# The design and structure of geex

#### 2020-02-17

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])
##   }
## <environment: 0x7fd9e9d1df58>
##
## $2 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9e9cef2a8> 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: 0x7fd9e7de2f50> ## 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 ## 24 1.165717760 0 ## 25 -0.198696160 1 ## 26 0.213533342 1 ## 27 -0.072493261 1 ## 28 0.736487513 1 ## 29 -0.625758090 1 ## 30 1.375465405 1 ## 31 0.264670535 0 ## 32 2.972649859 1 ## 33 0.215875121 1 ## 34 0.782782994 1 ## 35 -0.227084853 1 ## 36 0.442637449 1 ## 37 0.421447969 0 ## 38 0.882479555 0 ## 39 1.373000995 1 ## 40 1.864965592 1 ## 41 0.387733146 1 ## 42 1.943114799 1 ## 43 -1.474856978 0 ## 44 1.741072051 1 ## 45 0.024847168 1 ## 46 1.966803213 1 ## 47 0.239605022 0 ## 48 2.177764398 1 ## 49 1.088997768 1 ## 50 -0.964458223 1 ## 51 0.715242972 1 ## 52 -0.631970427 1 ## 53 0.996355205 0 ## 54 2.634852773 1 ## 55 -1.246686055 1 ## 56 1.764940768 0 ## 57 -0.173094497 1 ## 58 3.188926631 1 ## 59 2.321353405 1 ## 60 0.149069864 0 ## 61 0.842453670 1 ## 62 0.903578781 0 ## 63 0.542090297 1 ## 64 3.532272980 0 ## 65 1.088732578 1 ## 66 0.144233610 1 ## 67 1.470126269 0 ## 68 1.537177460 0 ## 69 0.708145014 1 ## 70 0.751337374 0 ## 71 1.535905791 1 ## 72 0.146399418 0 ## 73 -0.255543077 0 ## 74 3.055486628 0 ## 75 -1.205682549 1 ## 76 1.282809142 1 ## 77 0.050654962 1 ## 78 2.135029369 1 ## 79 1.812166070 1 ## 80 -0.886040754 1 ## 81 2.206165066 1 ## 82 1.037387368 1 ## 83 0.083754535 0 ## 84 -0.926108918 0 ## 85 2.078535519 1 ## 86 0.935458616 0 ## 87 1.393866742 0 ## 88 1.865718680 0 ## 89 2.601152645 0 ## 90 1.024876085 1 ## 91 0.412999035 1 ## 92 2.607900007 0 ## 93 0.195371813 1 ## 94 0.159654048 1 ## 95 0.403777090 0 ## 96 3.771937632 1 ## 97 -0.038425654 1 ## 98 1.609367331 0 ## 99 -0.135412360 1 ## 100 -0.245682938 0 ## ## Slot ".units": ## character(0) ## ## Slot ".weights": ## numeric(0) ## ## Slot ".psiFUN_list": ##$1
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
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## $22 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1ec698> ## ##$23
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## <environment: 0x7fd9ee1e9b28>
##
## $26 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1e4e40> ## ##$27
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1e2858>
##
## $28 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1e3d90> ## ##$29
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1e10e0>
##
## $30 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1de900> ## ##$31
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1dc008>
##
## $32 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1dd658> ## ##$33
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1da8c8>
##
## $34 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1dbe00> ## ##$35
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1d9268>
##
## $36 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1d67e8> ## ##$37
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1d7d20>
##
## $38 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1d0fc8> ## ##$39
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1ce7b0>
##
## $40 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1cc0e8> ## ##$41
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1cde00>
##
## $42 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1c9770> ## ##$43
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1c6f58>
##
## $44 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1c42e0> ## ##$45
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1c5540>
##
## $46 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1c0890> ## ##$47
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1c1b28>
##
## $48 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1bf0e0> ## ##$49
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1bc4a0>
##
## $50 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1bd8c0> ## ##$51
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1baee8>
##
## $52 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1b8858> ## ##$53
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1b9e38>
##
## $54 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1b78c0> ## ##$55
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1b33f0>
##
## $56 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1ae7b0> ## ##$57
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1afcb0>
##
## $58 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1ad2d8> ## ##$59
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1aacb8>
##
## $60 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1a6e78> ## ##$61
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1a4a50>
##
## $62 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1a5d90> ## ##$63
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1a3188>
##
## $64 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1a05f0> ## ##$65
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1a1a48>
##
## $66 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee19d230> ## ##$67
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee19a548>
##
## $68 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee19b770> ## ##$69
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee198ac0>
##
## $70 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee196190> ## ##$71
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee197498>
##
## $72 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee194970> ## ##$73
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1921c8>
##
## $74 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee193700> ## ##$75
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee190ac0>
##
## $76 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee18a238> ## ##$77
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee18b4d0>
##
## $78 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee188ac0> ## ##$79
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee1865f0>
##
## $80 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1877a8> ## ##$81
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee182cf0>
##
## $82 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1808c8> ## ##$83
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee181d58>
##
## $84 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee17d188> ## ##$85
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee17a7b0>
##
## $86 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee178468> ## ##$87
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee176388>
##
## $88 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1777a8> ## ##$89
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee175118>
##
## $90 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee1725f0> ## ##$91
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee173e70>
##
## $92 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee171380> ## ##$93
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee16c858>
##
## $94 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee16a0e8> ## ##$95
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee16b460>
##
## $96 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee168858> ## ##$97
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee169c78>
##
## $98 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee166f90> ## ##$99
## function(theta){
##     c(Y1 - theta[1],
##       (Y1 - theta[1])^2 - theta[2])
##   }
## <bytecode: 0x7fd9e9a54348>
## <environment: 0x7fd9ee164468>
##
## $100 ## function(theta){ ## c(Y1 - theta[1], ## (Y1 - theta[1])^2 - theta[2]) ## } ## <bytecode: 0x7fd9e9a54348> ## <environment: 0x7fd9ee165738> ## ## ## Slot ".GFUN": ## function (theta) ## { ## psii <- lapply(psi_list, function(psi) { ## do.call(psi, args = append(list(theta = theta), object@.inner_args)) ## }) ## compute_sum_of_list(psii, object@.weights) ## } ## <environment: 0x7fd9ee12f7a8> ## ## Slot ".control": ## An object of class "geex_control" ## Slot ".approx": ## An object of class "approx_control" ## Slot ".FUN": ## function () ## NULL ## <bytecode: 0x7fd9e35a06a0> ## ## Slot ".options": ## list() ## ## ## Slot ".root": ## An object of class "root_control" ## Slot ".object_name": ## [1] "root" ## ## Slot ".FUN": ## function (f, start, maxiter = 100, rtol = 1e-06, atol = 1e-08, ## ctol = 1e-08, useFortran = TRUE, positive = FALSE, jacfunc = NULL, ## jactype = "fullint", verbose = FALSE, bandup = 1, banddown = 1, ## parms = NULL, ...) ## { ## initfunc <- NULL ## if (is.list(f)) { ## if (!is.null(jacfunc) & "jacfunc" %in% names(f)) ## stop("If 'f' is a list that contains jacfunc, argument 'jacfunc' should be NULL") ## jacfunc <- f$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: 0x7fd9e35d20d0> ## <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: 0x7fd9e35b3f30>
## <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: 0x7fd9e9a4fa50>
##
## 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":
##              [,1] [,2]
## [1,]  1.00000e+02    0
## [2,] -1.65139e-11  100
##
## Slot ".A_i":
## $1 ## [,1] [,2] ## [1,] 1.000000 0 ## [2,] -2.752514 1 ## ##$2
##          [,1] [,2]
## [1,]  1.00000    0
## [2,] 10.81578    1
##
## $3 ## [,1] [,2] ## [1,] 1.000000 0 ## [2,] -3.842305 1 ## ##$4
##          [,1] [,2]
## [1,] 1.000000    0
## [2,] 6.653878    1
##
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## [1,]  3.113554 12.22408
## [2,] 12.224084 47.99281
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## [1,]  0.363852 -5.837414
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## [1,]   5.12201 -11.13313
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##
##
##
## Slot "GFUN":
## function ()
## NULL
## <bytecode: 0x7fd9e355f6a8>
##
## Slot "corrections":
## list()
##
## Slot "estimates":
## [1]  5.044563 10.041239
##
## Slot "vcov":
##            [,1]       [,2]
## [1,] 0.10041239 0.03667969
## [2,] 0.03667969 2.49219638`