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Data Formatting

There are a few data formatting issues to note when using the LRMoE package, which are summarized below. Throughout this page, we assume there are N = 100 insurance policyholders, who have two coverages y_1 and y_2 (dimension of response D = 2). Each policyholder has 5 covariates (number of covariates P = 5).

  • Covariate X: The dimension of X should be 100 by 5 (= N * P), which is relatively straightforward. If there are factor covariates (e.g. indicators for urban and rural region), the intercept term should be manually added, and the user needs to manually augment a column of zero-one indicator.

  • Response Y (Exact Case): If both y_1 and y_2 are observed exactly (i.e. the incurred losses are not censored or truncated), then we can set exact_Y = true in the initialization and fitting function. In this case, the dimension of Y should be 100 by 2 (= N * D): the first column is for y_1 and the second for y_2.

  • Response Y (Not Exact Case): If either y_1 or y_2 is not observed exactly, then exact_Y = false and the dimension of Y should be 100 by 8 (= N * 4D): the first four columns are for y_1 and the remaining for y_2. For each block of 4 columns, they should be structured as (tl, yl, yu, tu), corresponding to the lower bound of truncation, lower bound of censoring, upper bound of censoring and upper bound of truncation, respectively. Some typical cases are listed below:

    • Both y_1 and y_2 are observed exactly: Assume y_1 = 2.0 and y_2 = 3.0, then the first row of Y should be [0.0 2.0 2.0 Inf 0.0 3.0 3.0 Inf]. Alternatively, we can set exact_Y = true (see the previous case), and also set the first row of Y as [2.0 3.0].
    • y_1 is observed exactly, y_2 is left-truncated at 1.0 but observed exactly (e.g. an insurance deductible): Assume y_1 = 2.0 and y_2 = 3.0, then the first row of Y should be [0.0 2.0 2.0 Inf 1.0 3.0 3.0 Inf].
    • y_1 is observed exactly, y_2 is right-censored at 2.0 (e.g. a payment limit): Assume y_1 = 2.0 and y_2 = 3.0, then the first row of Y should be [0.0 2.0 2.0 Inf 0.0 2.0 Inf Inf].
    • y_1 is observed exactly, y_2 is only observed within a range: Assume y_1 = 2.0 and y_2 = 3.0, but we only observe 2.5 < y_2 < 3.5, then the first row of Y should be [0.0 2.0 2.0 Inf 0.0 2.5 3.5 Inf].

  • Logit Regression Coefficients α: Assume we would like to fit an LRMoE with three latent classes (g = 3), then the dimension of α should be 5 by 3 (= N * g). For example, a noninformative guess can be initialized as α = fill(0.0, 5, 3).

  • Component Distribution comp_dist: Assume we would like to fit an LRMoE with three lateht classes (g = 3), then the dimension of comp_dist should be 2 by 3 (= D * g). For example, if we assume y_1 is a mixture of lognormals and y_2 is a mixture of gammas, then comp_dist = [LogNormalExpert(1.0, 2.0) LogNormalExpert(1.5, 2.5) LogNormalExpert(2.0, 3.0); GammaExpert(1.0, 2.0) GammaExpert(1.5, 2.5) GammaExpert(2.0, 3.0)]. Note that the columns of comp_dist should be distinct, otherwise the model is not identifiable.

  • Penalty on Logit Regression Coefficients pen_α: It should be a single number (i.e. a uniform penalty imposed on all coefficients in α). For example, when pen_α = 2.0, then the penalty sum( (α ./ 2.0).^2 ) is subtracted from the loglikelihood as a penalty. In other words, we would not like the magnitude of α to be too large.

  • Penalty on Parameters of Expert Functions pen_params: Using the comp_dist mentioned above, pen_params should be a matrix of size 2 by 3 (= D * g), where each entry is a vector of real numbers penalizing the parameters of expert functions. Usually, the user can leave this argument as default. For more details, the user is referred to the package source code.