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*θ*, 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},*σ*_{u},*ξ*,*θ*) using maximum likelihood estimation. First, we make inferences about the model parameters.

```
library(lite)
cdata <- exdex::cheeseboro
# Each column of the matrix cdata corresponds to data from a different year
# flite() sets cluster automatically to correspond to column (year)
cfit <- flite(cdata, u = 45, k = 3)
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 × 24 = 744 for these data.

```
rl <- returnLevel(cfit, m = 100, level = 0.95, ny = 31 * 24)
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},*σ*_{u},*ξ*,*θ*), 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.

```
cpost <- blite(cdata, u = 45, k = 3, ny = 31 * 24)
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.