The design and structure of geex

2022-07-24

The details below are for those interested in how geex is organized. It is not necessary for using geex.

The Estimating Function

The design of geex starts with the key to M-estimation, the estimating function:

$\psi(O_i, \theta) .$

geex composes $$\psi$$ with two R functions: the “outer” estFUN and the “inner” psiFUN. In pseudocode, $$\psi(O_i, \theta) =$$:

estFUN <- function(O_i){
psiFUN <- function(theta){
psi(O_i, theta)
}
return(psiFUN)
}

The reason for composing the $$\psi$$ function in this way is that in order to do estimation (finding roots) and inference (computing the empirical sandwich variance estimator), $$\psi$$ needs to be function of $$\theta$$. M-estimation theory gives the following instructions:

• To estimate $$\hat{\theta}$$, we need to find roots of $$G_m = \sum_i \psi(O_i, \theta) = 0$$.
• To estimate the empirical sandwich variance estimator, we need two quantities for each unit: $$A_i = - (\partial \psi(O_i, \theta)/\partial \theta)|_{\theta = \hat{\theta}}$$ and $$B_i = \psi(O_i, \hat{\theta})\psi(O_i, \theta)^{\intercal}$$.

With $$\hat{\theta}$$ in hand, the quantity $$B_i$$ is simple to compute. The computational challenges of M-estimation, then, are finding roots of $$G_m$$ and calculating the derivative $$A_i$$. By composing $$\psi$$ of two functions in geex, one can first do all the manipulations of $$O_i$$ (data) that are independent of $$\theta$$. In a sense, estFUN “fixes” the data so that numerical routines only need deal with $$\theta$$ in psiFUN.

M-estimation basis

Before describing the mechanics of how geex finding roots of $$G_m$$ and computes derivatives of $$\psi$$, let’s look at the m_estimation_basis S4 object which forms the basis of all computations in geex.

An m_estimation_basis object, at a minimum needs two objects: an estFUN and a data.frame. Let’s use a simple estFUN that estimates the mean and variance of Y1 in the geexex dataset.

library(geex)
library(dplyr)

myee <- function(data){
Y1 <- data$Y1 function(theta){ c(Y1 - theta[1], (Y1 - theta[1])^2 - theta[2]) } } Now we can create a basis: mybasis <- new("m_estimation_basis", .estFUN = myee, .data = geexex) And look at what this object contains: slotNames(mybasis) ## [1] ".data" ".units" ".weights" ".psiFUN_list" ".GFUN" ## [6] ".control" ".estFUN" ".outer_args" ".inner_args" Two slots are worth examining. First, .psiFUN_list is a list of functions: mybasis@.psiFUN_list[1:2] ##$1
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
##
## $2 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b8adb7350> This object is essentially equivalent to: m <- nrow(geexex) lapply(split(geexex, f = 1:m), function(O_i){ myee(O_i) }) From this list of functions, we can compute $$A_i$$, and by summing across the list, form $$G_m$$. The latter is found in: mybasis@.GFUN ## function (theta) ## { ## psii <- lapply(psi_list, function(psi) { ## do.call(psi, args = append(list(theta = theta), object@.inner_args)) ## }) ## compute_sum_of_list(psii, object@.weights) ## } ## <environment: 0x7f8b8a8c2588> Finding roots Now that we have $$G_m$$ as a function of theta, we can found its roots using a root-finding algorithm such as rootSolve::multiroot: rootSolve::multiroot( f = mybasis@.GFUN, start = c(0, 0)) ##$root
## [1]  5.044563 10.041239
##
## $f.root ## [1] -2.131628e-14 4.654055e-13 ## ##$iter
## [1] 4
##
## $estim.precis ## [1] 2.433609e-13 Within geex this is done with the estimate_GFUN_roots function. To illustrate, I first need to update the .control slot in mybasis with starting values for multiroot. mycontrol <- new('geex_control', .root = setup_root_control(start = c(1, 1))) mybasis@.control <- mycontrol roots <- mybasis %>% estimate_GFUN_roots() roots ##$root
## [1]  5.044563 10.041239
##
## $f.root ## [1] -2.131628e-14 -2.238210e-13 ## ##$iter
## [1] 4
##
## $estim.precis ## [1] 1.225686e-13 Note that is bad form to assign S4 slot with someS4object@aslot <- something, but I do so here because I have not created a generic function for setting the .control slot. Computing the Empirical Sandwich Variance Estimator In the last section, we found $$\hat{\theta}$$, which we now use to compute the $$A_i$$ and $$B_i$$ matrices. geex uses the numDeriv::jacobian function to numerically evaluate derivatives. For example, $$A_1 = - (\partial \psi(O_1, \theta)/\partial \theta)|_{\theta = \hat{\theta}}$$ for this example is: -numDeriv::jacobian(func = mybasis@.psiFUN_list[[1]], x = roots$root)
##           [,1] [,2]
## [1,]  1.000000    0
## [2,] -2.752514    1

geex performs this operation for each $$i = 1, \dots, m$$ to yield a list of $$A_i$$ matrices. Then summing across this list yields $$A = \sum_i A_i$$. The estimate_sandwich_matrices function computes the list of $$A_i$$, $$B_i$$ and $$A$$ and $$B$$:

mats <- mybasis %>%
estimate_sandwich_matrices(.theta = roots$root) # Compare to the numDeriv computation above grab_bread_list(mats)[[1]] ## [,1] [,2] ## [1,] 1.000000 0 ## [2,] -2.752514 1 Finally, computing $$\hat{\Sigma} = A^{-1} B (A^{-1})^{\intercal}$$ is accomplished with the compute_sigma function. mats %>% {compute_sigma(A = grab_bread(.), B = grab_meat(.))} ## [,1] [,2] ## [1,] 0.10041239 0.03667969 ## [2,] 0.03667969 2.49219638 M-estimation with m_estimate All of the operations described above are wrapped and packaged in the m_estimate function: m_estimate( estFUN = myee, data = geexex, root_control = setup_root_control(start = c(0, 0)) ) ## An object of class "geex" ## Slot "call": ## m_estimate(estFUN = myee, data = geexex, root_control = setup_root_control(start = c(0, ## 0))) ## ## Slot "basis": ## An object of class "m_estimation_basis" ## Slot ".data": ## Y1 Y2 X1 Y3 W1 Z1 X2 ## 1 3.66830660 2.02817177 4.949316 16.345756 4.823768 8.921782 0 ## 2 10.45245483 1.64329659 7.851962 25.687417 7.884845 13.909474 0 ## 3 3.12341064 2.85262638 4.729075 16.108307 4.709346 9.014695 0 ## 4 8.37150253 2.51336525 2.564395 10.579970 2.786091 6.733378 0 ## 5 -0.83197489 3.01820300 4.782347 16.464013 4.811590 9.290492 0 ## 6 3.39877632 0.97852092 5.335713 18.325769 5.415370 10.322199 0 ## 7 1.89433086 1.43833173 1.386442 5.577536 1.240995 3.497873 0 ## 8 3.52281395 0.98744392 3.453377 13.074664 3.632010 7.894598 0 ## 9 9.96040583 -1.02081430 2.958662 10.050725 2.752347 5.612733 0 ## 10 4.57026477 2.33235027 7.591370 24.414247 7.501404 13.027192 0 ## 11 5.69037402 3.24051157 6.812940 22.528706 6.835412 12.309296 0 ## 12 6.01840507 2.67134960 2.481492 9.540750 2.505561 5.818512 0 ## 13 2.54186468 0.66996589 3.307246 11.720103 3.256837 6.759235 0 ## 14 -0.71686038 1.14941969 2.366527 9.839421 2.551487 6.289631 0 ## 15 3.67609826 0.21116926 6.308752 21.049635 6.339597 11.586507 0 ## 16 5.51354425 3.23152191 2.280638 8.812598 2.273309 5.391641 0 ## 17 9.07247997 1.66560033 2.872154 10.227607 2.774940 5.919377 0 ## 18 3.97770523 1.03267790 4.361465 15.595252 4.489179 9.053054 0 ## 19 3.78983596 2.87937035 3.573053 11.805345 3.344600 6.445765 0 ## 20 11.46076273 1.74642131 5.556376 20.979426 6.133951 12.644862 0 ## 21 1.90514658 0.48212421 7.752991 24.820884 7.643469 13.191397 0 ## 22 6.69600961 1.97611674 6.030068 20.854263 6.221083 11.809162 0 ## 23 2.66421207 2.02665947 4.213262 14.901747 4.278752 8.581854 0 ## 24 6.66014272 2.16368120 2.923132 11.542799 3.116483 7.158102 0 ## 25 -1.18104663 2.41000794 5.156830 16.656110 4.953235 8.920865 0 ## 26 2.92500198 1.37263740 5.519839 18.121067 5.410226 9.841308 0 ## 27 3.88083378 2.63691800 5.477283 17.711627 5.297228 9.495703 0 ## 28 9.02982953 0.79806522 4.055430 14.397234 4.113166 8.314089 0 ## 29 3.12172019 3.34654241 4.319714 13.801412 4.030281 7.321841 0 ## 30 6.19158815 1.40123269 10.283894 33.098758 10.345663 17.672917 0 ## 31 3.32882227 2.44220444 2.557841 9.582409 2.535063 5.745648 0 ## 32 1.59847689 2.61352641 11.152742 37.215603 11.592086 20.486489 0 ## 33 7.75618478 1.70090363 2.538047 9.476212 2.503565 5.669141 0 ## 34 3.15921522 0.39941190 7.939765 25.708101 7.911967 13.798454 0 ## 35 10.39273751 1.66053304 3.629295 12.197870 3.456791 6.753928 0 ## 36 6.77228554 1.41869225 5.644317 18.711156 5.588868 10.244681 0 ## 37 4.39629525 1.60963799 1.385403 6.339116 1.431130 4.261012 0 ## 38 6.82219543 2.84551436 3.651563 13.372011 3.755894 7.894667 0 ## 39 4.83938127 2.68472721 2.075987 9.293362 2.342337 6.179382 0 ## 40 6.82448417 2.23771308 7.947636 26.813109 8.190186 14.891656 0 ## 41 3.36629988 1.28937811 3.893624 13.579242 3.868217 7.738807 0 ## 42 -3.54597542 4.61331896 4.399113 16.600543 4.749914 10.001873 0 ## 43 5.62728767 0.37335265 2.019187 6.280784 1.574993 3.252004 0 ## 44 7.64019560 0.39269371 10.182047 33.169007 10.337763 17.895937 0 ## 45 1.07266235 2.34031745 4.471305 14.891632 4.340734 8.184674 0 ## 46 0.54542518 4.72788771 5.445723 19.659399 5.776280 11.490815 0 ## 47 3.25060929 1.67280996 5.030453 16.727920 4.939593 9.182240 0 ## 48 2.93555501 0.74310325 7.586987 26.080025 7.916753 14.699546 0 ## 49 6.67598396 1.56860189 9.452187 30.400340 9.463132 16.222060 0 ## 50 5.53662175 4.54885325 8.141977 24.547274 7.672313 12.334309 0 ## 51 9.13874582 1.22859200 5.623052 18.422092 5.511286 9.987515 1 ## 52 11.61401290 1.49265765 5.066275 15.460228 4.631626 7.860815 1 ## 53 4.92821273 1.72997742 2.174904 8.703576 2.219620 5.441220 1 ## 54 4.90318672 2.74811656 1.373871 8.019078 1.848237 5.958272 1 ## 55 6.00098760 2.66859381 4.252394 12.485257 3.684413 6.106666 1 ## 56 3.65150186 1.54470134 1.844766 8.514763 2.089882 5.747614 1 ## 57 4.54658518 0.07215478 6.257311 19.373108 5.907605 9.987141 1 ## 58 4.60446834 3.88197707 7.640542 26.746499 8.096760 15.285686 1 ## 59 6.05634729 0.75028887 3.400547 13.582939 3.745871 8.482119 1 ## 60 5.55593474 1.51065503 3.879217 12.798800 3.669504 6.979974 1 ## 61 4.03092200 2.21539129 5.044494 16.871488 4.978996 9.304746 1 ## 62 5.23612553 2.42210867 3.724228 13.103840 3.707017 7.517498 1 ## 63 4.29091253 0.77885172 3.209739 11.250332 3.115018 6.435724 1 ## 64 8.17872107 2.31222782 3.503141 15.091380 4.148630 9.836670 1 ## 65 5.02695115 2.88646213 3.588984 12.896787 3.621443 7.513311 1 ## 66 2.48083883 2.47481069 2.572586 9.004733 2.394330 5.145854 1 ## 67 3.99004087 2.86984135 2.321320 9.601955 2.480819 6.119975 1 ## 68 2.23831135 1.11347620 7.354859 24.266268 7.405282 13.233980 1 ## 69 5.81016858 1.87134447 1.780620 7.271942 1.763140 4.601012 1 ## 70 8.38552575 3.09651049 2.438272 9.222328 2.415150 5.564919 1 ## 71 7.52829625 2.51802955 4.870025 17.058979 4.982251 9.753941 1 ## 72 5.80565410 2.39803318 6.107551 19.258297 5.841462 10.096971 1 ## 73 4.63571743 3.06665941 3.068762 10.043868 2.778158 5.440724 1 ## 74 6.15793650 1.55045992 8.069649 27.857468 8.481779 15.752995 1 ## 75 4.78126024 2.62610198 2.564135 7.630308 2.048611 3.784106 1 ## 76 -3.16739941 1.18116405 6.700594 22.114532 6.703782 12.063641 1 ## 77 6.43347697 1.73648379 5.381833 17.057971 5.109951 8.985221 1 ## 78 3.50959659 2.15457529 12.644899 40.205236 12.712534 21.237888 1 ## 79 10.07323536 2.56844555 2.037142 9.119878 2.289255 6.064165 1 ## 80 13.67440127 -0.66015968 5.883640 17.576515 5.365039 8.751055 1 ## 81 0.04110863 3.13653254 7.093428 24.177106 7.317634 13.536964 1 ## 82 7.35949555 2.42177278 4.873831 16.571498 4.861332 9.260751 1 ## 83 5.49607715 3.35008260 8.291038 25.527766 7.954701 13.091208 1 ## 84 2.90516885 3.10375689 4.051026 12.221867 3.568223 6.145328 1 ## 85 7.48091201 2.64704611 7.689539 25.778200 7.866935 14.243891 1 ## 86 7.83288634 2.17563581 4.933636 16.643004 4.894160 9.242550 1 ## 87 4.62720660 2.65355779 5.774989 19.541334 5.829081 10.878851 1 ## 88 3.81921320 1.93450970 4.483566 16.268060 4.687907 9.542711 1 ## 89 0.65673908 2.64552217 2.739769 11.946482 3.171563 7.836829 1 ## 90 2.50073977 2.36429404 5.286464 17.755621 5.260521 9.825925 1 ## 91 4.06797383 2.84344157 3.701213 12.546517 3.561933 6.994698 1 ## 92 3.99673254 1.32352113 5.795986 20.816259 6.153061 12.122280 1 ## 93 8.81558134 1.60856710 4.883292 15.756919 4.660053 8.431981 1 ## 94 3.93610997 2.40494064 7.172253 22.359187 6.882860 11.600808 1 ## 95 12.58110379 0.89314130 3.340735 11.491910 3.208161 6.480807 1 ## 96 3.28003669 1.61669959 7.262549 26.233329 7.873969 15.339506 1 ## 97 11.30218798 2.29402025 1.940701 6.989609 1.732577 4.078556 1 ## 98 5.64776480 3.79306067 5.958475 20.288944 6.061855 11.351232 1 ## 99 0.65818837 2.81403217 4.432708 14.119440 4.138037 7.470379 1 ## 100 7.30774920 0.67997560 3.283518 10.676520 2.990010 5.751243 1 ## Y4 Y5 ## 1 0.092739260 1 ## 2 1.016727357 1 ## 3 0.493990392 0 ## 4 1.243224329 0 ## 5 0.695205988 1 ## 6 0.952201378 1 ## 7 -0.343146465 0 ## 8 1.159870423 0 ## 9 -0.429393276 0 ## 10 0.499274828 1 ## 11 0.871180147 1 ## 12 0.444423658 0 ## 13 0.229090617 1 ## 14 1.076493168 0 ## 15 0.854254673 1 ## 16 0.298747112 0 ## 17 -0.001638862 0 ## 18 1.047002780 1 ## 19 -0.456508875 1 ## 20 2.965934470 0 ## 21 0.437209150 0 ## 22 1.467067372 0 ## 23 0.783287466 0 ## 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##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a620dd8>
##
## $26 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a61e4e0> ## ##$27
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a620038>
##
## $28 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a61b900> ## ##$29
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a618e10>
##
## $30 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a616c88> ## ##$31
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a612668>
##
## $32 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a613b68> ## ##$33
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a611430>
##
## $34 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a60aef0> ## ##$35
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a606940>
##
## $36 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a607eb0> ## ##$37
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a605548>
##
## $38 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5fee10> ## ##$39
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5fcb38>
##
## $40 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5fa6d8> ## ##$41
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5fbd60>
##
## $42 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5f75b8> ## ##$43
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5f37e8>
##
## $44 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5f0e48> ## ##$45
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5ee898>
##
## $46 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5efdd0> ## ##$47
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5ed708>
##
## $48 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5e6cc0> ## ##$49
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5e2898>
##
## $50 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5e0438> ## ##$51
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5e1c80>
##
## $52 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5df190> ## ##$53
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5dc940>
##
## $54 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5de000> ## ##$55
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5dbd98>
##
## $56 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5d73c0> ## ##$57
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5d4dd8>
##
## $58 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5d3040> ## ##$59
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5c24e0>
##
## $60 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5c0668> ## ##$61
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5c1fc8>
##
## $62 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5bd938> ## ##$63
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5bb120>
##
## $64 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5b8898> ## ##$65
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5b9d98>
##
## $66 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5b7388> ## ##$67
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5b4c18>
##
## $68 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5b5f58> ## ##$69
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5af7b0>
##
## $70 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5ad1c8> ## ##$71
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5a8c50>
##
## $72 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5a64a8> ## ##$73
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5a7eb0>
##
## $74 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5a3cf0> ## ##$75
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5a1580>
##
## $76 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a59eac8> ## ##$77
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a59a240>
##
## $78 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a59b708> ## ##$79
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a599548>
##
## $80 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a596b00> ## ##$81
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a592240>
##
## $82 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a593858> ## ##$83
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a588a58>
##
## $84 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a586358> ## ##$85
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a587b30>
##
## $86 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a583e08> ## ##$87
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a5814d8>
##
## $88 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a57f0b0> ## ##$89
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a57ac50>
##
## $90 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a574b38> ## ##$91
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a575fc8>
##
## $92 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a571740> ## ##$93
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a56d4d8>
##
## $94 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a56aeb8> ## ##$95
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a566d68>
##
## $96 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a564320> ## ##$97
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a565dd0>
##
## $98 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a5635b8> ## ##$99
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7f8b8a93d120>
## <environment: 0x7f8b7a561238>
##
## $100 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7f8b8a93d120> ## <environment: 0x7f8b7a55eba8> ## ## ## Slot ".GFUN": ## function (theta) ## { ## psii <- lapply(psi_list, function(psi) { ## do.call(psi, args = append(list(theta = theta), object@.inner_args)) ## }) ## compute_sum_of_list(psii, object@.weights) ## } ## <environment: 0x7f8b7a511cb8> ## ## Slot ".control": ## An object of class "geex_control" ## Slot ".approx": ## An object of class "approx_control" ## Slot ".FUN": ## function () ## NULL ## <bytecode: 0x7f8b6da82970> ## ## Slot ".options": ## list() ## ## ## Slot ".root": ## An object of class "root_control" ## Slot ".object_name": ## [1] "root" ## ## Slot ".FUN": ## function (f, start, maxiter = 100, rtol = 1e-06, atol = 1e-08, ## ctol = 1e-08, useFortran = TRUE, positive = FALSE, jacfunc = NULL, ## jactype = "fullint", verbose = FALSE, bandup = 1, banddown = 1, ## parms = NULL, ...) ## { ## initfunc <- NULL ## if (is.list(f)) { ## if (!is.null(jacfunc) & "jacfunc" %in% names(f)) ## stop("If 'f' is a list that contains jacfunc, argument 'jacfunc' should be NULL") ## jacfunc <- f$jacfunc
##         initfunc <- f$initfunc ## f <- f$func
##     }
##     N <- length(start)
##     if (!is.numeric(start))
##         stop("start conditions should be numeric")
##     if (!is.numeric(maxiter))
##         stop("maxiter' must be numeric")
##     if (as.integer(maxiter) < 1)
##         stop("maxiter must be >=1")
##     if (!is.numeric(rtol))
##         stop("rtol' must be numeric")
##     if (!is.numeric(atol))
##         stop("atol' must be numeric")
##     if (!is.numeric(ctol))
##         stop("ctol' must be numeric")
##     if (length(atol) > 1 && length(atol) != N)
##         stop("atol' must either be a scalar, or as long as start'")
##     if (length(rtol) > 1 && length(rtol) != N)
##         stop("rtol' must either be a scalar, or as long as y'")
##     if (length(ctol) > 1)
##         stop("ctol' must be a scalar")
##     if (useFortran) {
##         if (!is.compiled(f) & is.null(parms)) {
##             Fun1 <- function(time = 0, x, parms = NULL) list(f(x,
##                 ...))
##             Fun <- Fun1
##         }
##         else if (!is.compiled(f)) {
##             Fun2 <- function(time = 0, x, parms) list(f(x, parms,
##                 ...))
##             Fun <- Fun2
##         }
##         else {
##             Fun <- f
##             f <- function(x, ...) Fun(n = length(start), t = 0,
##                 x, f = rep(0, length(start)), 1, 1)$f ## } ## JacFunc <- jacfunc ## if (!is.null(jacfunc)) ## if (!is.compiled(JacFunc) & is.null(parms)) ## JacFunc <- function(time = 0, x, parms = parms) jacfunc(x, ## ...) ## else if (!is.compiled(JacFunc)) ## JacFunc <- function(time = 0, x, parms = parms) jacfunc(x, ## parms, ...) ## else JacFunc <- jacfunc ## method <- "stode" ## if (jactype == "sparse") { ## method <- "stodes" ## if (!is.null(jacfunc)) ## stop("jacfunc can not be used when jactype='sparse'") ## x <- stodes(y = start, time = 0, func = Fun, atol = atol, ## positive = positive, rtol = rtol, ctol = ctol, ## maxiter = maxiter, verbose = verbose, parms = parms, ## initfunc = initfunc) ## } ## else x <- steady(y = start, time = 0, func = Fun, atol = atol, ## positive = positive, rtol = rtol, ctol = ctol, maxiter = maxiter, ## method = method, jacfunc = JacFunc, jactype = jactype, ## verbose = verbose, parms = parms, initfunc = initfunc, ## bandup = bandup, banddown = banddown) ## precis <- attr(x, "precis") ## attributes(x) <- NULL ## x <- unlist(x) ## if (is.null(parms)) ## reffx <- f(x, ...) ## else reffx <- f(x, parms, ...) ## i <- length(precis) ## } ## else { ## if (is.compiled(f)) ## stop("cannot combine compiled code with R-implemented solver") ## precis <- NULL ## x <- start ## jacob <- matrix(nrow = N, ncol = N, data = 0) ## if (is.null(parms)) ## reffx <- f(x, ...) ## else reffx <- f(x, parms, ...) ## if (length(reffx) != N) ## stop("'f', function must return as many function values as elements in start") ## for (i in 1:maxiter) { ## refx <- x ## pp <- mean(abs(reffx)) ## precis <- c(precis, pp) ## ewt <- rtol * abs(x) + atol ## if (max(abs(reffx/ewt)) < 1) ## break ## delt <- perturb(x) ## for (j in 1:N) { ## x[j] <- x[j] + delt[j] ## if (is.null(parms)) ## fx <- f(x, ...) ## else fx <- f(x, parms, ...) ## jacob[, j] <- (fx - reffx)/delt[j] ## x[j] <- refx[j] ## } ## relchange <- as.numeric(solve(jacob, -1 * reffx)) ## if (max(abs(relchange)) < ctol) ## break ## x <- x + relchange ## if (is.null(parms)) ## reffx <- f(x, ...) ## else reffx <- f(x, parms, ...) ## } ## } ## names(x) <- names(start) ## return(list(root = x, f.root = reffx, iter = i, estim.precis = precis[length(precis)])) ## } ## <bytecode: 0x7f8b6dab4ec0> ## <environment: namespace:rootSolve> ## ## Slot ".options": ##$start
## [1] 0 0
##
##
##
## Slot ".deriv":
## An object of class "deriv_control"
## Slot ".FUN":
## function (func, x, method = "Richardson", side = NULL, method.args = list(),
##     ...)
## UseMethod("jacobian")
## <bytecode: 0x7f8b6da97a20>
## <environment: namespace:numDeriv>
##
## Slot ".options":
## $method ## [1] "Richardson" ## ## ## ## ## Slot ".estFUN": ## function(data){ ## Y1 <- data$Y1
##   function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## }
## <bytecode: 0x7f8b8a93e710>
##
## Slot ".outer_args":
## list()
##
## Slot ".inner_args":
## list()
##
##
## Slot "rootFUN_results":
## $root ## [1] 5.044563 10.041239 ## ##$f.root
## [1] -2.131628e-14  4.654055e-13
##
## $iter ## [1] 4 ## ##$estim.precis
## [1] 2.433609e-13
##
##
## Slot "sandwich_components":
## An object of class "sandwich_components"
## Slot ".A":
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##
## Slot ".B":
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## Slot ".B_i":
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## $87 ## [,1] [,2] ## [1,] 0.1741867 4.118081 ## [2,] 4.1180809 97.358719 ## ##$88
##           [,1]     [,2]
## [1,]  1.501483 10.46419
## [2,] 10.464191 72.92743
##
## $89 ## [,1] [,2] ## [1,] 19.2530 -40.41960 ## [2,] -40.4196 84.85658 ## ##$90
##          [,1]     [,2]
## [1,] 6.471038  9.08196
## [2,] 9.081960 12.74633
##
## $91 ## [,1] [,2] ## [1,] 0.9537271 8.874769 ## [2,] 8.8747688 82.582870 ## ##$92
##          [,1]      [,2]
## [1,] 1.097949  9.371054
## [2,] 9.371054 79.982427
##
## $93 ## [,1] [,2] ## [1,] 14.22058 15.76036 ## [2,] 15.76036 17.46687 ## ##$94
##          [,1]      [,2]
## [1,] 1.228669  9.768323
## [2,] 9.768323 77.661390
##
## $95 ## [,1] [,2] ## [1,] 56.79944 352.3951 ## [2,] 352.39509 2186.3295 ## ##$96
##           [,1]     [,2]
## [1,]  3.113554 12.22408
## [2,] 12.224084 47.99281
##
## $97 ## [,1] [,2] ## [1,] 39.15787 182.2009 ## [2,] 182.20092 847.7780 ## ##$98
##           [,1]      [,2]
## [1,]  0.363852 -5.837414
## [2,] -5.837414 93.651817
##
## $99 ## [,1] [,2] ## [1,] 19.24029 -40.35047 ## [2,] -40.35047 84.62246 ## ##$100
##           [,1]      [,2]
## [1,]   5.12201 -11.13313
## [2,] -11.13313  24.19881
##
##
## Slot ".ee_i":
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##
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##
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##
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##
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##
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##
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##
## $93 ## [1] 3.771018 4.179338 ## ##$94
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##
## $95 ## [1] 7.53654 46.75820 ## ##$96
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##
## $97 ## [1] 6.257625 29.116627 ## ##$98
## [1]  0.6032015 -9.6773869
##
## $99 ## [1] -4.386375 9.199047 ## ##$100
## [1]  2.263186 -4.919229
##
##
##
## Slot "GFUN":
## function ()
## NULL
## <bytecode: 0x7f8b6da0a2f8>
##
## Slot "corrections":
## list()
##
## Slot "estimates":
## [1]  5.044563 10.041239
##
## Slot "vcov":
##            [,1]       [,2]
## [1,] 0.10041239 0.03667969
## [2,] 0.03667969 2.49219638`