The **lite** package performs likelihood-based inference
for stationary time series extremes. The general approach follows Fawcett and Walshaw (2012).
There are 3 independent parts to the inference.

- A Bernoulli (
*p*_{u}) model for whether a given observation exceeds the threshold \(u\). - A generalised Pareto, GP
(
*σ*_{u},*ξ*), model for the marginal distribution of threshold excesses. - The \(K\)-gaps model for the extremal index \(\theta\), based on inter-exceedance times.

For parts 1 and 2 it is necessary to adjust the inferences because we
expect that the data will exhibit cluster dependence. This is achieved
using the methodology developed in Chandler and Bate
(2007) to produce a log-likelihood that is adjusted for this
dependence. This is achieved using the chandwich
package. For part 3, the methodology described in Süveges and Davison (2010)
is used, implemented by the function `kgaps`

in the exdex package. The
(adjusted) log-likelihoods from parts 1, 2 and 3 are combined to make
inferences about return levels.

We illustrate the main functions in `lite`

using the
`cheeseboro`

wind gusts data from the exdex package, which
contains hourly wind gust data from each January over the 10-year period
2000-2009.

The function `flite`

makes frequentist inferences about
\((p_u, \sigma_u, \xi, \theta)\) using
maximum likelihood estimation. First, we make inferences about the model
parameters.

```
library(lite)
<- exdex::cheeseboro
cdata # Each column of the matrix cdata corresponds to data from a different year
# flite() sets cluster automatically to correspond to column (year)
<- flite(cdata, u = 45, k = 3)
cfit summary(cfit)
#>
#> Call:
#> flite(data = cdata, u = 45, k = 3)
#>
#> Estimate Std. Error
#> p[u] 0.02771 0.005988
#> sigma[u] 9.27400 2.071000
#> xi -0.09368 0.084250
#> theta 0.24050 0.023360
```

Then, we make inferences about the 100-year return level, including
95% confidence intervals. The argument `ny`

sets the number
of observations per year, which is \(31 \times
24 = 744\) for these data.

```
<- returnLevel(cfit, m = 100, level = 0.95, ny = 31 * 24)
rl
rl#>
#> Call:
#> returnLevel(x = cfit, m = 100, level = 0.95, ny = 31 * 24)
#>
#> MLE and 95% confidence limits for the 100-year return level
#>
#> Normal interval:
#> lower mle upper
#> 69.80 88.62 107.44
#> Profile likelihood-based interval:
#> lower mle upper
#> 75.89 88.62 125.43
```

The function `blite`

performs Bayesian inferences about
\((p_u, \sigma_u, \xi, \theta)\), based
on a likelihood constructed from the (adjusted) log-likelihoods detailed
above. First, we sample from the posterior distribution of the
parameters, using the default priors.

```
<- blite(cdata, u = 45, k = 3, ny = 31 * 24)
cpost summary(cpost)
#>
#> Call:
#> blite(data = cdata, u = 45, k = 3, ny = 31 * 24)
#>
#> Posterior mean Posterior SD
#> p[u] 0.02832 0.006008
#> sigma[u] 10.03000 2.435000
#> xi -0.07196 0.094030
#> theta 0.24250 0.024040
```

Then, we estimate a 95% highest predictive density (HPD) interval for the largest value \(M_{100}\) to be observed over a future time period of length \(100\) years.

```
predict(cpost, hpd = TRUE, n_years = 100)$short
#> lower upper n_years level
#> [1,] 73.09008 139.8616 100 95
```

Objects returned from `flite`

and `blite`

have
`plot`

methods to summarise graphically, respectively,
log-likelihoods and posterior distributions.

To get the current released version from CRAN:

`install.packages("lite")`

See `vignette("lite-1-frequentist", package = "lite")`

and
`vignette("lite-2-bayesian", package = "lite")`

.