library(dplyr)
X<-matrix(data = c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
1,1,1,216,283,237,203,259,374,342,301,365,
384,404,426,432,409,553,572,506,528,501,
628,677,602,630,652),
nrow = 24, ncol = 2, byrow = FALSE)
colnames(X) <-c("cte","x")
Y<-matrix(data = c(6.1,9.1,7.2,7.5,6.9,11.5,10.3,9.5,9.2,
10.6,12.5,12.9,13.6,12.8,16.5,17.1,15,
16.2,15.8,19,19.4,19.1,18,20.2),
nrow = 24)
print(X)## cte x
## [1,] 1 216
## [2,] 1 283
## [3,] 1 237
## [4,] 1 203
## [5,] 1 259
## [6,] 1 374
## [7,] 1 342
## [8,] 1 301
## [9,] 1 365
## [10,] 1 384
## [11,] 1 404
## [12,] 1 426
## [13,] 1 432
## [14,] 1 409
## [15,] 1 553
## [16,] 1 572
## [17,] 1 506
## [18,] 1 528
## [19,] 1 501
## [20,] 1 628
## [21,] 1 677
## [22,] 1 602
## [23,] 1 630
## [24,] 1 652print(Y)## [,1]
## [1,] 6.1
## [2,] 9.1
## [3,] 7.2
## [4,] 7.5
## [5,] 6.9
## [6,] 11.5
## [7,] 10.3
## [8,] 9.5
## [9,] 9.2
## [10,] 10.6
## [11,] 12.5
## [12,] 12.9
## [13,] 13.6
## [14,] 12.8
## [15,] 16.5
## [16,] 17.1
## [17,] 15.0
## [18,] 16.2
## [19,] 15.8
## [20,] 19.0
## [21,] 19.4
## [22,] 19.1
## [23,] 18.0
## [24,] 20.2txx<-solve(t(X)%*%(X))
A<-txx%*%t(X)
P<-X%*%A
Iden<-diag(x=1,24,24)
M<-Iden-P
Ui<-(M%*%Y)
ui<-matrix(sort(Ui))
print(ui)## [,1]
## [1,] -1.83297302
## [2,] -0.99733747
## [3,] -0.98441350
## [4,] -0.90437184
## [5,] -0.90043126
## [6,] -0.50716765
## [7,] -0.22115126
## [8,] -0.11721068
## [9,] -0.08157512
## [10,] -0.04979501
## [11,] -0.03093888
## [12,] 0.05512008
## [13,] 0.15503494
## [14,] 0.19969645
## [15,] 0.30859470
## [16,] 0.32537412
## [17,] 0.36804405
## [18,] 0.46007774
## [19,] 0.50270510
## [20,] 0.57689973
## [21,] 0.64215354
## [22,] 0.72736569
## [23,] 1.02732313
## [24,] 1.27897644Varianza
n<-24
varJB<-sqrt((1/n)*(sum(ui^2)))
print(varJB)## [1] 0.7105005Encontrar la asimetria \(\mu_3\) y curtosis \(\mu_4\)
#asimetria = As
As<-(1/n)*(sum(ui^3))
#curtosis = Ct
Ct<-(1/n)*(sum(ui^4))Asimetria
print(As)## [1] -0.2229066Curtosis
print(Ct)## [1] 0.7963161Encontrando \(\alpha\) de asimetria y curtosis
#alpha de asimetria = Aas
Aas<-(As/varJB^3)
#alpha de curtosis = Act
Act<-(Ct/varJB^4)Alpha de asimetria
print(Aas)## [1] -0.6214835Alpha de curtosis
print(Act)## [1] 3.124841Calcular estadistico de prueba \(JB\)
JB<-(n/6)*(Aas^2)+(n/24)*(Act-3)^2Estadistico de Jarque Bera
(JB)## [1] 1.560552JB es menor que el V.C 5.9915 Hay evidencia de que le modelo sigue una distribucion normal.
Varianza
i<-1:n
varKS <- sqrt((1/n)*(sum(ui^2)))
print(varKS)## [1] 0.7105005Calculo de \(Z_{i}\): \[Z_{i}=\frac{u_{i}}{\sigma}\] Valor \(Z_{i}\)
z <- (ui-(0.000000020000000766629))
zi<- z/varKS
print(zi)## [,1]
## [1,] -2.57983343
## [2,] -1.40371109
## [3,] -1.38552114
## [4,] -1.27286584
## [5,] -1.26731963
## [6,] -0.71381743
## [7,] -0.31126125
## [8,] -0.16496919
## [9,] -0.11481362
## [10,] -0.07008444
## [11,] -0.04354521
## [12,] 0.07757919
## [13,] 0.21820521
## [14,] 0.28106443
## [15,] 0.43433419
## [16,] 0.45795053
## [17,] 0.51800669
## [18,] 0.64754028
## [19,] 0.70753652
## [20,] 0.81196238
## [21,] 0.90380440
## [22,] 1.02373698
## [23,] 1.44591460
## [24,] 1.80010620Valor \(P_{i}\)
pi <- pnorm((zi))
print(pi)## [,1]
## [1,] 0.004942399
## [2,] 0.080202449
## [3,] 0.082946587
## [4,] 0.101532827
## [5,] 0.102520513
## [6,] 0.237670040
## [7,] 0.377801012
## [8,] 0.434484117
## [9,] 0.454296425
## [10,] 0.472063225
## [11,] 0.482633462
## [12,] 0.530918602
## [13,] 0.586365387
## [14,] 0.610669509
## [15,] 0.667977110
## [16,] 0.676506007
## [17,] 0.697773197
## [18,] 0.741358833
## [19,] 0.760383438
## [20,] 0.791593390
## [21,] 0.816950437
## [22,] 0.847020237
## [23,] 0.925899422
## [24,] 0.964078064Valor \(D^{+}\)
Dmas <- abs((i/n)-pi)
print(Dmas)## [,1]
## [1,] 0.036724268
## [2,] 0.003130884
## [3,] 0.042053413
## [4,] 0.065133840
## [5,] 0.105812820
## [6,] 0.012329960
## [7,] 0.086134346
## [8,] 0.101150783
## [9,] 0.079296425
## [10,] 0.055396558
## [11,] 0.024300129
## [12,] 0.030918602
## [13,] 0.044698720
## [14,] 0.027336176
## [15,] 0.042977110
## [16,] 0.009839341
## [17,] 0.010560136
## [18,] 0.008641167
## [19,] 0.031283229
## [20,] 0.041739944
## [21,] 0.058049563
## [22,] 0.069646429
## [23,] 0.032433911
## [24,] 0.035921936Valor \(D^{-}\)
Dmenos <- abs(pi - ((i-1)/n))
print(Dmenos)## [,1]
## [1,] 4.942399e-03
## [2,] 3.853578e-02
## [3,] 3.867465e-04
## [4,] 2.346717e-02
## [5,] 6.414615e-02
## [6,] 2.933671e-02
## [7,] 1.278010e-01
## [8,] 1.428175e-01
## [9,] 1.209631e-01
## [10,] 9.706323e-02
## [11,] 6.596680e-02
## [12,] 7.258527e-02
## [13,] 8.636539e-02
## [14,] 6.900284e-02
## [15,] 8.464378e-02
## [16,] 5.150601e-02
## [17,] 3.110653e-02
## [18,] 3.302550e-02
## [19,] 1.038344e-02
## [20,] 7.327683e-05
## [21,] 1.638290e-02
## [22,] 2.797976e-02
## [23,] 9.232755e-03
## [24,] 5.744731e-03Cuadro completo
Tab<-cbind(i,i/n,zi,pi,Dmas,Dmenos)
colnames(Tab)<-c("i","i/n","zi","P(i)","Dmas","Dmenos")
round(Tab,3)## i i/n zi P(i) Dmas Dmenos
## [1,] 1 0.042 -2.580 0.005 0.037 0.005
## [2,] 2 0.083 -1.404 0.080 0.003 0.039
## [3,] 3 0.125 -1.386 0.083 0.042 0.000
## [4,] 4 0.167 -1.273 0.102 0.065 0.023
## [5,] 5 0.208 -1.267 0.103 0.106 0.064
## [6,] 6 0.250 -0.714 0.238 0.012 0.029
## [7,] 7 0.292 -0.311 0.378 0.086 0.128
## [8,] 8 0.333 -0.165 0.434 0.101 0.143
## [9,] 9 0.375 -0.115 0.454 0.079 0.121
## [10,] 10 0.417 -0.070 0.472 0.055 0.097
## [11,] 11 0.458 -0.044 0.483 0.024 0.066
## [12,] 12 0.500 0.078 0.531 0.031 0.073
## [13,] 13 0.542 0.218 0.586 0.045 0.086
## [14,] 14 0.583 0.281 0.611 0.027 0.069
## [15,] 15 0.625 0.434 0.668 0.043 0.085
## [16,] 16 0.667 0.458 0.677 0.010 0.052
## [17,] 17 0.708 0.518 0.698 0.011 0.031
## [18,] 18 0.750 0.648 0.741 0.009 0.033
## [19,] 19 0.792 0.708 0.760 0.031 0.010
## [20,] 20 0.833 0.812 0.792 0.042 0.000
## [21,] 21 0.875 0.904 0.817 0.058 0.016
## [22,] 22 0.917 1.024 0.847 0.070 0.028
## [23,] 23 0.958 1.446 0.926 0.032 0.009
## [24,] 24 1.000 1.800 0.964 0.036 0.006Valor D \[D=max(D^{+},D^{-})\]
D1 <- max(Dmas)
D2 <- max(Dmenos)
maxD<-max(Dmas, Dmenos)
print(maxD)## [1] 0.1428175V.C en tabla es igual a 0.1788
**Identificar p(i) y mi
p_i <- matrix((i-0.375)/(24+0.25))
m_i <- qnorm(mean=0, sd=1, lower.tail = FALSE, p_i)*-1
matriz_m <- matrix(m_i)
m <- (sum(m_i^2))
xSW <- matriz_m[n,1]
print(xSW)## [1] 1.946903print(p_i)## [,1]
## [1,] 0.02577320
## [2,] 0.06701031
## [3,] 0.10824742
## [4,] 0.14948454
## [5,] 0.19072165
## [6,] 0.23195876
## [7,] 0.27319588
## [8,] 0.31443299
## [9,] 0.35567010
## [10,] 0.39690722
## [11,] 0.43814433
## [12,] 0.47938144
## [13,] 0.52061856
## [14,] 0.56185567
## [15,] 0.60309278
## [16,] 0.64432990
## [17,] 0.68556701
## [18,] 0.72680412
## [19,] 0.76804124
## [20,] 0.80927835
## [21,] 0.85051546
## [22,] 0.89175258
## [23,] 0.93298969
## [24,] 0.97422680print(matriz_m)## [,1]
## [1,] -1.94690278
## [2,] -1.49843365
## [3,] -1.23590240
## [4,] -1.03864671
## [5,] -0.87524006
## [6,] -0.73241136
## [7,] -0.60317579
## [8,] -0.48332361
## [9,] -0.37005675
## [10,] -0.26136061
## [11,] -0.15567569
## [12,] -0.05170609
## [13,] 0.05170609
## [14,] 0.15567569
## [15,] 0.26136061
## [16,] 0.37005675
## [17,] 0.48332361
## [18,] 0.60317579
## [19,] 0.73241136
## [20,] 0.87524006
## [21,] 1.03864671
## [22,] 1.23590240
## [23,] 1.49843365
## [24,] 1.94690278calculo de ai paso a paso
ted <- 1/sqrt(24)
print(ted)## [1] 0.2041241an <- (((ted)^5)*-2.706056)+(4.434685*(ted)^4)-(2.071190*(ted)^3)-(0.147981*(ted)^2)+(0.2211570*(ted))+(matriz_m[24,1]/sqrt(m))
an_1 <- (((ted)^5)*-3.582633)+(5.682633*(ted)^4)-(1.752461*(ted)^3)-(0.293762*(ted)^2)+(0.042981*(ted))+(matriz_m[(23),1]/sqrt(m))
w <- (m-(2*(matriz_m[24,1])^2)-(2*(matriz_m[23,1])^2))/(1-(2*(an)^2)-2*(an_1)^2)
print(w)## [1] 23.48625ai <- matriz_m/sqrt(w)
a_i<-matrix(data = c(-an, -an_1, ai[3,1],ai[4,1],ai[5,1],ai[6,1],ai[7,1],ai[8,1],ai[9,1],
ai[10,1],ai[11,1],ai[12,1],ai[13,1],ai[14,1],ai[15,1],ai[16,1],
ai[17,1],ai[18,1],ai[19,1],ai[20,1],ai[21,1],ai[22,1],an_1,an),
nrow = 24, ncol = 1, byrow = FALSE)
print(a_i)## [,1]
## [1,] -0.44751326
## [2,] -0.31302439
## [3,] -0.25502179
## [4,] -0.21431914
## [5,] -0.18060106
## [6,] -0.15112913
## [7,] -0.12446207
## [8,] -0.09973122
## [9,] -0.07635921
## [10,] -0.05393035
## [11,] -0.03212284
## [12,] -0.01066927
## [13,] 0.01066927
## [14,] 0.03212284
## [15,] 0.05393035
## [16,] 0.07635921
## [17,] 0.09973122
## [18,] 0.12446207
## [19,] 0.15112913
## [20,] 0.18060106
## [21,] 0.21431914
## [22,] 0.25502179
## [23,] 0.31302439
## [24,] 0.44751326Encontrando El producto de los residuos y W
au_i<- a_i*ui
print(au_i)## [,1]
## [1,] 0.8202797359
## [2,] 0.3121909499
## [3,] 0.2510468930
## [4,] 0.1938242002
## [5,] 0.1626188434
## [6,] 0.0766478070
## [7,] 0.0275249440
## [8,] 0.0116895637
## [9,] 0.0062290121
## [10,] 0.0026854625
## [11,] 0.0009938446
## [12,] -0.0005880911
## [13,] 0.0016541100
## [14,] 0.0064148168
## [15,] 0.0166426203
## [16,] 0.0248453117
## [17,] 0.0367054816
## [18,] 0.0572622273
## [19,] 0.0759733859
## [20,] 0.1041887046
## [21,] 0.1376257968
## [22,] 0.1854941022
## [23,] 0.3215771919
## [24,] 0.5723589192uu_i<- (ui)^2
print(uu_i)## [,1]
## [1,] 3.3597901075
## [2,] 0.9946820249
## [3,] 0.9690699314
## [4,] 0.8178884330
## [5,] 0.8107764500
## [6,] 0.2572190250
## [7,] 0.0489078817
## [8,] 0.0137383430
## [9,] 0.0066545004
## [10,] 0.0024795434
## [11,] 0.0009572141
## [12,] 0.0030382229
## [13,] 0.0240358324
## [14,] 0.0398786721
## [15,] 0.0952306868
## [16,] 0.1058683157
## [17,] 0.1354564223
## [18,] 0.2116715247
## [19,] 0.2527124189
## [20,] 0.3328132949
## [21,] 0.4123611645
## [22,] 0.5290608535
## [23,] 1.0553928037
## [24,] 1.6357807424W <- (sum(au_i)^2)/sum(uu_i)
print(W)## [1] 0.9574588Construir matriz
tabla<-cbind(i,p_i,matriz_m,a_i,ui,au_i,uu_i)
colnames(tabla)<-c("i", "p(i)", "mi", "ai", "ui", "ai*ui", "ui^2")
round(tabla,6)## i p(i) mi ai ui ai*ui ui^2
## [1,] 1 0.025773 -1.946903 -0.447513 -1.832973 0.820280 3.359790
## [2,] 2 0.067010 -1.498434 -0.313024 -0.997337 0.312191 0.994682
## [3,] 3 0.108247 -1.235902 -0.255022 -0.984413 0.251047 0.969070
## [4,] 4 0.149485 -1.038647 -0.214319 -0.904372 0.193824 0.817888
## [5,] 5 0.190722 -0.875240 -0.180601 -0.900431 0.162619 0.810776
## [6,] 6 0.231959 -0.732411 -0.151129 -0.507168 0.076648 0.257219
## [7,] 7 0.273196 -0.603176 -0.124462 -0.221151 0.027525 0.048908
## [8,] 8 0.314433 -0.483324 -0.099731 -0.117211 0.011690 0.013738
## [9,] 9 0.355670 -0.370057 -0.076359 -0.081575 0.006229 0.006655
## [10,] 10 0.396907 -0.261361 -0.053930 -0.049795 0.002685 0.002480
## [11,] 11 0.438144 -0.155676 -0.032123 -0.030939 0.000994 0.000957
## [12,] 12 0.479381 -0.051706 -0.010669 0.055120 -0.000588 0.003038
## [13,] 13 0.520619 0.051706 0.010669 0.155035 0.001654 0.024036
## [14,] 14 0.561856 0.155676 0.032123 0.199696 0.006415 0.039879
## [15,] 15 0.603093 0.261361 0.053930 0.308595 0.016643 0.095231
## [16,] 16 0.644330 0.370057 0.076359 0.325374 0.024845 0.105868
## [17,] 17 0.685567 0.483324 0.099731 0.368044 0.036705 0.135456
## [18,] 18 0.726804 0.603176 0.124462 0.460078 0.057262 0.211672
## [19,] 19 0.768041 0.732411 0.151129 0.502705 0.075973 0.252712
## [20,] 20 0.809278 0.875240 0.180601 0.576900 0.104189 0.332813
## [21,] 21 0.850515 1.038647 0.214319 0.642154 0.137626 0.412361
## [22,] 22 0.891753 1.235902 0.255022 0.727366 0.185494 0.529061
## [23,] 23 0.932990 1.498434 0.313024 1.027323 0.321577 1.055393
## [24,] 24 0.974227 1.946903 0.447513 1.278976 0.572359 1.635781Calcular \(W_n\) usando las siguientes formulas:
mu<- 0.0038915*(log(24)^3)-0.083751*(log(24)^2)-0.31082*(log(24))-1.5861
varSW<- 2.718281828^{(0.0030302*log(24)^2)-0.082676*(log(24))-0.4803}
Wn<-(log(1-W)-mu)/varSW
print(Wn)## [1] 0.2805541