ex09. Causal (No X, CRP Backend) - Gamma Kernel

Website workflow note. This page reflects the current exported API and recommended wrapper-first usage. Last updated: 2026-02-19.

For the full package narrative, see the main package vignettes (basic, unconditional, conditional, and causal).

Causal Inference: No Covariates (CRP)

What you’ll learn

• How causal fits are structured when no covariates are available (X = NULL).
• How to interpret and report marginal estimands: ate(), att(), qte(), qtt().
• How tail augmentation (GPD = TRUE) changes effect behavior in the extremes.

Key limitation (important)

Because X = NULL, conditional estimands cate() and cqte() are unavailable. This is a feature: it prevents interpreting conditional effects without covariates.

What this page does

This vignette fits two no-X causal models where the treatment arms are modeled independently using unconditional outcome models:

• Model A: CRP bulk-only (GPD = FALSE)
• Model B: CRP with GPD tail (GPD = TRUE)

---

Data Setup (No X)

data("causal_alt_real500_p4_k2")
y <- abs(causal_alt_real500_p4_k2$y) + 0.01
A <- causal_alt_real500_p4_k2$A

summary_tbl <- tibble(
  statistic = c("N", "Mean", "SD", "Min", "Max"),
  value = c(length(y), mean(y), sd(y), min(y), max(y))
)

summary_tbl %>%
  mutate(value = signif(value, 4)) %>%
  print()

# A tibble: 5 × 2
  statistic    value
  <chr>        <dbl>
1 N         500     
2 Mean        1.43  
3 SD          1.08  
4 Min         0.0126
5 Max         8.10  

u_threshold <- as.numeric(stats::quantile(y, 0.8, names = FALSE))
y_eval <- y[seq_len(if (isTRUE(FAST)) 20L else 40L)]

---

Model A: CRP Bulk-only (Gamma)

bundle_crp_bulk <- bundle(
  y = y,
  treat = A,
  X = NULL,
  kernel = "gamma",
  backend = "crp",
  PS = FALSE,
  GPD = FALSE,
  components = components_ex09,
  mcmc_outcome = bulk_mcmc_ex09
)

bundle_crp_bulk

CausalMixGPD causal bundle
PS model: disabled 
      Field                    Treated                    Control
    Backend Chinese Restaurant Process Chinese Restaurant Process
     Kernel                      gamma                      gamma
 Components                          2                          2
   GPD tail                      FALSE                      FALSE
    Epsilon                      0.025                      0.025

Outcome PS included: FALSE 
n (control) = 232 | n (treated) = 268 

fit_crp_bulk <- load_or_fit(
  tag_crp_bulk,
  quiet_mcmc(dpmix.causal(bundle_crp_bulk, mcmc = causal_run_mcmc))
)
summary(fit_crp_bulk)

-- Outcome fits --
[control]
MixGPD summary | backend: Chinese Restaurant Process | kernel: Gamma Distribution | GPD tail: FALSE | epsilon: 0.025
n = 232 | components = 2
Summary
Initial components: 2 | Components after truncation: 2

WAIC: 557.032
lppd: -234.382 | pWAIC: 44.135

Summary table
  parameter  mean    sd q0.025 q0.500 q0.975
 weights[1] 0.603 0.055  0.513  0.612  0.705
 weights[2] 0.397 0.055  0.295  0.388  0.487
      alpha 0.279 0.171  0.018   0.25  0.589
   shape[1] 2.936 0.803  1.142  3.033  4.146
   shape[2] 1.013 0.429  0.632  0.919  2.369
   scale[1] 1.835 0.414  1.025  1.929  2.453
   scale[2] 1.099 0.269  0.641   1.12  1.658

[treated]
MixGPD summary | backend: Chinese Restaurant Process | kernel: Gamma Distribution | GPD tail: FALSE | epsilon: 0.025
n = 268 | components = 2
Summary
Initial components: 2 | Components after truncation: 1

WAIC: 715.194
lppd: -344.829 | pWAIC: 12.768

Summary table
  parameter  mean    sd q0.025 q0.500 q0.975
 weights[1] 0.971 0.052  0.844      1      1
      alpha 0.242 0.169  0.013   0.23  0.577
   shape[1]  1.83 0.048  1.776  1.815   1.95
   scale[1] 1.182 0.055  1.083  1.185  1.273

pred_mean_bulk <- predict(fit_crp_bulk, type = "mean", interval = "credible", nsim_mean = nsim_mean_ex09, workers = predict_workers_ex09)
head(pred_mean_bulk)

  id  estimate      lower    upper ps
1  1 -1.766581 -0.1960662 -3.51657 NA

plot(pred_mean_bulk)

pred_q_bulk <- predict(fit_crp_bulk, type = "quantile", p = 0.5, interval = "credible", workers = predict_workers_ex09)
head(pred_q_bulk)

  id index  estimate     lower     upper ps
1  1   0.5 -1.353012 -3.373245 0.1303204 NA

plot(pred_q_bulk)

pred_d_bulk <- predict(fit_crp_bulk, y = y_eval, type = "density", interval = "credible", workers = predict_workers_ex09)
head(pred_d_bulk)

  id         y ps trt_estimate trt_lower trt_upper con_estimate  con_lower
1  1 0.9001906 NA            1 0.2911910 0.3757297            1 0.08295440
2  2 1.3517565 NA            1 0.2968176 0.3399002            1 0.04857509
3  3 1.1475287 NA            1 0.2994682 0.3611678            1 0.05932362
4  4 1.9323578 NA            1 0.2594861 0.2667790            1 0.03948270
5  5 3.3439817 NA            1 0.1094912 0.1426296            1 0.04830250
6  6 0.9493979 NA            1 0.2940181 0.3742134            1 0.07701585
  con_upper
1 0.3126950
2 0.2660926
3 0.2873544
4 0.2088758
5 0.1386561
6 0.3077044

plot(pred_d_bulk)

pred_surv_bulk <- predict(fit_crp_bulk, y = y_eval, type = "survival", interval = "credible", workers = predict_workers_ex09)
head(pred_surv_bulk)

  id         y ps trt_estimate trt_lower trt_upper con_estimate con_lower
1  1 0.9001906 NA            1 0.7375929 0.8241993            1 0.6614518
2  2 1.3517565 NA            1 0.5744480 0.6895195            1 0.5526746
3  3 1.1475287 NA            1 0.6461257 0.7506967            1 0.6054908
4  4 1.9323578 NA            1 0.3990952 0.5258520            1 0.4260355
5  5 3.3439817 NA            1 0.1471727 0.2432305            1 0.2055285
6  6 0.9493979 NA            1 0.7191382 0.8097823            1 0.6493800
  con_upper
1 0.8119139
2 0.7577686
3 0.7771960
4 0.7109987
5 0.6224437
6 0.8045888

plot(pred_surv_bulk)

ate_bulk <- ate(fit_crp_bulk, interval = "credible", nsim_mean = nsim_mean_ex09)
head(ate_bulk)

$fit
[1] -1.766581

$fit_df
   estimate     lower        upper
1 -1.766581 -3.797965 -0.009391505

$lower
[1] -3.797965

$upper
[1] -0.009391505

$ate
$ate$fit
   estimate     lower        upper
1 -1.766581 -3.797965 -0.009391505

$ate$draws
              [,1]
 [1,] -1.215383873
 [2,] -0.792846979
 [3,] -0.968611779
 [4,] -0.992906205
 [5,] -0.528059171
 [6,] -1.025130201
 [7,] -0.694915620
 [8,] -0.632898705
 [9,] -0.328284726
[10,]  0.199826473
[11,]  0.008289534
[12,] -0.028933705
[13,] -0.240875710
[14,] -0.310493015
[15,] -0.383423142
[16,] -0.825870699
[17,] -0.653630001
[18,] -0.615501293
[19,] -0.561349643
[20,] -0.560815115
[21,] -0.766125698
[22,] -0.774846882
[23,] -0.591425269
[24,] -0.652303210
[25,] -0.943773980
[26,] -0.887665047
[27,] -0.573967241
[28,] -0.355066845
[29,] -0.445651982
[30,] -1.302114729
[31,] -2.140486667
[32,] -1.922263967
[33,] -3.305118115
[34,] -3.212736427
[35,] -3.817283492
[36,] -3.895973409
[37,] -2.939332658
[38,] -2.315554690
[39,] -2.286879545
[40,] -2.472765785
[41,] -2.199136380
[42,] -2.323326499
[43,] -2.439926831
[44,] -2.542809056
[45,] -3.097899690
[46,] -2.534866893
[47,] -2.688779960
[48,] -2.802058535
[49,] -3.119008098
[50,] -2.988293817
[51,] -3.532526089
[52,] -2.967840153
[53,] -2.859174005
[54,] -2.860106704
[55,] -2.993087731
[56,] -3.553712871
[57,] -3.362614044
[58,] -3.042202332
[59,] -3.557749237
[60,] -3.776613448


$grid
NULL

plot(ate_bulk)

qte_bulk <- qte(fit_crp_bulk, probs = c(0.25, 0.5, 0.75), interval = "credible")
head(qte_bulk)

$fit
  index   estimate      lower      upper
1  0.25  0.1237246 -0.4915869  0.7020959
2  0.50 -1.3286725 -3.8227786  0.3664909
3  0.75 -3.1335342 -6.6520262 -0.1865833

$fit_df
  index   estimate      lower      upper
1  0.25  0.1237246 -0.4915869  0.7020959
2  0.50 -1.3286725 -3.8227786  0.3664909
3  0.75 -3.1335342 -6.6520262 -0.1865833

$lower
[1] -0.4915869 -3.8227786 -6.6520262

$upper
[1]  0.7020959  0.3664909 -0.1865833

$qte
$qte$fit
  index   estimate      lower      upper
1  0.25  0.1237246 -0.4915869  0.7020959
2  0.50 -1.3286725 -3.8227786  0.3664909
3  0.75 -3.1335342 -6.6520262 -0.1865833

$qte$draws
, , 1

              [,1]
 [1,] -0.554440342
 [2,] -0.283676647
 [3,] -0.422117218
 [4,] -0.403513209
 [5,]  0.154347918
 [6,] -0.227688809
 [7,]  0.014147107
 [8,]  0.116037621
 [9,]  0.441863606
[10,]  0.425650919
[11,]  0.245309463
[12,]  0.190484236
[13,]  0.309299009
[14,]  0.106960562
[15,]  0.162885499
[16,] -0.010890040
[17,]  0.055210120
[18,]  0.394793550
[19,]  0.238412139
[20,]  0.071395424
[21,]  0.151551372
[22,] -0.059210257
[23,]  0.142218640
[24,]  0.030883629
[25,] -0.237594778
[26,] -0.058722741
[27,]  0.105488934
[28,]  0.281047030
[29,]  0.269776848
[30,]  0.503954431
[31,] -0.009014745
[32,] -0.144123948
[33,]  0.745139399
[34,]  0.631451898
[35,] -0.606326243
[36,] -0.020408155
[37,]  0.557183681
[38,]  0.689316332
[39,]  0.713658355
[40,]  0.442170507
[41,]  0.161470745
[42,] -0.111325699
[43,]  0.098583645
[44,]  0.547805054
[45,]  0.053788704
[46,]  0.447996993
[47,]  0.338753132
[48,]  0.365327916
[49,] -0.329336640
[50,] -0.331787898
[51,]  0.068591246
[52,] -0.009719949
[53,]  0.344083744
[54,]  0.460920569
[55,]  0.463370533
[56,] -0.233573426
[57,]  0.168802101
[58,]  0.110979087
[59,] -0.154603561
[60,] -0.189560409

, , 2

             [,1]
 [1,] -1.15602099
 [2,] -0.41019833
 [3,] -1.00236206
 [4,] -0.90069376
 [5,] -0.45965163
 [6,] -0.60648672
 [7,] -0.39311821
 [8,] -0.56123904
 [9,]  0.35920752
[10,]  0.41705720
[11,] -0.04080695
[12,]  0.04987707
[13,]  0.37308060
[14,] -0.09772410
[15,] -0.37490660
[16,] -0.48441689
[17,] -0.22872893
[18,] -0.16210743
[19,] -0.24539357
[20,] -0.33672423
[21,] -0.05552412
[22,] -0.90897300
[23,] -0.23567718
[24,] -0.50367251
[25,] -0.69586382
[26,] -0.40405655
[27,] -0.50987180
[28,] -0.19301837
[29,] -0.02227993
[30,] -0.10756470
[31,] -1.81393595
[32,] -1.56540654
[33,] -2.82632628
[34,] -3.59323379
[35,] -3.95108657
[36,] -3.60666087
[37,] -2.30070934
[38,] -1.79835821
[39,] -1.16480202
[40,] -2.51315414
[41,] -2.04712669
[42,] -2.07881464
[43,] -2.49061956
[44,] -1.91132803
[45,] -3.73903708
[46,] -2.78316720
[47,] -2.32799846
[48,] -1.95755462
[49,] -3.02134536
[50,] -2.02024135
[51,] -1.89317829
[52,] -1.80091070
[53,] -1.78060730
[54,] -0.70802810
[55,] -1.03355192
[56,] -2.98815995
[57,] -2.27462667
[58,] -1.93403772
[59,] -1.99993916
[60,] -3.89854473

, , 3

             [,1]
 [1,] -2.11349261
 [2,] -1.04490277
 [3,] -1.37413523
 [4,] -1.43372491
 [5,] -0.84041626
 [6,] -1.79037015
 [7,] -1.21694441
 [8,] -1.48715741
 [9,] -0.51140852
[10,] -0.14250159
[11,] -0.56501757
[12,]  0.06819229
[13,] -0.50004158
[14,] -0.23530515
[15,] -1.13192376
[16,] -1.30208212
[17,] -1.04714650
[18,] -1.28850336
[19,] -0.71735077
[20,] -1.14603004
[21,] -0.99715893
[22,] -1.56019207
[23,] -1.27297967
[24,] -0.96151552
[25,] -1.52375089
[26,] -1.38495650
[27,] -1.10740700
[28,] -1.43385790
[29,] -0.74325187
[30,] -2.03955493
[31,] -4.17212198
[32,] -4.28522618
[33,] -6.32852063
[34,] -5.83118155
[35,] -6.61931320
[36,] -6.18878378
[37,] -5.61821155
[38,] -4.81992001
[39,] -3.83957352
[40,] -4.37818937
[41,] -4.05853495
[42,] -3.98372112
[43,] -4.66046975
[44,] -4.26676156
[45,] -5.99307558
[46,] -5.17776961
[47,] -4.75262880
[48,] -4.14913790
[49,] -4.79450853
[50,] -4.76703628
[51,] -4.17410840
[52,] -4.68832460
[53,] -5.71892922
[54,] -5.05642458
[55,] -5.61008220
[56,] -7.46008386
[57,] -6.66637939
[58,] -4.86752269
[59,] -5.60446250
[60,] -6.63616220



$probs
[1] 0.25 0.50 0.75

plot(qte_bulk)

---

Model B: CRP with GPD Tail (Gamma)

param_specs_gpd <- list(
  gpd = list(
    threshold = list(
      mode = "dist",
      dist = "lognormal",
      args = list(meanlog = log(max(u_threshold, .Machine$double.eps)), sdlog = 0.25)
    )
  )
)

bundle_crp_gpd <- bundle(
  y = y,
  treat = A,
  X = NULL,
  kernel = "gamma",
  backend = "crp",
  PS = FALSE,
  GPD = TRUE,
  components = components_ex09,
  param_specs = param_specs_gpd,
  mcmc_outcome = gpd_mcmc_ex09
)

bundle_crp_gpd

CausalMixGPD causal bundle
PS model: disabled 
      Field                    Treated                    Control
    Backend Chinese Restaurant Process Chinese Restaurant Process
     Kernel                      gamma                      gamma
 Components                          2                          2
   GPD tail                       TRUE                       TRUE
    Epsilon                      0.025                      0.025

Outcome PS included: FALSE 
n (control) = 232 | n (treated) = 268 

fit_crp_gpd <- load_or_fit(
  tag_crp_gpd,
  quiet_mcmc(dpmgpd.causal(bundle_crp_gpd, mcmc = causal_run_mcmc_gpd))
)
summary(fit_crp_gpd)

-- Outcome fits --
[control]
MixGPD summary | backend: Chinese Restaurant Process | kernel: Gamma Distribution | GPD tail: TRUE | epsilon: 0.025
n = 232 | components = 2
Summary
Initial components: 2 | Components after truncation: 2

WAIC: 576.226
lppd: -280.598 | pWAIC: 7.515

Summary table
  parameter   mean    sd q0.025 q0.500 q0.975
 weights[1]  0.748 0.088  0.614  0.761  0.889
 weights[2]  0.252 0.088  0.111  0.239  0.386
      alpha  0.272 0.248  0.036   0.16  0.811
 tail_scale  0.669 0.141  0.454  0.648  0.969
 tail_shape -0.011 0.076 -0.149  0.045  0.045
  threshold  2.222 0.098  2.073    2.2  2.295
   shape[1]  1.085 0.031  1.042  1.101   1.11
   shape[2]  1.043  0.12      1      1  1.353
   scale[1]  1.392 0.117  1.169  1.394  1.566
   scale[2]  1.618 0.544  0.621   1.75  2.325

[treated]
MixGPD summary | backend: Chinese Restaurant Process | kernel: Gamma Distribution | GPD tail: TRUE | epsilon: 0.025
n = 268 | components = 2
Summary
Initial components: 2 | Components after truncation: 2

WAIC: 726.257
lppd: -355.22 | pWAIC: 7.908

Summary table
  parameter  mean    sd q0.025 q0.500 q0.975
 weights[1]  0.79 0.066  0.675  0.795  0.886
 weights[2]  0.21 0.066  0.114  0.205  0.325
      alpha 0.351  0.25  0.065  0.293  0.966
 tail_scale 1.091  0.16  0.878  1.069  1.415
 tail_shape 0.067 0.056   0.01  0.021  0.126
  threshold 1.782 0.541  1.416   1.49  3.218
   shape[1] 1.338 0.009  1.317  1.342  1.342
   shape[2] 1.745 0.531      1  1.787  2.566
   scale[1] 1.226 0.119  0.986  1.216  1.411
   scale[2] 1.018   0.5  0.582   0.76  2.198

pred_mean_gpd <- predict(fit_crp_gpd, type = "mean", interval = "credible", nsim_mean = nsim_mean_ex09, workers = predict_workers_ex09)
head(pred_mean_gpd)

  id  estimate     lower     upper ps
1  1 0.1869545 0.2051496 0.2354198 NA

plot(pred_mean_gpd)

pred_q_gpd <- predict(fit_crp_gpd, type = "quantile", p = 0.5, interval = "credible", workers = predict_workers_ex09)
head(pred_q_gpd)

  id index  estimate       lower     upper ps
1  1   0.5 0.1466194 -0.06411202 0.3451832 NA

plot(pred_q_gpd)

pred_d_gpd <- predict(fit_crp_gpd, y = y_eval, type = "density", interval = "credible", workers = predict_workers_ex09)
head(pred_d_gpd)

  id         y ps trt_estimate  trt_lower  trt_upper con_estimate  con_lower
1  1 0.9001906 NA            1 0.36894678 0.43335498            1 0.34662672
2  2 1.3517565 NA            1 0.29581696 0.35984669            1 0.26624070
3  3 1.1475287 NA            1 0.32917748 0.38218819            1 0.30069059
4  4 1.9323578 NA            1 0.20139009 0.30596584            1 0.17709079
5  5 3.3439817 NA            1 0.05850158 0.08458721            1 0.04530071
6  6 0.9493979 NA            1 0.36116550 0.42360808            1 0.33705241
   con_upper
1 0.39009447
2 0.28357319
3 0.32707542
4 0.23595487
5 0.08576576
6 0.37679814

plot(pred_d_gpd)

pred_surv_gpd <- predict(fit_crp_gpd, y = y_eval, type = "survival", interval = "credible", workers = predict_workers_ex09)
head(pred_surv_gpd)

  id         y ps trt_estimate  trt_lower trt_upper con_estimate  con_lower
1  1 0.9001906 NA            1 0.57626683 0.6585675            1 0.51729488
2  2 1.3517565 NA            1 0.41386430 0.5052801            1 0.36778880
3  3 1.1475287 NA            1 0.48138003 0.5707501            1 0.42919823
4  4 1.9323578 NA            1 0.25293776 0.3260639            1 0.23709518
5  5 3.3439817 NA            1 0.05507868 0.1215358            1 0.02287802
6  6 0.9493979 NA            1 0.55620173 0.6403787            1 0.49845493
   con_upper
1 0.60397049
2 0.46388110
3 0.52213072
4 0.33218656
5 0.06925259
6 0.58677461

plot(pred_surv_gpd)

ate_gpd <- ate(fit_crp_gpd, interval = "credible", nsim_mean = nsim_mean_ex09)
head(ate_gpd)

$fit
[1] 0.1869545

$fit_df
   estimate      lower     upper
1 0.1869545 0.00247693 0.3694645

$lower
[1] 0.00247693

$upper
[1] 0.3694645

$ate
$ate$fit
   estimate      lower     upper
1 0.1869545 0.00247693 0.3694645

$ate$draws
              [,1]
 [1,]  0.288861602
 [2,]  0.183679455
 [3,]  0.266421412
 [4,]  0.018388137
 [5,]  0.264019770
 [6,]  0.272065595
 [7,]  0.096270120
 [8,]  0.105668407
 [9,]  0.225863024
[10,]  0.236462690
[11,]  0.261280341
[12,]  0.228366475
[13,]  0.158020588
[14,]  0.210702253
[15,]  0.109864191
[16,]  0.186648903
[17,]  0.172555441
[18,]  0.279785264
[19,]  0.263260302
[20,]  0.370615518
[21,] -0.012857383
[22,]  0.267989462
[23,]  0.138574801
[24,]  0.369027881
[25,]  0.168167906
[26,]  0.050695749
[27,]  0.167827893
[28,]  0.053649032
[29,]  0.008293394
[30,]  0.198467541


$grid
NULL

plot(ate_gpd)

qte_gpd <- qte(fit_crp_gpd, probs = c(0.25, 0.5, 0.75), interval = "credible")
head(qte_gpd)

$fit
  index   estimate      lower     upper
1  0.25 0.29898088 -0.1825482 0.7470433
2  0.50 0.11268025 -0.4875149 0.4597343
3  0.75 0.06120091 -0.3150973 0.5973389

$fit_df
  index   estimate      lower     upper
1  0.25 0.29898088 -0.1825482 0.7470433
2  0.50 0.11268025 -0.4875149 0.4597343
3  0.75 0.06120091 -0.3150973 0.5973389

$lower
[1] -0.1825482 -0.4875149 -0.3150973

$upper
[1] 0.7470433 0.4597343 0.5973389

$qte
$qte$fit
  index   estimate      lower     upper
1  0.25 0.29898088 -0.1825482 0.7470433
2  0.50 0.11268025 -0.4875149 0.4597343
3  0.75 0.06120091 -0.3150973 0.5973389

$qte$draws
, , 1

             [,1]
 [1,] -0.04424420
 [2,] -0.34357095
 [3,]  0.10016723
 [4,] -0.12147058
 [5,]  0.10536952
 [6,]  0.27784154
 [7,]  0.31335264
 [8,]  0.48410805
 [9,]  0.72781246
[10,]  0.30157102
[11,]  0.27367318
[12,]  0.55805014
[13,]  0.28857033
[14,]  0.44174321
[15,]  0.38244942
[16,]  0.28669254
[17,]  0.09016803
[18,]  0.18870477
[19,]  0.24364153
[20,]  0.25687455
[21,]  0.07750228
[22,]  0.42134872
[23,]  0.17148001
[24,]  0.67435278
[25,]  0.56093327
[26,]  0.23318932
[27,]  0.79774278
[28,]  0.38005589
[29,]  0.42026171
[30,]  0.42105534

, , 2

             [,1]
 [1,]  0.06008338
 [2,] -0.65131666
 [3,]  0.01406248
 [4,] -0.42538320
 [5,]  0.04114661
 [6,]  0.23819769
 [7,] -0.03388199
 [8,] -0.11240625
 [9,]  0.38273775
[10,]  0.26996707
[11,]  0.31249804
[12,]  0.27915585
[13,] -0.02180782
[14,]  0.36514756
[15,]  0.52371499
[16,]  0.30571620
[17,]  0.32070360
[18,]  0.43546577
[19,]  0.22474131
[20,]  0.17806529
[21,] -0.23760038
[22,]  0.08746259
[23,] -0.42097239
[24,]  0.40774002
[25,]  0.26177540
[26,] -0.03460771
[27,]  0.14558284
[28,]  0.23214276
[29,]  0.24932697
[30,] -0.01705033

, , 3

              [,1]
 [1,]  0.872624953
 [2,]  0.305966261
 [3,]  0.492920088
 [4,]  0.118619105
 [5,]  0.009408430
 [6,] -0.070962552
 [7,] -0.274250766
 [8,] -0.067824182
 [9,]  0.438526926
[10,] -0.252866917
[11,]  0.001475438
[12,]  0.228013384
[13,]  0.035196105
[14,] -0.014533672
[15,]  0.174094439
[16,]  0.094837314
[17,]  0.144629835
[18,]  0.344853387
[19,]  0.091699368
[20,]  0.071899548
[21,] -0.326197085
[22,]  0.154996168
[23,] -0.096351210
[24,]  0.458705431
[25,]  0.078314644
[26,] -0.309049541
[27,] -0.033185670
[28,] -0.253095319
[29,] -0.310887047
[30,] -0.271549619



$probs
[1] 0.25 0.50 0.75

plot(qte_gpd)

---

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