Quantile and Log-Quantile Least Squares for Robust-Efficient Fitting and Validation of Log-Location-Scale Loss Models

Many insurance loss models, such as lognormal and Pareto I distributions, are special cases of the log-location-scale family. This paper uses two asymptotic results—the joint normality of sample quantiles and the delta method—to develop nonlinear and linear regression frameworks for estimation and validation of log-location-scale distributions. It proposes four equally robust estimators: ordinary and generalized quantile least squares (oQLS and gQLS) and their log-quantile counterparts (log-oQLS and log-gQLS). For such models, logarithmic transformation of quantiles approximates the nonlinear least squares solution exactly, yielding more accurate estimators. Also, the log-linear regression framework facilitates studying the estimators’ properties and designing a residuals-based goodness-of-fit test. Moreover, log-oQLS and log-gQLS have explicit formulas and are easily computed for medium- (n = 10 3 ), large- (n = 10 4 ), and very-large size (n > 10 6 ) samples. This paper illustrates estimators’ computational and statistical performances, outlier-labeling rules, and goodness-of-fit using simulated and real datasets.