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vignettes/lmls.Rmd

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The `lmls` package implements two inference algorithms: a frequentist maximum likelihood (ML) algorithm, which it uses by default, and a Bayesian Markov chain Monte Carlo (MCMC) algorithm. The derivatives that are required for these inference algorithms are derived in the Appendix in Section \@ref(sec:appendix).
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To simplify the notation in this and the next section, we first define the response vector $\yvec = [y_1, \dots, y_n]'$, the design matrices $\xmat = [\xvec_1', \dots, \xvec_n']'$ and $\zmat = [\zvec_1', \dots, \zvec_n']'$, and the standard deviation of the $i$-th observation $\sigma_i = \exp(\zvec_i'\gammavec)$. Finally, let $\wmat$ be the weight matrix
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\begin{equation}
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\wmat = \begin{bmatrix}
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which implies the following multivariate normal distribution:
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$$\betavec \mid \cdot\, \sim \N(\betawls, (\xmat'\wmat\xmat)^{-1}).$$
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# Comparison with other R packages
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There are a number of R packages with similar capabilities as *lmls*. The popular *mgcv* package [@Wood2017], which is typically used to estimate generalized additive models (GAMs), added support for multi-predictor models including the LMLS with a normally distributed response variable in version 1.8. The *gamlss* package implements generalized additive models for location, scale and shape [GAMLSS, @Rigby2005] in a very general way, and the *bamlss* package [@Umlauf2018] is a Bayesian alternative to the *gamlss* package.

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