), such that exact reproducibility of results from R versions 3.5.3 and earlier requires setting RNGversion("3.5.0"). In any case, do use set.seed() additionally for reproducibility!

Author

Eduardo L. T. Conceicao; compatibility (to nlrob) tweaks and generalizations, inference, by Martin Maechler.

Source

For "MTL": Maronna, Ricardo A., Martin, R. Douglas, and Yohai, Victor J. (2006). Robust Statistics: Theory and Methods Wiley, Chichester, p. 133.

References

"MM":

Yohai, V.J. (1987) High breakdown-point and high efficiency robust estimates for regression. The Annals of Statistics 15, 642–656.

"tau":

Yohai, V.J., and Zamar, R.H. (1988). High breakdown-point estimates of regression by means of the minimization of an efficient scale. Journal of the American Statistical Association 83, 406–413.

"CM":

Mendes, B.V.M., and Tyler, D.E. (1996) Constrained M-estimation for regression.

In: Robust Statistics, Data Analysis and Computer Intensive Methods, Lecture Notes in Statistics 109, Springer, New York, 299–320.

"MTL":

Hadi, Ali S., and Luceno, Alberto (1997). Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms. Computational Statistics & Data Analysis 25, 251–272.

Gervini, Daniel, and Yohai, Victor J. (2002). A class of robust and fully efficient regression estimators. The Annals of Statistics 30, 583–616.

Examples


DNase1 <- DNase[DNase$Run == 1,]
form <- density ~ Asym/(1 + exp(( xmid -log(conc) )/scal ))
pnms <- c("Asym", "xmid", "scal")
set.seed(47) # as these by default use randomized optimization:

fMM <- robustbase:::nlrob.MM(form, data = DNase1,
           lower = setNames(c(0,0,0), pnms), upper = 3,
           ## call to nlrob.control to pass 'optim.control':
           ctrl = nlrob.control("MM", optim.control = list(trace = 1),
                                optArgs = list(trace = TRUE)))
#> 1 : < 0.05069941 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 2 : < 0.04523961 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 3 : < 0.02986584 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 4 : < 0.0264096 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 5 : < 0.02894298 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 6 : < 0.02603563 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 7 : < 0.01986311 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 8 : < 0.01609618 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 9 : < 0.01455267 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 10 : < 0.01243339 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 11 : < 0.01072001 > ( 0.01920318 ) 2.575602 1.693522 1.07722
#> 12 : < 0.009677451 > ( 0.01920318 ) 2.575602 1.693522 1.07722
#> 13 : < 0.009602394 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 14 : < 0.008188723 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 15 : < 0.006119826 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 16 : < 0.005292673 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 17 : < 0.003710545 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 18 : < 0.002748527 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 19 : < 0.002485876 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 20 : < 0.002063535 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 21 : < 0.001470831 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 22 : < 0.001417322 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 23 : < 0.001409855 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 24 : < 0.001389385 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 25 : < 0.001389385 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 26 : < 0.001286905 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 27 : < 0.001286905 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 28 : < 0.001123711 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 29 : < 0.001123711 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 30 : < 0.0009308617 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 31 : < 0.0009173112 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 32 : < 0.0007617063 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 33 : < 0.0007617063 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 34 : < 0.0007703871 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 35 : < 0.0007703871 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 36 : < 0.0007703871 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 37 : < 0.0007572125 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 38 : < 0.0007273438 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 39 : < 0.0006588138 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 40 : < 0.0006588138 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 41 : < 0.0006588138 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 42 : < 0.0006490691 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 43 : < 0.0006064914 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 44 : < 0.0006064914 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 45 : < 0.000550097 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 46 : < 0.0005228653 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 47 : < 0.0006788435 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 48 : < 0.0006788435 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 49 : < 0.0006788435 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 50 : < 0.000645026 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 51 : < 0.0006275806 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 52 : < 0.0005851229 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 53 : < 0.0005123287 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 54 : < 0.0007941211 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 55 : < 0.0007760146 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 56 : < 0.0007760146 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 57 : < 0.0007760146 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 58 : < 0.0007607864 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 59 : < 0.0007607864 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 60 : < 0.0008091312 > ( 0.01080434 ) 2.500988 1.681037 1.100433
#> 61 : < 0.000736377 > ( 0.01080434 ) 2.500988 1.681037 1.100433
#> 62 : < 0.000736377 > ( 0.01080434 ) 2.500988 1.681037 1.100433
#> 63 : < 0.0008071798 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 64 : < 0.0007727917 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 65 : < 0.0007727917 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 66 : < 0.0006622312 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 67 : < 0.0005160917 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 68 : < 0.0005086453 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 69 : < 0.0005760943 > ( 0.009832042 ) 2.554514 1.741773 1.121975
#> 70 : < 0.0006680199 > ( 0.009093679 ) 2.570388 1.745766 1.122237
#> 71 : < 0.00059494 > ( 0.009093679 ) 2.570388 1.745766 1.122237
#> 72 : < 0.0005516235 > ( 0.008927132 ) 2.551371 1.734223 1.117755
#> 73 : < 0.000576669 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 74 : < 0.0005456564 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 75 : < 0.0004743475 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 76 : < 0.0004739582 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 77 : < 0.0004378026 > ( 0.008456503 ) 2.542438 1.722437 1.116707
#> 78 : < 0.0003832678 > ( 0.008389518 ) 2.58577 1.767961 1.129992
#> 79 : < 0.0003323723 > ( 0.008389518 ) 2.58577 1.767961 1.129992
#> 80 : < 0.0002282994 > ( 0.008389518 ) 2.58577 1.767961 1.129992
#> 81 : < 0.000294096 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 82 : < 0.0002188404 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 83 : < 0.0001875213 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 84 : < 0.0001721344 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 85 : < 0.0001539168 > ( 0.007984676 ) 2.564217 1.743265 1.124157
#> 86 : < 0.0001072988 > ( 0.007984676 ) 2.564217 1.743265 1.124157
#> 87 : < 9.464617e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 88 : < 8.127913e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 89 : < 6.538715e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 90 : < 5.824443e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 91 : < 4.474242e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 92 : < 4.474242e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 93 : < 4.002027e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 94 : < 3.115745e-05 > ( 0.007891114 ) 2.568015 1.748257 1.12596
#> 95 : < 2.346785e-05 > ( 0.007891114 ) 2.568015 1.748257 1.12596
#> 96 : < 1.890454e-05 > ( 0.007891114 ) 2.568015 1.748257 1.12596
#> 97 : < 1.776247e-05 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 98 : < 1.08226e-05 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 99 : < 8.733677e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 100 : < 8.070519e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 101 : < 7.682144e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 102 : < 6.803753e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 103 : < 6.803753e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 104 : < 6.803753e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 105 : < 4.361337e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 106 : < 4.050931e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 107 : < 3.540563e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 108 : < 3.188359e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 109 : < 2.853233e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 110 : < 2.464473e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 111 : < 2.279735e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 112 : < 2.279735e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 113 : < 2.16731e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 114 : < 2.137293e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 115 : < 1.728817e-06 > ( 0.007882499 ) 2.569528 1.749486 1.126152
#> 116 : < 1.550326e-06 > ( 0.007882499 ) 2.569528 1.749486 1.126152
#> 117 : < 1.314956e-06 > ( 0.007882499 ) 2.569528 1.749486 1.126152
#> 118 : < 8.032411e-07 > ( 0.007882499 ) 2.569528 1.749486 1.126152
#> iter   10 value 599.703996
#> final  value 599.699450 
#> converged

## The same via nlrob() {recommended; same random seed to necessarily give the same}:
set.seed(47)
gMM  <- nlrob(form, data = DNase1, method = "MM",
              lower = setNames(c(0,0,0), pnms), upper = 3, trace = TRUE)
#> 1 : < 0.05069941 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 2 : < 0.04523961 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 3 : < 0.02986584 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 4 : < 0.0264096 > ( 0.05299046 ) 2.006964 1.450969 1.199327
#> 5 : < 0.02894298 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 6 : < 0.02603563 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 7 : < 0.01986311 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 8 : < 0.01609618 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 9 : < 0.01455267 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 10 : < 0.01243339 > ( 0.02079043 ) 2.705921 1.706303 1.068746
#> 11 : < 0.01072001 > ( 0.01920318 ) 2.575602 1.693522 1.07722
#> 12 : < 0.009677451 > ( 0.01920318 ) 2.575602 1.693522 1.07722
#> 13 : < 0.009602394 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 14 : < 0.008188723 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 15 : < 0.006119826 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 16 : < 0.005292673 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 17 : < 0.003710545 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 18 : < 0.002748527 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 19 : < 0.002485876 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 20 : < 0.002063535 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 21 : < 0.001470831 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 22 : < 0.001417322 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 23 : < 0.001409855 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 24 : < 0.001389385 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 25 : < 0.001389385 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 26 : < 0.001286905 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 27 : < 0.001286905 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 28 : < 0.001123711 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 29 : < 0.001123711 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 30 : < 0.0009308617 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 31 : < 0.0009173112 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 32 : < 0.0007617063 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 33 : < 0.0007617063 > ( 0.0154993 ) 2.443425 1.612181 1.07251
#> 34 : < 0.0007703871 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 35 : < 0.0007703871 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 36 : < 0.0007703871 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 37 : < 0.0007572125 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 38 : < 0.0007273438 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 39 : < 0.0006588138 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 40 : < 0.0006588138 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 41 : < 0.0006588138 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 42 : < 0.0006490691 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 43 : < 0.0006064914 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 44 : < 0.0006064914 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 45 : < 0.000550097 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 46 : < 0.0005228653 > ( 0.01483469 ) 2.569391 1.749319 1.108162
#> 47 : < 0.0006788435 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 48 : < 0.0006788435 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 49 : < 0.0006788435 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 50 : < 0.000645026 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 51 : < 0.0006275806 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 52 : < 0.0005851229 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 53 : < 0.0005123287 > ( 0.0133622 ) 2.528541 1.713568 1.103098
#> 54 : < 0.0007941211 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 55 : < 0.0007760146 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 56 : < 0.0007760146 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 57 : < 0.0007760146 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 58 : < 0.0007607864 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 59 : < 0.0007607864 > ( 0.01118383 ) 2.500988 1.681037 1.099034
#> 60 : < 0.0008091312 > ( 0.01080434 ) 2.500988 1.681037 1.100433
#> 61 : < 0.000736377 > ( 0.01080434 ) 2.500988 1.681037 1.100433
#> 62 : < 0.000736377 > ( 0.01080434 ) 2.500988 1.681037 1.100433
#> 63 : < 0.0008071798 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 64 : < 0.0007727917 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 65 : < 0.0007727917 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 66 : < 0.0006622312 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 67 : < 0.0005160917 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 68 : < 0.0005086453 > ( 0.01019707 ) 2.566007 1.746734 1.11744
#> 69 : < 0.0005760943 > ( 0.009832042 ) 2.554514 1.741773 1.121975
#> 70 : < 0.0006680199 > ( 0.009093679 ) 2.570388 1.745766 1.122237
#> 71 : < 0.00059494 > ( 0.009093679 ) 2.570388 1.745766 1.122237
#> 72 : < 0.0005516235 > ( 0.008927132 ) 2.551371 1.734223 1.117755
#> 73 : < 0.000576669 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 74 : < 0.0005456564 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 75 : < 0.0004743475 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 76 : < 0.0004739582 > ( 0.008467724 ) 2.544567 1.723901 1.116786
#> 77 : < 0.0004378026 > ( 0.008456503 ) 2.542438 1.722437 1.116707
#> 78 : < 0.0003832678 > ( 0.008389518 ) 2.58577 1.767961 1.129992
#> 79 : < 0.0003323723 > ( 0.008389518 ) 2.58577 1.767961 1.129992
#> 80 : < 0.0002282994 > ( 0.008389518 ) 2.58577 1.767961 1.129992
#> 81 : < 0.000294096 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 82 : < 0.0002188404 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 83 : < 0.0001875213 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 84 : < 0.0001721344 > ( 0.008033429 ) 2.58098 1.762339 1.13023
#> 85 : < 0.0001539168 > ( 0.007984676 ) 2.564217 1.743265 1.124157
#> 86 : < 0.0001072988 > ( 0.007984676 ) 2.564217 1.743265 1.124157
#> 87 : < 9.464617e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 88 : < 8.127913e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 89 : < 6.538715e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 90 : < 5.824443e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 91 : < 4.474242e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 92 : < 4.474242e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 93 : < 4.002027e-05 > ( 0.007892267 ) 2.574925 1.755196 1.127971
#> 94 : < 3.115745e-05 > ( 0.007891114 ) 2.568015 1.748257 1.12596
#> 95 : < 2.346785e-05 > ( 0.007891114 ) 2.568015 1.748257 1.12596
#> 96 : < 1.890454e-05 > ( 0.007891114 ) 2.568015 1.748257 1.12596
#> 97 : < 1.776247e-05 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 98 : < 1.08226e-05 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 99 : < 8.733677e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 100 : < 8.070519e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 101 : < 7.682144e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 102 : < 6.803753e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 103 : < 6.803753e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 104 : < 6.803753e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 105 : < 4.361337e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 106 : < 4.050931e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 107 : < 3.540563e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 108 : < 3.188359e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 109 : < 2.853233e-06 > ( 0.007884938 ) 2.573635 1.753783 1.1275
#> 110 : < 2.464473e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 111 : < 2.279735e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 112 : < 2.279735e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 113 : < 2.16731e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 114 : < 2.137293e-06 > ( 0.007882932 ) 2.569874 1.750024 1.126498
#> 115 : < 1.728817e-06 > ( 0.007882499 ) 2.569528 1.749486 1.126152
#> 116 : < 1.550326e-06 > ( 0.007882499 ) 2.569528 1.749486 1.126152
#> 117 : < 1.314956e-06 > ( 0.007882499 ) 2.569528 1.749486 1.126152
#> 118 : < 8.032411e-07 > ( 0.007882499 ) 2.569528 1.749486 1.126152
gMM
#> Robustly fitted nonlinear regression model (method MM)
#>   model:  density ~ Asym/(1 + exp((xmid - log(conc))/scal)) 
#>    data:  DNase1 
#>     Asym     xmid     scal 
#> 2.536678 1.702704 1.103871 
#>  status:  converged 
summary(gMM)
#> 
#> Call:
#> nlrob(formula = form, data = DNase1, lower = setNames(c(0, 0, 
#>     0), pnms), upper = 3, method = "MM", trace = TRUE)
#> 
#> Method "MM", init = "S", psi = "bisquare"
#> Residuals:
#>       Min        1Q    Median        3Q       Max 
#> -0.017710 -0.007115  0.003025  0.007331  0.067414 
#> 
#> Parameters:
#>      Estimate Std. Error t value Pr(>|t|)    
#> Asym  2.53668    0.08489   29.88 6.96e-12 ***
#> xmid  1.70270    0.08590   19.82 5.88e-10 ***
#> scal  1.10387    0.02758   40.03 2.87e-13 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Robust residual standard error: 0.007882 
#> Convergence after 23 function and 23 gradient evaluations
#> 
#> Robustness weights: 
#>  2 observations c(11,12) are outliers with |weight| = 0 ( < 0.0063); 
#>  The remaining 14 ones are summarized as
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  0.4523  0.8529  0.9380  0.8571  0.9724  0.9966 
## and they are the same {apart from 'call' and 'ctrl' and new stuff in gMM}:
ni <- names(fMM); ni <- ni[is.na(match(ni, c("call","ctrl")))]
stopifnot(all.equal(fMM[ni], gMM[ni]))