# KoutParamsEstim: Iterative Koutrouvelis regression method In StableEstim: Estimate the Four Parameters of Stable Laws using Different Methods

## Description

Iterative Koutrouvelis regression method with different spacing schemes (points where the eCF is computed).

## Usage

 ```1 2 3``` ```KoutParametersEstim(x, theta0 = NULL, spacing = c("Kout", "UniformSpac", "ArithSpac", "free"), pm = 0, tol = 0.05, NbIter = 10, PrintTime = FALSE, ...) ```

## Arguments

 `x` Data used to perform the estimation: vector of length n. `theta0` Initial guess for the 4 parameters values: vector of length 4 `spacing` Scheme used to select the points where the moment conditions are evaluated. `Kout` is the scheme suggested by Koutrouvelis, `UniformSpac` and `ArithSpac` are the uniform and arithmetic spacing schemes over the informative interval [ε,A_n]. If user choose free, he needs to provide a set of points `t_points` and `u_points` in `...`. `pm` Parametrisation, an integer (0 or 1); default: `pm=0` (the Nolan ‘S0’ parametrisation). `tol` The loop stops if the relative error between two consecutive estimation is smaller then `tol`; default=0.05 `NbIter` Maximum number of iteration. The loop stops when `NbIter` is reached; default=10 `PrintTime` Logical flag; if set to TRUE, the estimation duration is printed out to the screen in a readable format (h/min/sec). `...` Other arguments to pass to the function. See details

## Details

spacing 4 options for the spacing scheme are implemented as described above. In particular:

`UniformSpac`, `ArithSpac`:

user can specify the number of points to choose in both regression by inputting `nb_t` and `nb_u`. Otherwise the Koutrouvelis table will be used to compte them.

`free`:

User is expected to provide `t_points` and `u_points` otherwise the `Kout` scheme will be used.

## Value

Returns a list with the following elements:

 `Estim` `list` containing the vector of 4 parameters estimate (`par`), the 2 regressions objects (`reg1` and `reg2`) and the matrix of iterations estimate (`vals`). `duration` estimation duration in a numerical format `method` `character` describing the method used

## References

Koutrouvelis IA (1980). “Regression-type estimation of the parameters of stable laws.” Journal of the American Statistical Association, 75(372), pp. 918–928.

Koutrouvelis IA (1981). “An iterative procedure for the estimation of the parameters of stable laws: An iterative procedure for the estimation.” Communications in Statistics-Simulation and Computation, 10(1), pp. 17–28.

`Estim`

## Examples

 ```1 2 3 4 5 6 7 8``` ``` pm=0 theta <- c(1.45,0.5,1.1,0.4) set.seed(1235);x <- rstable(200,theta[1],theta[2],theta[3],theta[4],pm=pm) theta0=theta-0.1 spacing="Kout" KoutParametersEstim(x=x,theta0=theta0, spacing=spacing,pm=pm) ```

### Example output

```Loading required package: Matrix
\$Estim
\$Estim\$par
[1] 1.3344151 0.4676414 1.0085928 0.6678007

\$Estim\$regObject1
\$Estim\$regObject1\$alpha
[1] 1.334415

\$Estim\$regObject1\$gamma
[1] 1.008593

\$Estim\$regObject1\$obj
\$coefficients
(Intercept)           w
0.7088417   1.3344151

\$residuals
[1] -0.05787720  0.04029226 -0.07402443 -0.03429568  0.05751061  0.06609260
[7]  0.02759895  0.01291000  0.06764385  0.07514438 -0.04056695 -0.09603401
[13] -0.05689624 -0.09900973 -0.18281535 -0.20263143 -0.23564454 -0.27082373
[19] -0.33518429 -0.41815004 -0.34564423

\$effects
(Intercept)            w
-2.88909651 -12.75249112   0.03718116  -1.04345420  -0.36569028  -0.17901914

-0.72032817  -0.44130671  -0.04961527   0.99193952  -0.63024592   0.49598630

0.69735376  -1.80494979   1.31693622   1.74037573  -1.35024588   0.72165765

0.24167200   0.45964172  -0.07459873

\$rank
[1] 2

\$fitted.values
[1] -2.05892986 -1.13398383 -0.59292508 -0.20903779  0.08872832  0.33202096
[7]  0.53772194  0.71590824  0.87307970  1.01367436  1.14085770  1.25696699
[13]  1.36377719  1.46266798  1.55473311  1.64085428  1.72175269  1.79802574
[19]  1.87017385  1.93862040  2.00372673

\$assign
NULL

\$qr
\$qr
(Intercept)             w
[1,] -9.558134e+00  2.912218e+00
[2,] -7.396163e-01 -9.556615e+00
[3,] -4.586215e-01  8.569227e-01
[4,] -2.265525e-01 -4.769813e-01
[5,] -1.974310e-01 -2.727447e-02
[6,]  5.561188e-02 -1.062211e-01
[7,] -1.598650e-02 -8.328343e-02
[8,]  1.983991e-02  2.173686e-02
[9,]  1.252809e-02 -1.661564e-03
[10,] -3.035077e-03  1.179246e-02
[11,]  3.476299e-04 -7.863231e-03
[12,]  1.014034e-03  5.110691e-03
[13,]  6.466102e-04  1.996900e-03
[14,] -3.170199e-04 -7.325344e-04
[15,]  7.200185e-05  2.244465e-04
[16,]  6.891658e-07 -1.169513e-04
[17,] -9.251990e-06  8.004342e-05
[18,] -3.525780e-06  4.940021e-05
[19,] -4.625320e-07  2.360839e-05
[20,] -1.199750e-07 -8.561639e-06
[21,]  5.921626e-08  2.143827e-06

\$qraux
[1] 1.385252 1.136024

\$pivot
[1] 1 2

\$tol
[1] 1e-07

\$rank
[1] 2

attr(,"class")
[1] "qr"

\$df.residual
[1] 19

\$terms
y ~ w
attr(,"variables")
list(y, w)
attr(,"factors")
w
y 0
w 1
attr(,"term.labels")
[1] "w"
attr(,"order")
[1] 1
attr(,"intercept")
[1] 1
attr(,"response")
[1] 1
attr(,".Environment")
<environment: 0x846fd08>
attr(,"predvars")
list(y, w)
attr(,"dataClasses")
y         w
"numeric" "numeric"

\$call
lm.gls(formula = y ~ w, W = sig, inverse = TRUE)

\$xlevels
named list()

attr(,"class")
[1] "lm.gls"

\$Estim\$regObject1\$updatedData
[1]  -0.626933424  -0.722570038  -2.383259156   2.403082886   0.993929693
[6]  -1.179618588  -0.823781073   0.342432858  -0.544732252  -2.636008744
[11]   3.006516247  -0.494420604  -1.131950271  -0.260840472   0.319892188
[16]  -8.871491763   1.612402224  -0.383010903  -0.067098228   1.280735952
[21]  -2.418525878   4.357222048   1.351933392  -1.111651724  -2.381504635
[26]   1.230201551  -1.838725853  -0.840261119  -2.849381002  -0.681011824
[31]  -0.105437015  -0.988982776   0.164135986  -0.425440079  -2.109310242
[36]  -0.421339197  -1.355627428  -2.592407493  -0.284291044  -1.075824274
[41]  -0.924269723   2.930037067  -3.587158102  -1.192462096   3.076546141
[46]   2.026704817  -1.805413153  -0.848286649  -0.021069395  -1.778134358
[51]  -1.831695318  -1.348231297  -1.258149400  -2.313893859  -0.860540285
[56]  -0.040732426  -2.030051766   1.200154039  -0.495555913  -2.535110841
[61]  -0.830010838  -0.773541122  -2.292878431  -0.089297603  13.373256152
[66]  -0.810038048  -1.962963696   2.235511791   2.161293706  -3.536976857
[71]   1.778112999  -0.568805990   0.005438781  -2.294454376  -0.908567864
[76]   1.534477240  -7.130534721  -0.944800156   2.919490321  -3.529773840
[81]  -0.557014475   0.181842975  -2.999932316  -1.579776309  -0.345316355
[86]   0.644682788  -0.854012470   1.165438366  -2.764356968  -1.487143921
[91]  -0.633522156  -3.495262901  -1.893289735  -1.175698406  -0.184887331
[96]  -1.270711831   0.135692892   0.280685554 115.160789010  -0.344421118
[101]  -0.271014960   1.038545633  -5.639841670  -0.095688616  -0.511490911
[106]  -1.611364715  -0.743067270   2.464408490   4.604983727  -4.116897352
[111]   3.923263909  -2.178734707  -0.250667027  -1.606707628  -1.301101620
[116]  -1.283885028  -1.504903136   2.803195437  -1.606649302   7.159408263
[121]   0.928531765  -0.200074912  -0.706378788   3.430007742  -0.937316767
[126]  -0.868270444  -3.312635562  -1.636513940   0.075269659  -0.611841973
[131]   0.247478644   1.425506684  -3.261735414   0.333644252   1.057669745
[136]   0.357680615  -2.308920534  -1.019030988  -1.277398464  -2.164374185
[141]  -0.510920464  -0.149071819  -0.082272347  -0.106027976  -1.286002082
[146]   6.011522834   0.901826329  -2.509077140  -1.708738376 -13.184319957
[151]  -0.124998067  -1.501454459  -1.454259201   0.743984706  -2.212461930
[156]  -1.261411484  -1.596635311  -0.628765150  -1.226249012  -1.105750477
[161]  -0.327003174   3.751433032   1.416651709  -0.566832174   0.011391106
[166]  -0.626765893  -0.290205685   0.281344241  -3.353261333  -0.051925272
[171]   0.657999833  -1.038381031  -0.014177686   0.767003357   0.886416745
[176]  -1.159231571   0.840368069  -2.836240756  -3.635908314  -0.095890406
[181]  -3.479619778  -1.336547594  -0.555365153  -1.964800649  -0.097427363
[186]  -2.441086333  -0.465432481   1.914286202   0.350360387  -0.428384672
[191]   1.774025059   1.127742215  -1.553577895  -0.151862620  12.068249607
[196]  -1.680149738  11.823836616  -1.205903417  -1.719171620   3.172317401

\$Estim\$regObject2
\$Estim\$regObject2\$beta
[1] 0.4676414

\$Estim\$regObject2\$delta
[1] 0.6678007

\$Estim\$regObject2\$obj
\$coefficients
u          Om
-0.02991856 -0.80680953

\$residuals
[1] -0.0003928949  0.0013324999 -0.0032343258 -0.0113183750 -0.0138728714
[6] -0.0044447134  0.0144084290  0.0326832941  0.0391892862  0.0287596496
[11]  0.0075947863 -0.0097960422 -0.0131611042

\$effects
u          Om
-6.14625911  2.35525590 -1.62845436 -0.09577137 -1.46623619  0.26650545

-0.12977734 -0.15814915  0.16767628  0.20991726 -0.19669805 -0.17729849

-0.15574861

\$rank
[1] 2

\$fitted.values
[1] -0.02197281 -0.05442877 -0.09268012 -0.13529322 -0.18149101 -0.23077213
[7] -0.28278109 -0.33725019 -0.39396921 -0.45276787 -0.51350491 -0.57606088
[13] -0.64033324

\$assign
NULL

\$qr
\$qr
u          Om
[1,]  9.0864682745  7.28103091
[2,]  0.1347664608 -2.91922173
[3,] -0.1473166164 -0.29669835
[4,] -0.0020693168 -0.37701712
[5,]  0.0526670375  0.15972862
[6,]  0.0055423598 -0.21230884
[7,]  0.0244398529  0.10568262
[8,]  0.0005303477 -0.12716215
[9,]  0.0089739577  0.09016390
[10,]  0.0042369355  0.09035207
[11,] -0.0041972146 -0.08227781
[12,] -0.0033774937 -0.07550927
[13,] -0.0026543150 -0.06679224

\$qraux
[1] 1.978057 1.799172

\$pivot
[1] 1 2

\$tol
[1] 1e-07

\$rank
[1] 2

attr(,"class")
[1] "qr"

\$df.residual
[1] 11

\$terms
z ~ -1 + u + Om
attr(,"variables")
list(z, u, Om)
attr(,"factors")
u Om
z  0  0
u  1  0
Om 0  1
attr(,"term.labels")
[1] "u"  "Om"
attr(,"order")
[1] 1 1
attr(,"intercept")
[1] 0
attr(,"response")
[1] 1
attr(,".Environment")
<environment: 0x85ba480>
attr(,"predvars")
list(z, u, Om)
attr(,"dataClasses")
z         u        Om
"numeric" "numeric" "numeric"

\$call
lm.gls(formula = z ~ -1 + u + Om, W = sig2, inverse = TRUE)

\$xlevels
named list()

attr(,"class")
[1] "lm.gls"

\$Estim\$regObject2\$updatedData
[1]  -0.597014868  -0.692651483  -2.353340601   2.433001442   1.023848248
[6]  -1.149700033  -0.793862517   0.372351413  -0.514813697  -2.606090188
[11]   3.036434803  -0.464502048  -1.102031716  -0.230921916   0.349810743
[16]  -8.841573208   1.642320780  -0.353092348  -0.037179672   1.310654508
[21]  -2.388607322   4.387140603   1.381851948  -1.081733168  -2.351586079
[26]   1.260120107  -1.808807297  -0.810342563  -2.819462446  -0.651093268
[31]  -0.075518460  -0.959064220   0.194054542  -0.395521523  -2.079391686
[36]  -0.391420641  -1.325708872  -2.562488937  -0.254372488  -1.045905719
[41]  -0.894351168   2.959955623  -3.557239547  -1.162543541   3.106464696
[46]   2.056623373  -1.775494597  -0.818368093   0.008849161  -1.748215802
[51]  -1.801776763  -1.318312741  -1.228230844  -2.283975303  -0.830621729
[56]  -0.010813870  -2.000133210   1.230072594  -0.465637358  -2.505192285
[61]  -0.800092282  -0.743622566  -2.262959875  -0.059379047  13.403174708
[66]  -0.780119493  -1.933045140   2.265430346   2.191212262  -3.507058302
[71]   1.808031555  -0.538887434   0.035357337  -2.264535820  -0.878649308
[76]   1.564395796  -7.100616165  -0.914881600   2.949408877  -3.499855285
[81]  -0.527095919   0.211761531  -2.970013760  -1.549857753  -0.315397799
[86]   0.674601344  -0.824093914   1.195356922  -2.734438413  -1.457225366
[91]  -0.603603600  -3.465344345  -1.863371180  -1.145779851  -0.154968775
[96]  -1.240793275   0.165611448   0.310604109 115.190707565  -0.314502562
[101]  -0.241096404   1.068464189  -5.609923114  -0.065770060  -0.481572356
[106]  -1.581446160  -0.713148714   2.494327045   4.634902283  -4.086978796
[111]   3.953182465  -2.148816152  -0.220748472  -1.576789072  -1.271183064
[116]  -1.253966473  -1.474984581   2.833113993  -1.576730746   7.189326819
[121]   0.958450321  -0.170156357  -0.676460232   3.459926298  -0.907398211
[126]  -0.838351888  -3.282717007  -1.606595384   0.105188214  -0.581923418
[131]   0.277397200   1.455425240  -3.231816858   0.363562807   1.087588300
[136]   0.387599171  -2.279001978  -0.989112432  -1.247479908  -2.134455629
[141]  -0.481001908  -0.119153264  -0.052353791  -0.076109421  -1.256083526
[146]   6.041441390   0.931744885  -2.479158585  -1.678819820 -13.154401401
[151]  -0.095079511  -1.471535903  -1.424340646   0.773903261  -2.182543375
[156]  -1.231492928  -1.566716755  -0.598846594  -1.196330457  -1.075831921
[161]  -0.297084618   3.781351587   1.446570265  -0.536913619   0.041309662
[166]  -0.596847338  -0.260287130   0.311262797  -3.323342777  -0.022006717
[171]   0.687918388  -1.008462475   0.015740870   0.796921912   0.916335300
[176]  -1.129313015   0.870286625  -2.806322200  -3.605989758  -0.065971851
[181]  -3.449701222  -1.306629038  -0.525446598  -1.934882094  -0.067508807
[186]  -2.411167777  -0.435513925   1.944204757   0.380278943  -0.398466116
[191]   1.803943615   1.157660771  -1.523659339  -0.121944064  12.098168163
[196]  -1.650231182  11.853755171  -1.175984861  -1.689253064   3.202235956

\$Estim\$vals
[,1]      [,2]      [,3]      [,4]
[1,] 1.350000 0.4000000 1.0000000 0.3000000
[2,] 1.254499 0.5118235 0.9543142 0.9918242
[3,] 1.311907 0.4495955 0.9967998 0.6979764
[4,] 1.334415 0.4676414 1.0085928 0.6678007

\$duration
elapsed
0.87

\$method
[1] "Koutrouvelis_spacing=Kout"
```

StableEstim documentation built on May 30, 2017, 12:25 a.m.