sgt_dashboard.Rmd

---
title: "Skewed Generalised t - Distribution"
output: 
  flexdashboard::flex_dashboard:
    orientation: rows
runtime: shiny
---

```{r setup, include=FALSE}
library(tidyverse)
library(plotly)
library(shiny)
library(flexdashboard)

sgt_skewness = function(l, p, q){
  v = q^(-1/p) *
    (
      (3*l^2+1) *
        (beta(3/p,q-2/p)/beta(1/p,q)) -
        4*l^2 *
        (beta(2/p,q-1/p)/beta(1/p,q))^2
    )^(-1/2)
  
  skewness = 2 * q^(3/p) * l * v^3 / beta(1/p, q)^3 *
    (8*l^2*beta(2/p,q-1/p)^3 - 
       3*(1+3*l^2) * beta(1/p,q) * beta(2/p, q-1/p) * beta(3/p,q-2/p) + 
       2 * (1+l^2) * beta(1/p,q)^2 * beta(4/p, q-3/p)
       )
  
  return(skewness)
}

sgt_kurtosis = function(l, p, q){
  v = q^(-1/p) *
    (
      (3*l^2+1) *
        (beta(3/p,q-2/p)/beta(1/p,q)) -
        4*l^2 *
        (beta(2/p,q-1/p)/beta(1/p,q))^2
    )^(-1/2)
  
  kurtosis = q^(4/p) * v^4 / beta(1/p, q)^4 *
    (48*l^4*beta(2/p,q-1/p)^4 +
       24 * l^2 * (1+3*l^2) * beta(1/p,q) * beta(2/p, q-1/p)^2 * beta(3/p,q-2/p) -
       32 * l^2 * (1+l^2) * beta(1/p,q)^2 * beta(2/p, q-1/p) * beta(4/p,q-3/p) + 
       (1+10*l^2+5*l^4) * beta(1/p,q)^3 * beta(5/p, q-4/p)
    )
  
  return(kurtosis)
}
```



Inputs {.sidebar}
==========================================================

```{r}
sliderInput("lambda", label = h3("Skewness: lambda "), 
min = -0.999, max = 0.999, value = 0.01)
```

```{r}
sliderInput("p", label = h3("Kurtosis 1: p "), 
min = 0.01, max = 16.99, value = 1.5)
```

```{r}
sliderInput("q", label = h3("Kurtosis 2: q "), 
min = 0.01, max = 99.99, value = 4)
```


SGT
==================

Rows {data-height=100}
-----------------------------------------------------------------------

### Skewness

```{r}
renderValueBox({
  valueBox(value = sgt_skewness(input$lambda, input$p, input$q) %>% round(digits = 2))
})
```

### Kurtosis

```{r}
renderValueBox({
  valueBox(value = sgt_kurtosis(input$lambda, input$p, input$q) %>% round(digits = 2))
})
```

### Number of finite moments

```{r}
renderValueBox({
  valueBox(value = (input$p * input$q) %>% round())
})
```

### Value of $\lambda$

```{r}
renderValueBox({
  valueBox(value = input$lambda %>% round(digits = 2))
})
```

### Value of $p$

```{r}
renderValueBox({
  valueBox(value = input$p %>% round(digits = 2))
})
```

### Value of $q$

```{r}
renderValueBox({
  valueBox(value = input$q %>% round(digits = 2))
})
```


Row {data-height=650}
-----------------------------------------------------------------------

### Plot

```{r}
renderPlot({
  dd = tibble(x = seq(-3, 3, length.out = 1000)) %>% 
    mutate(SGT = sgt::dsgt(x, mu = 0, sigma = 1, mean.cent = TRUE, var.adj = TRUE,
                             lambda = input$lambda, p = input$p, q = input$q)) %>% 
    mutate(Gaussian = stats::dnorm(x)) %>% 
    pivot_longer(-x, names_to = "Distribution")
    
  
  dd %>% 
    ggplot(aes(x = x, y = value, color = Distribution)) + 
    geom_line() + 
    ggtitle("Density of SGT Distribution compared to Gaussian")
})  
```


### Density

The SGT family of densities whose mean is equal to $0$ and whose variance is equal to $1$ is parametrised as

$$
f_{SGT}\left(x; \lambda, p, q \right) = \frac{p}{2 \cdot v \cdot q^{\frac{1}{p}} \cdot B \left( \frac{1}{p}, q \right) \cdot \left( \frac{|x + m|^p}{q \cdot v^p \cdot \left( \lambda \cdot sign(x + m) + 1 \right)^p} + 1 \right)^{\frac{1}{p}+q}}
$$

where

$$
m = 
\frac{2 v \lambda q^{\frac{1}{p}} B \left( \frac{2}{p}, q - \frac{1}{p} \right)}{B \left( \frac{1}{p}, q \right)}
$$

where 

$$
v = q^{-\frac{1}{p}}
\left[
\left(
3\lambda^2 + 1
\right)
\left(
\frac{
B \left( \frac{3}{p}, q - \frac{2}{p} \right)
}{
B \left( \frac{1}{p}, q \right)
}
\right)
-
4 \lambda^2
\left(
\frac{
B \left( \frac{2}{p}, q - \frac{1}{p} \right)
}{
B \left( \frac{1}{p}, q \right)
}
\right)
\right]^{-\frac{1}{2}}
$$

and $B \left( \cdot, \cdot \right)$ is the Beta function.

For $p = 2$, we obtain the **skewed t distribution**.

For $q = \infty$, we obtain the **skewed generalised error distribution**.

**References**

* [Panayiotis Theodossiou. Financial data and the skewed generalized t distribution. Management Science, 44(12):1650–1661, 1998](https://www.jstor.org/stable/2634700)
* [SGT package vignette](https://cran.r-project.org/web/packages/sgt/vignettes/sgt.pdf)


Row {data-height=250}
-----------------------------------------------------------------------

### Skewness and Kurtosis

The skewness is equal to 

$$
\frac{2 \cdot q^{\frac{3}{p}} \cdot \lambda \cdot v^3}{B\left(\frac{1}{p}, q\right)^3} \cdot
  \left(
    8 \cdot \lambda^2 \cdot B\left(\frac{2}{p}, q-\frac{1}{p}\right)^3 - 
    3 \cdot \left(1 + 3 \cdot \lambda^2\right) \cdot B\left(\frac{1}{p}, q\right) \cdot B\left(\frac{2}{p}, q-\frac{1}{p}\right) \cdot B\left(\frac{3}{p}, q-\frac{2}{p}\right) + 
    2 \cdot \left(1 + \lambda^2\right) \cdot B\left(\frac{1}{p}, q\right)^2 \cdot B\left(\frac{4}{p}, q-\frac{3}{p}\right)
  \right)
$$

and the kurtosis is equal to


$$
\frac{q^{\frac{4}{p}} \cdot v^4}{B\left(\frac{1}{p}, q\right)^4} \cdot
  \left( 
    48 \cdot \lambda^4 \cdot B\left(\frac{2}{p}, q-\frac{1}{p} \right)^4 +
    24 \cdot \lambda^2 \cdot \left(1+3 \cdot \lambda^2\right) \cdot B\left(\frac{1}{p}, q\right) \cdot B\left(\frac{2}{p}, q-\frac{1}{p} \right)^2 \cdot B\left(\frac{3}{p}, q-\frac{2}{p} \right) -
    32 \cdot \lambda^2 \cdot \left(1+\lambda^2\right) \cdot B\left(\frac{1}{p}, q\right)^2 \cdot B\left(\frac{2}{p}, q-\frac{1}{p} \right) \cdot B\left(\frac{4}{p},q-\frac{3}{p} \right) + 
    \left(1+10 \cdot \lambda^2+5 \cdot \lambda^4\right) \cdot B\left(\frac{1}{p}, q \right)^3 \cdot B\left(\frac{5}{p}, q-\frac{4}{p}\right)
  \right)
$$