Dirichlet Process Log Skew-Normal Mixture with a Missing-at-Random-Covariate in Insurance Claim Analysis

Round 1

Reviewer 1 Report

The authors provide a novel framework, based on BNP, to deal with the observations missing at random in data from insurance.

The method is presented in a clear way and the empirical results are quite satisfactory. Two concerns regarding:

i) discussion of the literature: BNP is not new in economics and finance. Nevertheless, many relevant references are missing. Here below is a non-exhaustive list

ii) figures: it is difficult to understand the contents of the plots. I would strongly suggest revising all captions to make the figures self-explaining.

References:

Bassetti, F., Casarin, R., and Leisen, F. (2014), Beta-Product DependentPitman-Yor Processes for Bayesian Inference, Journal of Econometrics,180, 49–72

Billio, M., Casarin, R., and Rossini, L. (2019), Bayesian NonparametricSparse VAR Models, Journal of Econometrics, 212, 97–115.

Casarin, R., Costantini, M., Osuntuyi, A. (2023), Bayesian nonparametric panel Markov-switching GARCH models,  Journal of Business and Economic Statistics, 41(2), 429-439.

Griffin, J., and Kalli, M. (2018),  Bayesian Nonparametric Vector Autoregressive Models, Journal of Econometrics, 203, 267–282.

Griffin, J. E., and Steel, M. F. J. (2011),  Stick-Breaking AutoregressiveProcesses, Journal of Econometrics, 162, 383–396.

Hirano, K. (2002), Semiparametric Bayesian Inference in AutoregressivePanel Data Models, Econometrica, 70, 781–799

Jensen, J. M., and Maheu, M. J. (2010),  Bayesian Semiparametric StochasticVolatility Modeling, Journal of Econometrics, 157, 306–316

none

Author Response

Please see the attachment. Thank you!

Author Response File: Author Response.docx

Reviewer 2 Report

The paper proposes a Dirichlet Process (DP) mixture model framework for actuarial practice to model insurance loss. The model includes an interesting proposal for dealing with missing at random covariates (MAR covariates). It also contains a discussion on two different choices of measures that enter into the definition of the base measure of the DP, specifically focusing on the use of the log-normal and log skew-normal distributions.

The paper is well-written and clear. In my opinion, the most interesting part is the section related to the use of the DP structure to impute missing covariates, although I believe that the clarity of this section could be improved.

Below, I list some specific suggestions and questions

A)  Eq. (3) should be explained in a better way. The role of J (later on misspelled j) has to be clarified. Strictly speaking  if eq. (3) has to be understood as the conditional density of $S_h$ given the covariates and the parameters,  $J$ is infinity. See subsequent (4a)-(4c). Why the Author use $J<+\infty$? 

Clearly, due to the predictive structure of the DP,  the number of clusters when only $h$ observations 

have been drawn is $J_h$ with $J_h \leq h$, and probably this is what the Authors want to emphasise. 

As it is written is not clear and I think that this part should be recast in a clearer way.  Similarly, in the predictive distribution (5) it must be clarified that $J=J_h$, as well in (6a)-(6b) I suppose that $H=h$.  This part must be revised carefully. 

Note that in eq. (7) the index h can be different to H, since in this case what is described is part of the full-conditional, and the full-conditional of $s_h$, once all the information for the remaining $H-1$ latent variable is fixed, is the same for each $h=1,\dots,H$ by exchangeability.  

B) I do not completely understand if the Imputation step in Section 3.4 is coherent with the probabilistic model described in Section 3.3.  It seems that at this level the Author choose a prior for the missing covariates that depends on the observations without missing covariates (and on the posterior of the parameter given these observations). On the other hand in Section 3.2 the Authors include a likelihood for the covariates and consequently the the parameters entering in this part of the likelihood are included in the parameter drawn from the DP prior. Is the resulting procedure the same as drawing from the posterior distribution of the parameters and the non-observed covariates given the observations and the observed covariates? 

 The Author must clarify this point , since this is essential to justify  the approach in a fully Bayesian setting. 

This should be possible starting from model (1), where the covariates have their own distribution in the hierarchical model. 

In other words, the model described in Section 3.2-3.3 is the model that one obtain taking the covariates as an observation and hence the likelihood of these covariates should appear in the posterior or is the marginal model which is obtained ignoring the distribution of the covariates? 

Author Response

Please see the attachment. Thank you!

Author Response File: Author Response.docx

Reviewer 3 Report

This paper uses DPM framework to approximate the log-normal sum and enable the imputation of missing covariates under the assumption of random missingness. The use of that framework is interesting. Overall, the paper makes contributions to the literature in line with Econometrics. My specific comments appear below.

1. Elaborate on model limitations: Provide some concise explanation of the limitations of traditional parametric models. 

2. Line 471: Provide the formula of SSPE and SAPE and explain the reason to choose these evaluation metrics.

3. Line 527: SSPE of LogSN-DPM is not the smallest one. Need to address that.

Author Response

Please see the attachment. Thank you!

Author Response File: Author Response.docx