Let R be the random variable of daily rainfall amount at a given station. R is zero with probability p0 and, for any r>0, we have the following decomposition: pr(R≤r)=p0+1-p0G(r), where G is the CDF of nonzero rainfall at the considered station. Choice of G is an issue. One of the difficulties is that we wish to model adequately both the bulk of the distribution of nonzero rainfall and its tail, i.e., the probability of extreme rainfall occurring. The most common models for nonzero rainfall include the Gamma, Weibull and lognormal models , whose CDF G(r) or densities g(r)=∂G(r)/∂r, r>0 are given in Table . Although less common, another family of models for nonzero rainfall relies on univariate extreme value theory, which tells that probabilities of the form pr(R≤r|r>q), with q large, can be approximated by either an exponential or a generalized Pareto tail Sect. 4. This led to propose the extended exponential and extended generalized Pareto distributions, whose CDF is given in Table . Note that less parsimonious models are given in , but they are not considered in the present study. The extended exponential and extended generalized Pareto distributions of Table  ensure that the occurrence probability of small (but nonzero) rainfall amounts is driven by κ, while the upper tail of nonzero rainfall is equivalent to a generalized Pareto tail. The extended exponential model is also called “generalized exponential” and has been used previously for extreme rainfall in and .

In the models of Table , rainfall is implicitly assumed to come from a single distribution. This assumption may be questioned. Indeed, different climatological processes trigger precipitation, leading to the occurrence of rainfall of different natures and intensities (e.g., convective vs. stratiform precipitation). Furthermore, rainfall occurrence and intensities often vary with season, reflecting both variations in temperature and in storm tracks, for example. For this reason, proposed for the same region the use of subsampling based on seasons and weather patterns (WP) see alsorespectively, in Canada and Norway. Each day of the record period is assigned to a WP. If S seasons and K WP are considered, then days are classified into S×K subclasses. The law of total probability gives, for all r>0, pr(R≤r)=∑s=1S∑k=1Kpr(R≤r|season=s,WP=k)ps,k, where ps,k is the probability that a given day is in season s and in WP k (thus ∑s∑kps,k=1). Following Eq. (), R in season s and WP k is zero with probability ps,k0 and, for any r>0, we have the decomposition pr(R≤r|season=s,WP=k)=ps,k0+1-ps,k0Gs,k(r), where Gs,k is the CDF of nonzero rainfall at the considered station for a day in season s and WP k. This gives in Eq. (), for all r>0, pr(R≤r)=p0+∑s=1S∑k=1Kps,k1-ps,k0Gs,k(r), where p0=∑s=1S∑k=1Kps,kps,k0 is the probability of any day being dry. Nonzero precipitation amounts defined by Eq. () have CDF G(r)=pr(R≤r|R>0)=∑s=1S∑k=1Kps,k′Gs,k(r), where ps,k′=ps,k(1-ps,k0)/(1-p0). Equation () defines a mixture of S×K distributions, e.g., a mixture of S×K Gamma distributions. Analogously, the CDF of nonzero precipitation amounts in a given season s is written as Gs(r)=pr(R≤r|R>0,season=s)=∑k=1Kps,k′′Gs,k(r), where ps,k′′=ps,k(1-ps,k0)/(1-∑k=1Kps,kps,k0). A similar idea is used in for example, but considering a mixture of two (exponential) distributions in an unsupervised way, i.e., without relying on a priori subsampling. It shows the advantage of not requiring prior knowledge on the classification, but it is at the same time more difficult to estimate, in particular if the models for different seasons and WP do not differ much.

In this article, we will consider the supervised case (Eq. ), with S=2 seasons and K=3 WP, considering the five models of Table  for the distribution of precipitation amounts Gs,k (see Sect. ). This implies that estimation of Gs,k can be made independently of each others, by considering only the days of the record belonging to season s and WP k. A variety of inference methods exists. For rainfall analysis, two options are popular: maximum likelihood (ML) estimation and a method of moments based on probability weighted moments (PWMs). However, as noted in , ML estimation may fail for rainfall because the discretization due to instrumental precision strongly affects low values, which biases ML estimation if not accounted for. One way to circumvent this issue is to resort to censored likelihood but choice of the censoring threshold is in itself an issue. Results on our data (not shown) reveal that the threshold has to be no smaller than 5 mm. PWM, on the other side, is much more robust against discretization since it is based on summary statistics, rather than on the exact values of observations . For this reason, we estimate in this study the distributions of precipitation amounts Gs,k by PWM, while ps,k0 at a given station is estimated empirically as p^s,k0=ds,k0/d where d is the number of observations and ds,k0 is the number of zero values in season s and WP k. Combining estimations in Eq. () gives an estimation of the rainfall CDF at the considered station, and in Eq. () an estimation of the CDF of nonzero rainfall.

Estimates of return levels are then obtained as follows. The T-year return level rT is the level expected to be exceeded on average once every T years. It satisfies the relationship pr(R≤rT|R>0)=1-1/(Tδ), where δ is the mean number of nonzero rainfall per year at the considered station. When subsampling Eq. () is considered, there is no explicit formulation, and estimation of rT is obtained numerically by solving pr(R≤rT|R>0)=1-1/(Tδ) in Eq. ().

The goal of this evaluation is to assess which marginal model performs better at the regional scale, i.e., for a set of n stations taken as a whole, rather than individually. We follow the split sample evaluation proposed in and . We divide the data for each station i into two subsamples, Ci(1) and Ci(2), and consider nonzero rainfall for these two subsamples. We fit a given competing model on each of the subsamples, giving two estimated distributions of G in Eq. (): G^i(1), estimated on Ci(1), and G^i(2), estimated on Ci(2). Our goal is to test the consistency between validation data and predictions of the estimates, both for the core and tail of the distributions, and the stability of the estimates when calibration data changes, focusing particularly on the tail which is usually less stable.

As shown in Table , three families of scores are computed, assessing, respectively, (i) accuracy of the estimations along the full range of observations (MEAN(NRMSE)), (ii) reliability of the tail of the estimated distribution, checking in particular systematic over- or under-estimation of the observations (AREA(FF) and AREA(NT)), and (iii) stability of the tail at extrapolation (MEAN(SPAN)). The scores relating the tail of the distribution have been proposed and used in , and . In the split sample evaluation framework, four scores can be derived of a given score Sc: Sc(12) is the regional score when Gi(2) is validated on the nonzero rainfall subsample Ci(1). Sc(21), Sc(11) and Sc(22) are obtained symmetrically. Sc(11) and Sc(22) are calibration scores, while Sc(12) and Sc(21) are cross-validation scores. For the sake of conciseness, we detail below the case of Sc(12) for the different scores.

The NRMSE (normalized root mean squared error) evaluates the reliability of the fits in the whole observed range of nonzero rainfall, by comparing observed and predicted return levels of daily rainfall. For a given station i∈{1, …, Q}, NRMSEi(12)=1ni(1)∑k=1ni(1)ri,Tk(1)-r^i,Tk(2)21/21ni(1)∑k=1ni(1)ri,Tk(1), where ni(1) is the number of nonzero rainfall in Ci(1) for station i, Tk ranges the observed return periods of nonzero rainfall in Ci(1), ri,Tk(1) is the observed daily rainfall associated with the return period Tk for the subsample Ci(1) and r^i,Tk(2) is the Tk-year return period derived from the estimated G^i(2). Without loss of generality we assume T1, …, Tni(1) to be sorted in descending order (so T1 is associated with the maximum over Ci(1)). If station i has δi nonzero rainfall per year on average, usual practice is to consider the kth largest return period as Tk=(ni(1)+1)/(δik), k=1, …, ni(1), and to estimate ri,Tk(1) as the kth largest observed rainfall over Ci(1). Estimate r^i,Tk(2) is obtained numerically from G^i(2) as described in Sect. . The normalization by the mean rainfall of Ci(1) in Eq. () allows comparison of NRMSE over stations with different pluviometry. The smaller NRMSEi(12), the better G^i(2) fits the rainfalls over Ci(1). For the set of Q stations, we obtain a vector of NRMSE(12) of length Q which should remain reasonably close to zero. A regional score is obtained by computing the mean of the Q values: MEANNRMSE(12)=1Q∑i=1QNRMSEi(12). For competing models, the closer the mean is to 0, the better the goodness-of-fit.

NRMSE assesses goodness-of fit of the whole distribution in the observed range. Now let us have a closer look at the tail of the distribution, and in particular at the maximum over Ci(1); i.e., at ri,T1(1) in Eq. (), that for shortness we denote mi(1). If Gi is the true distribution of nonzero rainfall, then the corresponding random variable Mi(1) has distribution Gi to the power ni(1), whose variance is large. Thus computing error based on the single realization mi(1) would be very uncertain. For this reason, proposed to make evaluation by pulling together the maxima of the Q stations, after transformation to make them on the same scale. It is based on the idea that if X has CDF F, then F(X) follows the uniform distribution on (0, 1). Taking X=Mi(1) and F=(Gi)ni(1) implies that, if G^i(2) is a perfect estimate of Gi, then ffi(12)={G^i(2)(mi(1))}ni(1) should be a realization of the uniform distribution. For the set of Q stations, this gives a uniform sample ff(12) of size Q. Hypothesis testing for assessing the validity of the uniform assumption is challenging because the ffi(12) are not independent from site to site, due to the spatial dependence between data. Thus proposed to base comparison on the divergence of the density of the ff(12) to the uniform density. A reasonable estimate of the latter is obtained by computing the empirical histogram of the ff(12) with 10 equal bins between 0 and 1. As illustrated in Fig. , if G^i(2) are good estimates of Gi, i=1, …, Q, the histogram of ff(12) should be reasonably uniform on (0, 1). If the histogram is left-skewed, then G^i(2)(mi(1)) tends to overestimate the true Gi(1)(mi(1)), or in other words the return period of the maximum over Ci(1) tends to be underestimated (case of over-estimated risk). If the histogram is right-skewed, the return period of the maximum over Ci(1) tends to be over-estimated (case of under-estimated risk). Although any scenario of misfitting could theoretically be possible, in practice the histograms of ff(12) show mainly the three above alternatives: either a good fit (flat histogram), or a tendency towards a systematic under- or over-estimation (left- or right-skewed histograms). By focusing of maximum values, the histogram of ff(12) can be seen as a way of assessing systematic bias in the far tail of the distribution. For a more quantitative assessment, we compute the area between the density of the ff(12) and the uniform density as follows: AREAFF(12)=118∑c=11010cardBcn-1, where card(Bc) is the number of ffi(12) in the cth bin, for c=1, …, 10. The term inside the absolute value in Eq. () is the difference between densities in the cth bin. The division by 18 forces the score to lie in the range (0, 1), with lower values indicating better fits (the worst case being all values lying in the same bin). Figure  shows that, when Q=42 stations are considered, a value of AREA(FF(12)) around 0.2 corresponds to no systematic bias in the very tail of the distribution at regional scale, whereas a value around 0.5 corresponds to a strong over- or under-estimation. In the latter case, only looking at the histogram can inform about whether over- or under-estimation applies.

The NT criterion is an alternative to FF assessing reliability of the fit of the tail but focusing on prescribed (large) quantiles rather than on the overall maximum. It applies the same principle as FF, involving a transformation of X to F(X), but considering X as Ki,T(1), the random variable of the number of exceedances over Ci(1) of the T-year return level, i.e., Ki,T(1)=card({Ri,j∈Ci(1); Gi(Rj)>1-1/(Tδi)}), in which case F is the binomial distribution Bi(1) with parameters (ni(1), 1/(Tδi)). Thus, if G^i(2) is a perfect estimate of Gi, then ni,T(12)=Bi(1)(ki,T(12)), where ki,T(12)=cardri,j∈Ci(1);G^i(2)left(ri,j>1-1/δiT), should be a realization of the discrete uniform distribution. Randomization to transform ni,T(12) to a continuous uniform variate on (0, 1) is proposed in and extensively described in . For i ranging over the set of Q stations, we thus obtain a sample of Q uniform variates. Scores are calculated as for FF by comparing the empirical densities of NT(12) to the theoretical uniform density, giving the scores AREA(NT(12)). Taking T as, e.g., half to one-quarter the length of the observations allows assessment of the reliability of the close tail of the distribution. As such, it is a good complement to FF that focuses on the far tail (i.e., on the maximum).

Last but not least, the SPAN criterion evaluates the stability of the return level estimation, when using data for each of the two subsamples. More precisely, for a given return period T and station i, SPANi,T=|r^i,T(1)-r^i,T(2)|1/2r^i,T(1)+r^i,T(2), where r^i,T(1); e.g., is the T-year return level for the distribution G estimated on subsample Ci(1) of station i, i.e., such that G^i(1){r^i,T(1)}=1-1/(Tδi). SPANi,T is the relative absolute difference in T-year return levels estimated on the two subsamples. It ranges between 0 and 2; the closer to 0, the more stable the estimations for station i. For the set of Q stations, we obtain a vector of SPANT of length Q with a distribution which should remain reasonably close to zero. A rough summary of this information is obtained by computing the mean of the Q values of SPANi,T, i=1, …, Q: MEANSPANT=1Q∑i=1QSPANi,T. For competing models, the closer the mean is to 0, the more stable the model is. When T is larger than the observed range of return periods, MEAN(SPANT) evaluates the stability of the return levels in extrapolation. Note that it is by definition 0 in calibration, and thus it is only useful in cross-validation.

For the sake of concision, in the rest of this article the scores MEAN(NRMSE), AREA(FF), AREA(NT) and MEAN(SPANT) will be referred to as the NRMSE, FF, NT and SPANT scores.

Let Ri be the random variable of daily rainfall at station i, i=1, …, Q. Applying Sect.  at station i gives an estimate G^i(r) of the CDF Gi(r)=pr(Ri≤r|Ri>0). Our goal is to derive an estimate of the CDF of nonzero daily rainfall at any location l of the region, pr(R(l)≤r|R(l)>0), based on the Q estimated CDFs G^i(r). Location l refers here to the three coordinates of ground projection coordinates and altitude, that we write l=(x, y, z). Let θ^i be the set of estimated parameters for station i and θ^i,j its jth element. θ^i is composed of the S×K probability of zero rainfall ps,k0 and the 2×S×K or 3×S×K parameters of the distributions Gs,k, depending on the marginal distribution (see Table ). We assume the θi,j ordered so that the first S×K elements are the ps,k0. We aim at estimating the surface response θj(l) at any l of the region, knowing θj(li)=θ^i,j. In this study we consider three of the most popular method: kriging interpolation, linear regression methods and thin plate spline regressions. However, the parameters θjs are constrained, whereas these models apply the unbounded variables: the probabilities ps,k0 lie in (0, 1), while the parameters of Table  are all positive. Therefore we apply the mapping models to transformations of θj, i.e., to ψj=transf(θj), where “transf” maps the range of values of θj to (-∞, +∞). In this study we consider ψj(l)=Φ{θj(l)} if j≤S×K (i.e., if θj is any ps,k0), where Φ is the standard Gaussian CDF, and to ψj(l)=log⁡{θj(l)} otherwise. Other transformations would be possible, in particular ps,k0 may be transformed with the logit function, but will not be considered here for the sake of concision. Thus we aim at estimating ψ̃j(l) given values ψj(li)=ψ^i,j at station locations, with obvious notations. If l≤S×K, estimates of θj(l) are then obtained as θ̃j(l)=Φ-1(ψ̃j(l)). Otherwise surface response estimates are obtained as θ̃j(l)=exp⁡(ψ̃j(l)). For the sake of clarity, we omit below the index j, considering a surface ψ(l) to be estimated for all l in the region, given values ψ(li)=ψ^i.

The considered mapping models are listed in Table . Three families of method are considered: kriging, linear regression and thin plate spline. Additionally to how they map values, there is a fundamental difference between these models: kriging is an exact interpolation, i.e., ψ̃(li)=ψ^i at any station location li used to estimate the model. By contrast, the linear regression models and thin plate splines provide inexact interpolations: in the great majority of the time, ψ̃(li)≠ψ^i (the goal being obviously to minimize the overall error).

For the kriging interpolation, cases with and without external drift are tested Sect. 3.6 of. The external drift, if any, is modeled as a linear function of altitude (i.e., of the form a0+a1ζ), considering ζ as either the altitude of the station (z) or, following , as a smoothed altitude (Z) derived by smoothing a 1 km digital elevation model (DEM) with 5 km moving windows (i.e., taking Z as the average altitude of 25 DEM grid points). The results that will be presented below correspond to the case of an exponential covariance function of the form ρ(h)=e-h/β, with β>0. We also considered the case of a powered exponential covariance function ρ(h)=e-(h/β)ν with 0<ν≤2, but this resulted in a slight loss of stability due to the additional degree of freedom, without improving the accuracy at regional scale. For the sake of concision, these results are not presented here. Combining alternatives for the drift part gives a total of three kriging interpolation models with two to three unknown parameters each, for each ψ. Estimations of the kriging models are made by maximizing the likelihood associated with the ψ^i, assuming that ψ(l) is a Gaussian process see Sect. 5.4 of. Alternatives are to estimate drifts and variograms by least squares in different steps, with the risk of biasing estimates Sect. 5.1 to 5.3 of. Both estimation methods are available in R package “geoR” (e.g., functions “likfit” and “variofit”). In the case without drift, prediction at any site l of the region is obtained as ψ̃(l)=∑i=1Qwihiψ^i, where the weights wi(hi) are derived from the kriging equations and satisfy ∑i=1Qwi(hi)=1. The weights depend on the estimated covariance function and on the distance hi between l and station location li in the (x, y) space (i.e., hi2=(x-xi)2+(y-yi)2). In the case with external drift, prediction at any location l of the region is then obtained as ψ̃(l)=a1ζ+∑i=1Qwihiψ^i-a1ζi, where ζ is the altitude at location l (i.e., either z or Z). Predictions (Eqs.  and ) are exact: ψ̃(li)=ψ^i, and consequently θ̃(li)=θ^i.

For the linear regression models, we start from regressions of the form ψ(l)=a0+a1x+a2y+a3ζ+a4x2+a5y2+a6xy+a7xζ+a8yζ+ϵ(l), where ϵ(l)∼N(0, σ2) and ζ is, as before, either the altitude of the station (z) or the smoothed altitude (Z). We consider the Akaike information criterion (AIC), defined as AIC = 2η-2log⁡L, where η is the number of parameters (10 at the start) and L is the maximum likelihood value of the regression model. The lower AIC, the better the model. Then we repeatedly drop the variable that increases most the AIC – if any –, and add the variable that decreases most the AIC – if any. This stepwise method is implemented in the “step” function of R package “stats”. At algorithm stop, the model may contain 1 to 10 parameters, for each ψ. Predictions θ̃(l) are then obtained as the back transformation of the estimated regressions.

Last but not least, bivariate and trivariate thin plate splines are considered for ψ(l) . These methods are implemented in function “Tps” of R package “fields”. In the bivariate case, ψ(l) is modeled as ψ(l)=u(x,y)+ϵ(x,y), where u is an unknown smooth function and ϵ(x,y) are uncorrelated errors with zero means and equal variances. The function u is estimated by minimizing ∑i=1Qψ^i-uxi,yi2+λ∫-∞+∞∫-∞+∞∂2u(x,y)∂x22+2∂2u(x,y)∂x∂y2+∂2u(x,y)∂y22dxdy, where λ is the so-called smoothing parameter, which controls the trade-off between smoothness of the estimated function and its fidelity to the observations. It can be estimated by generalized cross-validation. Then predictions are obtained as ψ̃(l)=a0+a1x+a2y+∑i=1Qbihi2log⁡hi, where hi is the Euclidean distance in the (x, y) space between l and station location li. The partial trivariate case assumes that ψ(l)-a3ζ is a bivariate thin plate spline, where a3 is fixed and ζ is either z or Z. To make the connection with kriging, ψ(l) can thus also be seen as a bivariate thin plate spline with (fixed) drift in ζ. The coefficient a3 is estimated in a preliminary step by regressing ψ^i against ζi. Estimation of the bivariate thin plate spline for ψ(l)-a3ζ is made as described above given the values of ψ^i-a3ζi. Predictions are obtained as ψ̃(l)=a0+a1x+a2y+a3ζ+∑i=1Qbihi2log⁡hi. Finally in the trivariate case, we have ψ(l)=u(x,y,ζ)+ϵ(x,y,ζ). The minimization problem is similar to Eq. () with a penalization enlarged by several terms . Predictions are then obtained as ψ̃(l)=a0+a1x+a2y+a3ζ+∑i=1Qbihi′, where hi′ is the Euclidean distance in the (x, y, ζ) space between l and station location li, scaling the altitude by a factor of 10 following and (i.e., h′i2=(x-xi)2+(y-yi)2+100(ζ-ζi)2). Coefficients ai and bi in Eqs. () to () are estimated by solving a linear system of order Q involving the smoothing parameter λ. Note that the trivariate case (Eq. ) differs from the bivariate case with drift (Eq. ) in two ways. First, Eq. () considers distance in the (x, y, ζ) space, whereas Eq. () considers distance in the (x, y) space. Second, in Eq. (), the weights associated with the stations are linear functions of the distance, unlike in Eq. () (see the term hi2log⁡(hi) vs. hi′).

Evaluation is performed in two ways. The first one is a leave-one-out cross-validation scheme aiming to test at regional scale how the interpolated distributions are able to fit the data of the stations when these data are left out for estimating the mapping model. The second step assesses spatial stability by comparing the interpolated distributions obtained at a given station whether the data of this station are used or not in the mapping estimation. In other words, it is a comparison between leave-one-out and leave-zero-out mappings. So the two evaluations differ in that the first one compares an interpolated distribution to data, while the second step compares two interpolated distributions.

First, let us consider a given parameter estimate θ^i,j(1) obtained at station i based on the subsample Ci(1) (for a given marginal model). We apply a leave-one-out cross-validation scheme: for each station i0 alternatively, we use the set of θ^i,j(1), i≠i0 to estimate the surface response θ̃j(1)(l). We store the value of this estimate at station location i0, i.e., θ̃j(1)(li0)=θ̃i0,j(1). Repeating this for every parameter θj gives an estimation of the full set of parameters at station i0, i.e., estimation G̃i0(1) of Gi0. Iterating over the stations, we obtain new estimates G̃1(1), …, G̃Q(1) of G1, …, GQ. Applying similarly for the subsample Ci(2) gives estimates G̃1(2), …, G̃Q(2). We can assess the reliability of these estimates at the regional scale by computing the scores Sc(11), Sc(22), Sc(12) and Sc(21) of Sect. , where G^i is replaced by G̃i. Note that all these scores are cross-validation scores since the estimates G̃1(1) and G̃1(2) are computed without using any data of station i.

Second, we consider the set of all θ^i,j(1), i=1, …, Q to estimate the surface response θ̃j(1)(l). We store the value of this function at every station location, giving new estimates G̃1*(1), …, G̃Q*(1). Note that in the particular case of kriging, G̃i*(1) is exactly G^i(1) since it is an exact interpolation method, so every θ^i,j(1) equals θ̃i,j*(1). We can assess the stability of the interpolated distributions at a given location li when observations are available or not at this location by comparing G̃i*(1)(r) and G̃i(1)(r) for all r. For this we discretize r between 0 and 450 mm (which is the overall maximum rainfall) with 1 mm step and we compute the total variation distance (TVD) between G̃i*(1) and G̃i(1) and the Kullback–Leibler divergence KLD, from G̃i*(1) to G̃i(1), which are given by TVDi(1)=sup⁡r|G̃i*(1)(r)-G̃i(1)(r)|,KLDi(1)=∫rg̃i*(1)(r)log⁡g̃i*(1)(r)g̃i(1)(r)dr, where, e.g., g̃i(1) is the density function associated with G̃i(1). Note that the KLD is not symmetric. Written as such, it can be interpreted as the amount of information lost when G̃i(1) is used to approximate G̃i*(1), so considering G̃i*(1) as the “true” distribution of data. TVD and KLD differ in that TVD focuses on the largest deviation between the two CDFs, whereas KLD somewhat integrates deviations along rainfalls. Obviously, one would like the interpolation to be as stable as possible when data are available or not at station i, i.e., that the lower TVDi and KLDi, the more stable the interpolation at station i.

Regional scores MEAN(TVDi(1)) and MEAN(KLDi(1)) are then obtained by computing the mean of the Q values. MEAN(TVDi(2)) and MEAN(KLDi(2)) are obtained similarly for the subsample C(2). For competing models, the closer the means are to 0, the more spatially stable is the interpolation. For shortness, we will refer to MEAN(TVD) and MEAN(KLD) as the TVD and KLD scores, respectively (Table ).

The full cross-validation procedure for selecting both the marginal and mapping models is summarized in Fig. . First we consider the marginal distributions of Table  and select the best of them at regional scale, as described in Sect. :

We divide the days of 1948-2013 into two subsamples of equal size, denoted C(1) and C(2). Given the weak temporal dependence of rainfall in the region (80 % of the wet periods have length lower than 3), division is made by randomly choosing blocks of five consecutive days to compose C(1), the remaining blocks composing C(2).

Second we consider the mapping models of Sect.  for interpolating the selected marginal model, and we select the best of them in two ways, as described in Sect. .

We consider the estimates G^i(1,t), i=1, …, Q, obtained at the tth iteration of the marginal selection procedure, and corresponding to the subsamples Ci(1,t), i=1, …, Q.

At this step we have selected the best marginal model and the best mapping model (among those tested) for our data.

Finally, we consider the whole sample of data and apply the selected marginal distribution and mapping model.

We estimate the selected marginal distribution G^i* based on the full data, giving parameters θ^ij*, i=1, …, Q.