1.1 Define marginal distribution

library(CoSMoS)

marginaldist <- 'burrXII'
para <- list(scale = 2,
             shape1 = .75,
             shape2 = .1)

1.2 Define prob. zero

p0 <- .9

1.3 Define model order and target ACS

order <- 100
targetcorr <- acfparetoII(t = 0:order, scale = 15, shape = .3)

1.4 Calculate ACTI points

p <- actpnts(marginaldist, para, p0 = p0)

1.5 Fit the points

f <- fitactf(p)
plot(f)

1.6 Run the AR(p) model

target <- ARp(margdist = marginaldist,
              margarg = para,
              actfpara = f,
              n = 100000,
              p0 = p0,
              p = order,
              acsvalue = targetcorr)

1.7 Check the gaussian empirical + target transformed and the target process ACF

gauss <- attr(target, 'gaussian')

ggplot() +
  geom_point(aes(x = 0:order,
                 y = as.vector(acf(gauss, lag.max = order, plot = F)$acf)), colour = 'red2', alpha = .2) +
  geom_line(aes(x = 0:order,
                y = attr(target, 'trACS')), colour = 'red4') +
  theme_bw() +
  labs(x = 'Lag',
       y = 'ACF',
       title = 'Gaussian ACS')

ggplot() +
  geom_point(aes(x = 0:order,
                 y = as.vector(acf(target, lag.max = order, plot = F)$acf)), colour = 'steelblue2', alpha = .2) +
  geom_line(aes(x = 0:order,
                y = targetcorr), colour = 'steelblue4') +
  theme_bw() +
  labs(x = 'Lag',
       y = 'ACF',
       title = 'Target ACS')

1.8 Check the simulation

checksimulation(target)

>             mean   sd  skew   p0 acf_t1 acf_t2 acf_t3
> expected    0.18 1.04 12.20 0.90   0.94   0.88   0.82
> simulation1 0.16 0.95 12.14 0.91   0.94   0.88   0.83

1.9 Or check the model in general

chck <- checkmodel(margdist = marginaldist,
                   margarg = para,
                   n = 100000,
                   p = order,
                   p0 = p0,
                   acsvalue = targetcorr,
                   TSn = 15,
                   plot = T)

knitr::kable(chck)

2.1 Burr type III distribution

distribution argument/name - burrIII
parameters - list(scale, shape1, shape2)

\[f_{\text{BrIII}}(x)=\frac{\left(\frac{x}{\beta }\right)^{-\frac{1}{\gamma _2}-1} \left(\frac{\left(\frac{x}{\beta }\right)^{-\frac{1}{\gamma _2}}}{\gamma _1}+1\right){}^{-\gamma _1 \gamma _2-1}}{\beta } \\ F_{\text{BrIII}}(x)=\left(\frac{\left(\frac{x}{\beta }\right)^{-\frac{1}{\gamma _2}}}{\gamma _1}+1\right){}^{-\gamma _1 \gamma _2} \\ Q_{\text{BrIII}}(u)=\beta \left(\gamma _1 \left(u^{-\frac{1}{\gamma _1 \gamma _2}}-1\right)\right){}^{-\gamma _2} \\ m_{\text{BrIII}}(q)=\frac{\beta ^q \gamma _1^{\gamma _2 (-q)} \Gamma \left(\left(q+\gamma _1\right) \gamma _2\right) \Gamma \left(1-q \gamma _2\right)}{\Gamma \left(\gamma _1 \gamma _2\right)}\]

2.2 Burr type XII distribution

distribution argument/name - burrXII
parameters - list(scale, shape1, shape2)

\[f_{\text{BrXII}}(x)=\frac{\left(\frac{x}{\beta }\right)^{\gamma _1-1} \left(\gamma _2 \left(\frac{x}{\beta }\right)^{\gamma _1}+1\right){}^{-\frac{1}{\gamma _1 \gamma _2}-1}}{\beta } \\ F_{\text{BrXII}}(x)=1-\left(\gamma _2 \left(\frac{x}{\beta }\right)^{\gamma _1}+1\right){}^{-\frac{1}{\gamma _1 \gamma _2}} \\ Q_{\text{BrXII}}(u)=\beta \left(-\frac{1-(1-u)^{-\gamma _1 \gamma _2}}{\gamma _2}\right){}^{\frac{1}{\gamma _1}} \\ m_{\text{BrXII}}(q)=\frac{\beta ^q \gamma _2^{-\frac{q}{\gamma _1}-1} B\left(\frac{q+\gamma _1}{\gamma _1},\frac{1-q \gamma _2}{\gamma _1 \gamma _2}\right)}{\gamma _1}\]

2.3 Generalized gamma distribution

distribution argument/name - ggamma
parameters - list(scale, shape1, shape2)

\[f_{\mathcal{G}\mathcal{G}}(x)=\frac{\gamma _2 e^{-\left(\frac{x}{\beta }\right)^{\gamma _2}} \left(\frac{x}{\beta }\right)^{\gamma _1-1}}{\beta \Gamma \left(\frac{\gamma _1}{\gamma _2}\right)} \\ F_{\mathcal{G}\mathcal{G}}(x)=Q\left(\frac{\gamma _1}{\gamma _2},0,\left(\frac{x}{\beta }\right)^{\gamma _2}\right) \\ Q_{\mathcal{G}\mathcal{G}}(u)=\beta Q^{-1}\left(\frac{\gamma _1}{\gamma _2},0,u\right){}^{\frac{1}{\gamma _2}} \\ m_{\mathcal{G}\mathcal{G}}(q)=\frac{\beta ^q \Gamma \left(\frac{q}{\gamma _2}+\frac{\gamma _1}{\gamma _2}\right)}{\Gamma \left(\frac{\gamma _1}{\gamma _2}\right)}\]

2.4 Paretto type II distribution

distribution argument/name - burrIII
parameters - list(scale, shape)

\[f_{\text{PII}}(x)=\frac{\left(\frac{\gamma x}{\beta }+1\right)^{-\frac{1}{\gamma }-1}}{\beta } \\ F_{\text{PII}}(x)=1-\left(\frac{\gamma x}{\beta }+1\right)^{-1/\gamma } \\ Q_{\text{PII}}(u)=\frac{\beta \left((1-u)^{-\gamma }-1\right)}{\gamma } \\ m_{\text{PII}}(q)=\frac{\Gamma (q+1) \left(\frac{\beta }{\gamma }\right)^q \Gamma \left(\frac{1}{\gamma }-q\right)}{\Gamma \left(\frac{1}{\gamma }\right)}\]