---
title: "Introducing revdbayes: Ratio-of-Uniforms Sampling for Bayesian Extreme Value Analysis"
author: "Paul J. Northrop"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{Introducing revdbayes: Ratio-of-Uniforms Sampling for Bayesian Extreme Value Analysis}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: revdbayes.bib
csl: taylor-and-francis-chicago-author-date.csl
---

```{r, include = FALSE}
knitr::opts_chunk$set(comment = "#>", collapse = TRUE)
```

This vignette focuses on random sampling from extreme value posterior 
distributions using the function `rpost`.  From version 1.2.0 onwards sampling can be achieved more quickly using the function `rpost_rcpp`. See the vignette [Faster simulation using revdbayes](revdbayes-b-using-rcpp-vignette.html) for details.

Performing posterior predictive extreme value inference using posterior samples
is covered in a separate vignette
[Posterior Predictive Extreme Value Inference using the revdbayes Package](revdbayes-c-predictive-vignette.html)

### The *revdbayes* package

The *revdbayes* package tackles the same problem as the *evdbayes* package
[@evdbayes], that is, sampling from posterior distributions that occur in 
some relatively simple Bayesian extreme value analyses.  The essential
difference between these two packages is that evdbayes performs sampling 
using Markov Chain Monte Carlo (MCMC) techniques, whereas revdbayes uses
the generalised ratio-of-uniforms method [@WGS1991], implemented using the
*rust* package [@rust].  Otherwise, these two packages have similar 
functionality and their functions have a similar syntax.  For example, a prior
distribution specified using one package may be used in posterior sampling
implemented by the other.  For details of setting prior distributions
see the `set_prior` function.

For details of Bayesian extreme value analyses see the review of 
@Stephenson2016 and the evdbayes user guide (available in the 
evdbayes doc directory).  Typically, MCMC is used to sample from extreme 
value posterior distributions.  This requires tuning parameters of the MCMC 
algorithm to be set and results in a dependent sample from the posterior.  Diagnostics checks, based on running multiple chains from different starting 
values, are used to check convergence and to decide which values should 
be discarded prior to approximate convergence.  An advantage of using a direct 
method of simulation, such as the generalised ratio-of-uniforms method, is that
it produces a random sample from the posterior distribution.  This method 
also involves tuning parameters, but the revdbayes package has been 
designed to set those automatically.

The current version of revdbayes samples from posterior distributions 
based on the Generalised Extreme Value (GEV) model, the Generalised Pareto 
(GP) model, the point process (PP) model and the *r*-largest order 
statistics (OS) model. The evdbayes package offers in addition some 
extreme value regression models. 

We consider how to use the two main functions in 
revdbayes, namely `set_prior` (used to define the prior distribution)
and `rpost` (used to sample from the posterior distribution).  We use the
examples in the evdbayes user guide for illustration. Please see the evdbayes user guide for details of the models underlying these analyses.

### The ratio-of-uniforms method

The ratio-of-uniforms method is an acceptance-rejection type of simulation
algorithm.  A crude measure of the efficiency of the algorithm is the 
probability $p_a$ that a proposed value is accepted.  In the *rust* package 
[@rust] variable transformation is used to increase $p_a$. In the current 
context two transformations are of interest: (i) marginal Box-Cox 
transformation of (positive functions of) extreme value parameters (using 
argument `trans = "BC"` to `rpost` instead of the default 
`trans = "none"`), and/or (ii) rotation of parameter axes about the 
estimated mode of the posterior (the default `rotate = TRUE` 
instead of the `rotate = FALSE`), based on the estimated Hessian of the
log-posterior at this mode.  Transformation (i) seeks to reduce 
skewness in the marginal posterior distributions.  Transformation
(ii) tends to reduce posterior dependence between the parameters.  The
ratio-of-uniforms algorithm is then performed after relocating the
mode of the posterior to the origin, because this tends to increase the
probability of acceptance [@WGS1991].

For details of the ratio-of-uniforms method and the transformations used see 
the vignette of the *rust* package [@rust].  In the *rust* vignette it 
is argued that the largest attainable value of $p_a$ for a typical 
$d$-dimensional problem is the value resulting from the special case of a
$d$-dimensional normal distribution with independent components.  For $d=2$ 
this value is 0.534, for $d=3$ the value is 0.316.  In an object `ru_object`
returned from `rpost` an estimate of this probability is given by
`ru_object$pa`.  The examples presented in this vignette suggest that often 
rotation to reduce dependence is more helpful than marginal transformation
to reduce skewness.  This (`trans = "none"` and `rotate = TRUE`) is
the default setting of `rpost` and seems to work well enough for most 
problems.

The computations in the following sections were performed using a notebook 
containing a 3.0GHz Intel processor and 8GB RAM.

```{r}
library(revdbayes)
# Set the number of posterior samples required.
n <- 10000
set.seed(46)
```

## Generalised Pareto (GP) model

The GP distribution is used to model *threshold excesses*, that is, the amounts
by which *threshold exceedances* exceed some threshold. It has two 
parameters: a scale parameter $\sigma_u$ and shape parameter $\xi$. 
We use the following example to illustrate the use of the transformation
strategies implemented by the arguments `rotate` and `trans`
to `rpost` and how the user can specify their own prior distribution.

### Gulf of Mexico Wave Height Data

The numeric vector `gom` contains 315 storm peak significant wave 
heights from a location in the Gulf of Mexico, from the years 1900 to 
2005.  These data are analysed in @NAJ2016.  We set the threshold at 
the 65\% sample quantile and use `set_prior` to set a prior.

```{r}
thresh <- quantile(gom, probs = 0.65)
fp <- set_prior(prior = "flat", model = "gp", min_xi = -1)
```

We sample first on the $(\sigma_u, \xi)$ scale, with mode relocation only.  

```{r}
t1 <- system.time(
gp1 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
             rotate = FALSE)
)[3]
```

Then we add a rotation of $(\sigma_u, \xi)$ about the estimated posterior mode.  

```{r}
t2 <- system.time(
gp2 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom)
)[3]
```

Now we add marginal Box-Cox transformation. We apply Box-Cox transformation
to the parameters $\phi_1 = \sigma_u$ and $\phi_2 = \xi + \sigma / x_{(m)}$, 
where $x_{(m)}$ is the largest threshold excess.  The parameters $\phi_1$ and 
$\phi_2$ are positive for all combinations of $(\sigma_u, \xi)$ for which the 
GP likelihood is positive.

```{r}
t3 <- system.time(
gp3 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
             rotate = FALSE, trans = "BC")
)[3]
t4 <- system.time(
gp4 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
             trans = "BC")
)[3]
```

We plot the samples obtained with the contours of the corresponding 
densities superimposed.  The plot on the top left is on the original
$(\sigma_u, \xi)$ scale.  The other plots are on the scale used for the
ratio-of-uniforms algorithm, that is, with relocation of the mode to the
origin.  In the following the $\rho_i, i = 1, \ldots, d$, 
$\rho_1$ and $\rho_2$ in this example, are the variables to which the
ratio-of-uniforms algorithm is applied, i.e. after any transformation
(Box-Cox and/or rotation of axes) and relocation of the mode to the 
origin.

```{r, fig.show='hold', out.width="45%"}
plot(gp1, ru_scale = FALSE, cex.main = 0.75, cex.lab = 0.75, 
  main = paste("no transformation \n pa = ", round(gp1$pa, 3), 
               ", time = ", round(t1, 2), "s"))
plot(gp2, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75, 
  main = paste("rotation \n pa = ", round(gp2$pa, 3),
               ", time = ", round(t2, 2), "s"))
plot(gp3, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75, 
  main = paste("Box-Cox \n pa = ", round(gp3$pa, 3),
               ", time = ", round(t3, 2), "s"))
plot(gp4, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75,
  main = paste("Box-Cox and rotation \n pa = ", round(gp4$pa, 3),
               ", time = ", round(t4, 2), "s"))
```

Comparison of the plots on the right to the plots on the left shows that
rotation of the parameter axes about the mode of the posterior has reduced
dependence between the components.  The estimated probabilities of acceptance
for the plots on the right are close to the 0.534 obtained for a 2-dimensional
normal distribution with independent components.  Box-Cox transformation has
increased the estimated value of $p_a$ but not by much.  In fact, the 
extra computation time required to calculate the Box-Cox transformation 
each time that the posterior density is evaluated means that posterior 
sampling is slower than the default setting of `rotate = TRUE` and 
`trans = "none"`.  That is, in this example at least, the increase is $p_a$ 
resulting from the addition of the Box-Cox transformation is not sufficient 
to offset the extra time needed to compute the posterior density.  

We repeat the rotation only and Box-Cox transformation plus rotation analyses 
for a much higher threshold, set at the 95\% sample quantile.  

```{r, echo = TRUE}
thresh <- quantile(gom, probs = 0.95)
fp <- set_prior(prior = "flat", model = "gp", min_xi = -1)
t2 <- system.time(
gp2 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom)
)[3]
t4 <- system.time(
gp4 <- rpost(n = n, model = "gp", prior = fp, thresh = thresh, data = gom,
             trans = "BC")
)[3]
```

It is optimistic to use such a high threshold in this example, because it results in only 16 threshold excesses.  In the rotation only case, a convergence warning is triggered by one of the optimisations used to created the bounding box of the ratio-of-uniforms method.  This appears to be a spurious warning, because the plots of the simulations below seem fine.  When a Box-Cox transformation is used there are no warnings, because the optimisations have greater stability and it is now easier for the convergence of the optimisations to be confirmed. 

Strong asymmetry in the posterior distribution means that the combination of marginal Box-Cox transformation and rotation produces a larger improvement in $p_a$ and means that now this strategy is competitive with using rotation alone in terms of computation time. 

```{r, fig.show='hold', echo = FALSE, out.width="45%"}
plot(gp2, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75, 
  main = paste("rotation \n pa = ", round(gp2$pa, 3),
               ", time = ", round(t2, 2), "s"))
plot(gp4, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75,
  main = paste("Box-Cox and rotation \n pa = ", round(gp4$pa, 3),
               ", time = ", round(t4, 2), "s"))
```

**A User-defined prior**

Suppose that we wish to use a prior for $(\sigma_u, \xi)$ with a density 
that is proportional to $\sigma_u^{-1} (1+\xi)^{\alpha-1} (1-\xi)^{\beta-1}$, 
for $\sigma_u > 0, -1 < \xi < 1$ and for some $\alpha > 0$ and $\beta > 0$.
This is an improper prior in which $\sigma_u$ and $\xi$ are independent 
*a priori*, $\log \sigma_u$ is uniform over the real line and $\xi$ has 
beta($\alpha, \beta$)-type distribution on the interval 
$(-1, 1)$.  We can do this by creating a function that returns the prior
log-density and passing this function to `set_prior`.
The first argument of the prior log-density function must be the parameter 
vector (the GP parameters $(\sigma_u, \xi)$ in this case), followed by any
hyperparameters.

```{r}
u_prior_fn <- function(x, ab) {
  #
  # Calculates the the log of the (improper) prior density for GP parameters 
  # (sigma_u, xi) in which log(sigma_u) is uniform on the real line and xi has 
  # a beta(alpha, beta)-type prior on the interval (-1, 1).
  #
  # Args:
  #   x  : A numeric vector.  GP parameter vector (sigma, xi).
  #   ab : A numeric vector.  Hyperparameter vector (alpha, beta), where
  #        alpha and beta must be positive.
  #
  # Returns : the value of the log-prior at x.
  #
  if (x[1] <= 0 | x[2] <= -1 | x[2] >= 1) {
    return(-Inf)
  }  
  return(-log(x[1]) + (ab[1] - 1) * log(1 + x[2]) +
           (ab[2] - 1) * log(1 - x[2]))
}
up <- set_prior(prior = u_prior_fn, ab = c(2, 2), model = "gp")
gp_u <- rpost(n = n, model = "gp", prior = up, thresh = thresh, data = gom)
```

See also the "Posterior predictive extreme value inference" section of the [Posterior Predictive Extreme Value Inference using the revdbayes Package](revdbayes-c-predictive-vignette.html) for an analysis that adds inferences about the probability of threshold exceedance to the analysis of threshold excesses using a GP distribution.

## Generalised Extreme Value (GEV) model

We consider two examples involving the GEV distribution, which is used
to model block maxima.  The GEV distribution has three parameters: a 
location parameter $\mu$, a scale parameter $\sigma$ and a
shape parameter $\xi$.  In the first example we illustrate the use of
`set_prior` to set a prior distribution for the GEV parameters and 
show that a prior set using `set_prior` (including user-defined
prior) can be used as an argument to the evdbayes function 
`posterior`.  In the second example we show that the reverse is 
also true, that is, a prior set using an evdbayes function 
(`prior.prob`) can be used an an argument to the revdbayes 
function `rpost`.

### Port Pirie Sea Level Data

The numeric vector `portpirie` contains 65 annual maximum sea levels 
(in metres) recorded at Port Pirie, South Australia, from 1923 to 1987.

```{r}
data(portpirie)
mat <- diag(c(10000, 10000, 100))
pn <- set_prior(prior = "norm", model = "gev", mean = c(0,0,0), cov = mat)
p1 <- rpost(n = n, model = "gev", prior = pn, data = portpirie)
p1$pa
```

The proportion of accepted values is close to the best we can hope for in this
3-dimensional example, i.e. 0.316.  The following plots illustrate the effect 
of `rotate = TRUE`.  The first set of plots below show the simulated values of 
the GEV parameters.  There is some association between the parameters 
*a posteriori*, albeit weak association.  The second set of plots show the
values simulated on the scale used in the ratio-of-uniforms algorithm.  The
association has been reduced by rotating the parameter axes.  The increase in 
the proportion of accepted values is only modest in this example (from 
approximately 0.26 to 0.30) but this improvement is more substantial in 
other examples.

```{r, fig.width = 7}
plot(p1, cex.main = 0.75, cex.lab = 0.75, 
  main = "original scale")
plot(p1, ru_scale = TRUE, cex.main = 0.75, cex.lab = 0.75,
     main = "sampling scale")
```

We produce plots of the estimated marginal posterior densities.

```{r, fig.width = 7}
oldpar <- par(mfrow = c(1,3))
plot(density(p1$sim_vals[, 1], adj = 2), main = "", xlab = expression(mu))
plot(density(p1$sim_vals[, 2], adj = 2), main = "", xlab = expression(sigma))
plot(density(p1$sim_vals[, 3], adj = 2), main = "", xlab = expression(xi))
par(oldpar)
```

### Oxford Temperature Data

The numeric vector `oxford` contains 80 annual maximum temperatures, 
in degrees Fahrenheit, from 1901 to 1980 at Oxford, England.  

```{r}
data(oxford)
prox <- set_prior(prior = "prob", model = "gev", quant = c(85, 88, 95), 
                  alpha = c(4, 2.5, 2.25, 0.25))
# revdbayes
ox_post <- rpost(n = 1000, model = "gev", prior = prox, data = oxford)
```

We produce plots of the estimated marginal posterior densities.

```{r, fig.width = 7}
oldpar <- par(mfrow = c(1,3))
plot(density(ox_post$sim_vals[, 1], adj = 2), main = "", xlab = expression(mu))
plot(density(ox_post$sim_vals[, 2], adj = 2), main = "", xlab = expression(sigma))
plot(density(ox_post$sim_vals[, 3], adj = 2), main = "", xlab = expression(xi))
par(oldpar)
```

We do not illustrate the use of marginal Box-Cox
transformation in the GEV case.  If `trans = "BC"` then we use 
$\phi_1 = \mu$, 
$\phi_2 = [\sigma + \xi (x_{(1)} - \mu) ] / (x_{(m)} - x_{(1)})^{1/2}$ and 
$\phi_3 = [\sigma + \xi (x_{(m)} - \mu) ] / (x_{(m)} - x_{(1)})^{1/2}$, 
where $x_{(1)}$ and $x_{(m)}$ are the smallest and largest sample block maxima
respectively.  This transformation has been standardised to have Jacobian 1.
We apply Box-Cox transformation only to the (positive) 
parameters $\phi_2$ and $\phi_3$, because $\phi_1 = \mu$ is not constrained 
to be positive and tends to have a marginal posterior distribution that is
not strongly skewed.

## Point Process (PP) model

Like the GP model the PP model involves exceedances of a threshold `thresh`.
However, the PP model has one more parameter than the GP model, three 
parameters $(\mu, \sigma, \xi)$ say, because it considers the number of 
threshold exceedances (that is, in addition to the sizes of the threshold excesses.
The interpretation of the parameters $(\mu, \sigma, \xi)$ is determined
by the number of periods of observation, or blocks, into which the 
data are grouped.  This is set using the argument `noy` to `rpost`.
Under the PP model the distribution of the maximum over each of these
blocks has approximately a GEV distribution with parameters 
$(\mu, \sigma, \xi)$, provided that the numbers of observations within
each block are large.  Although the user may require a posterior sample 
based on a particular value of `noy` it may be that the sampling itself
is more efficient when a different value of `noy` is used.

We illustrate the use of a prior created by the evdbayes package as 
an argument to the revdbayes function `rpost`.  We 
also show how the choice of parameterisation of the model affects the
strength posterior dependence between those parameters and hence the 
efficiency of simulation from the posterior.
@WTJ2010 note that posterior dependence between the location parameter 
and the other two parameters tends to be reduced if we use a 
parameterisation in which the number of blocks is equal to the number 
of threshold excesses, rather the typical parameterisation where the 
number of blocks is equal to the number of years of data.  This is 
because, in the context of the likelihood alone, $\mu$ is orthogonal 
to ($\sigma, \xi$) under this parameterisation.  Suppose that
$(\mu, \sigma, \xi)$ is the parameter vector on the scale of interest,
that is, when the number of blocks is equal to `noy` and 
$(\mu_e, \sigma_e, \xi)$ is the parameter vector when the number of
blocks is set equal to the number of threshold excesses $n_{exc}$.  Then
$\mu_e = \mu + \sigma (c ^ \xi - 1) / \xi$, and $\sigma_e = \sigma c ^ \xi$
where $c=$`noy`$/n_{exc}$.

The `rotate = TRUE` argument of `rpost` produces a similar effect, 
but it can reduce posterior dependence between all parameters.
Also, rotating the posterior accounts for the influence of 
the prior in the posterior, whereas the argument for setting `noy` 
equal to the number of threshold excesses is based on the likelihood
alone.

### Rainfall Data

We illustrate the use of `rpost` to sample from a posterior distribution 
under the PP using the `rainfall` dataset, which contains 20820 daily 
rainfall totals, in mm, recorded at a rain gauge in England over a 
57 year period. Three of these years contain only missing values so if
we wish to parameterise the model in terms of the GEV parameters of
annual maxima we should set `noy = 54` when calling `rpost`.
We use the prior from the evdbayes user's guide and a threshold
of 40mm.

```{r}
data(rainfall)
rthresh <- 40
prrain <- set_prior(prior = "quant", model = "gev", prob = 10^-(1:3), 
                    shape = c(38.9, 7.1, 47), scale = c(1.5, 6.3, 2.6))
```

The first set of plots below relate to the standard parameterisation.
There is posterior dependence between all parameters.  The second set of plots
show that rotating the posterior using `rotate = TRUE` reduces the dependence.

```{r, fig.width = 7}
# Annual maximum parameterisation (number of blocks = number of years of data)
rn0 <- rpost(n = n, model = "pp", prior = prrain, data = rainfall, 
  thresh = rthresh, noy = 54, rotate = FALSE)
plot(rn0)
rn0$pa
# Rotation about maximum a posteriori estimate (MAP)
system.time(
rn1 <- rpost(n = n, model = "pp", prior = prrain, data = rainfall, 
  thresh = rthresh, noy = 54)
)
plot(rn1, ru_scale = TRUE)
rn1$pa
```

Using the @WTJ2010 parameterisation reduces the posterior dependence
between $\mu$ and the other two parameters, but not between $\sigma$ and
$\xi$ (see plot on the right labelled with $\rho_1$ and $\rho_2$).  Using the
@WTJ2010 parameterisation and `rotate = TRUE` also reduces posterior dependence
between all parameters but the earlier plot suggests that using `rotate = TRUE`
alone is sufficient in this example.

```{r, fig.width = 7}
# Number of blocks = number of threshold excesses
n_exc <- sum(rainfall > rthresh, na.rm = TRUE)
rn2 <- rpost(n = n, model = "pp", prior = prrain, data = rainfall, 
  thresh = rthresh, noy = 54, use_noy = FALSE, rotate = FALSE)
rn2$pa
plot(rn2, ru_scale = TRUE)
# Number of blocks = number of threshold excesses, rotation about MAP
system.time(
rn3 <- rpost(n = n, model = "pp", prior = prrain, data = rainfall,
  thresh = rthresh, noy = 54, use_noy = FALSE)
)
plot(rn3, ru_scale = TRUE)
rn3$pa
```

In this example, rotation of the posterior about its mode using `rotate = TRUE`
produces the greatest increases in $p_a$ and, if this is performed, there is 
little to choose between the two choices of the number of blocks.  Setting
the number of blocks equal to the number of threshold excesses does reduce 
posterior dependence between $\mu$ and the other parameters but enough 
dependence between $\sigma$ and $\xi$ remains that the value of $p_a$ 
doesn't increase substantially.  The reparameterisation resulting from 
the rotation is linear in the parameters $(\mu, \sigma, \xi)$, which is 
simpler than the non-linear transformation from one value of the number 
of blocks to another.  Using this non-linear transformation in addition to 
the rotation has increased the computation time in this example, that is,
`rn1` takes a little longer to produce than `rn3` and this is also the case
for other choices of prior.

We produce plots of the estimated marginal posterior densities.

```{r, fig.width = 7}
oldpar <- par(mfrow = c(1,3))
plot(density(rn1$sim_vals[, 1], adj = 2), main = "", xlab = expression(mu))
plot(density(rn1$sim_vals[, 2], adj = 2), main = "", xlab = expression(sigma))
plot(density(rn1$sim_vals[, 3], adj = 2), main = "", xlab = expression(xi))
par(oldpar)
```

We do not illustrate the use of marginal Box-Cox
transformation in the PP case.  If `trans = "BC"` then we use 
$\phi_1 = \mu$, 
$\phi_2 = [\sigma + \xi (u - \mu) ] / (x_{(m)} - u)^{1/2}$ and 
$\phi_3 = [\sigma + \xi (x_{(m)} - \mu) ] / (x_{(m)} - u)^{1/2}$, 
where $u$ is the threshold `thresh` and $x_{(m)}$ is the largest sample
value.  As in the GEV case we apply Box-Cox transformation only to the (positive) 
parameters $\phi_2$ and $\phi_3$.

## $r$-largest Order Statistics (OS) model

The OS model is an extension of the GEV model in which the $r$ largest sample 
values (the $r$ largest order statistics) in each block are modelled, rather 
than just the block maximum.  More generally, if there are $m$ blocks then 
we can use the $r_i$ largest order statistics in block $i$.  For the purposes
of illustrating how `rpost` can be used to sample from a posterior 
distribution under the OS model we use `venice` dataset, which (apart from
one year) contains the 10 largest sea levels in Venice for the years 
1931-1981.  Our analysis is flawed because, as demonstrated in the 
evdbayes user guide, it is clear that a time trend in the
location parameter $\mu$ should be considered.

### Venice Sea Level Data

```{r, fig.width = 7}
data(venice)
mat <- diag(c(10000, 10000, 100))
pv <- set_prior(prior = "norm", model = "gev", mean = c(0,0,0), cov = mat)
osv <- rpost(n = n, model = "os", prior = pv, data = venice)
plot(osv)
```

```{r, fig.width = 7}
plot(osv, ru_scale = TRUE)
```

We do not illustrate the use of marginal Box-Cox
transformation in the OS case.  If `trans = "BC"` then we use 
$\phi_1 = \mu$, 
$\phi_2 = [\sigma + \xi (x_{(1)} - \mu) ] / (x_{(m)} - x_{(1)})^{1/2}$ and 
$\phi_3 = [\sigma + \xi (x_{(m)} - \mu) ] / (x_{(m)} - x_{(1)})^{1/2}$, 
where $x_{(1)}$ and $x_{(m)}$ are the smallest and largest sample order
statistics respectively.  Again we apply Box-Cox transformation only 
to the (positive) parameters $\phi_2$ and $\phi_3$.

## References

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