Generalized extreme value regression for binary response data: An application to B2B electronic payments system adoption

• Logistic regression assumes a symmetric response curve, an assumption that may be violated by imbalanced, skewed adoption data and risk link misspecification.Source: §1 · pp. 2–3
• False logistic-link assumptions can increase bias and mean squared error in parameter estimates and predicted probabilities, especially when misspecification involves skewness rather than kurtosis.Source: §1 · p. 3
• Earlier skewed-link approaches faced posterior-propriety or intercept-confounding problems, while the GT-link shape constraint narrowed its available skewness.Source: §1 · p. 3
• The paper proposes the generalized extreme value (GEV) distribution as a flexible skewed link for binary data, with an unconstrained shape parameter that accommodates a wide range of skewness.Source: §1 · pp. 3–4
• The paper compares models using DIC, BIC, marginal likelihood, and holdout posterior predictive deviance to assess out-of-sample accuracy and guard against overfitting.Source: §1 · p. 4

2. The electronic payment system data set.

• The data include payment choices and firm- and transaction-specific characteristics, including firm size, prior bill amount, PayDex credit risk, financing option, region, and industry.The observations concern firms financing large commercial equipment purchases with the vendor before 2005.Source: §2 · pp. 4–5
• 10,784 of 15,175 transactions used traditional payment methods (EPS = 1), accounting for more than 70% of observations.The vendor offered ACH debits as a voluntary free service alongside conventional methods such as checks.Source: §2 · p. 5
• In 1,618 Midwest manufacturing lease transactions, 1,222 used traditional methods; the reduced-sample model predicts EPS = 1 from normalized log firm size.The subset was selected to show response curves under different link models.Source: §2 · p. 5
• The positively skewed Cloglog link fit the reduced-sample data better than logit and probit, but still showed a large discrepancy from observed values.The paper attributes the remaining discrepancy to data skewness exceeding the fixed skewness of the Cloglog link.Source: §2 · p. 6

3. Generalized extreme value link model.

• The model maps a binary response probability through a link defined by a cumulative distribution function, with logit and probit symmetric and Cloglog asymmetric.The section specifies independent binary responses and covariate vectors within the generalized linear model framework.Source: §3 · pp. 6, 8

3.1. The generalized extreme value distribution.

• ξ controls the GEV distribution’s tail behavior, and ξ → 0 yields the Gumbel distribution.The GEV family includes the Gumbel, Fréchet, and Weibull classes.Source: §3.1 · pp. 8–9

3.2. The generalized extreme value link model.

• GEV links provide a wider range of skewness than GT links with a specified distribution, even for ξ ∈ [−0.5, 0.5].The comparison uses the Arnold and Groeneveld skewness measure; Figure 3 contrasts the GEV and GT link curves.Source: Figure 3 · p. 10, §3.2 · p. 10
• The GEV link can approximate a symmetric probit link between the 0.02 and 0.98 quantiles, with the main discrepancy in the tails.The approximation matches the first three moments of a standard normal distribution.Source: §3.2 · p. 10, Figure 4 · p. 11

3.3. Prior and posterior distributions for the generalized extreme value link model.

• Unlike Stukel’s generalized logistic regression with covariates, GEV posteriors are proper under many noninformative priors, including Jeffreys’ and the improper uniform prior for β.The authors favor Bayesian analysis because maximum-likelihood regularity conditions are not satisfied when ξ < −0.5.Source: §3.3 · pp. 11–12
• Under the stated design-matrix condition, the posterior is proper with an improper uniform prior on the regression coefficients.With ξ restricted to [−1, 1), the link retains skewness in [−0.7293, 1).Source: §3.3 · p. 12

4.1. Model comparison criterion.

• DIC, marginal likelihood, and BIC assess model fit with complexity considerations, whereas posterior predictive deviance evaluates prediction on holdout data.The simulation setup uses data with around 70% response values of 1; the authors caution that imbalance is only an indicator of possible skewness, since it can also occur with a symmetric link.Source: §4.1 · pp. 12–13, §4.2 · pp. 13–14

4.2. Simulated data examples.

• In Simulation ♯1, GEV outperforms logit even at n = 200, while in Simulation ♯2 it approximates probit and gains model-selection support mainly at larger sample sizes.Under Simulation ♯2, BIC and marginal likelihood select GEV over logit only at n = 5000.Source: §4.2 · p. 14

4.3. The motivation subset of the electronic payments system data.

• The GEV response curve stretches to fit the observed EPS adoption values in the motivation subset, and holdout predictions are especially good for firms with normalized log sizes above 0.The holdout contains 173 transactions, including 135 with EPS = 1.Source: §4.3 · pp. 14, 16, Figure 5 · p. 15, Table 5 · p. 16

4.4. The electronic payment system data set.

• 446.68 DIC for GEV is lower than 660.79 for Cloglog on the full EPS data, and both skewed links outperform symmetric logit and probit links.Marginal likelihood and BIC give consistent model comparisons.Source: §4.4 · p. 17, Table 6 · p. 17
• GEV and logit estimates differ on EPS covariate effects, including the regional comparison and whether construction-industry transactions are significantly more likely to use EPS.GEV estimates a 7% higher probability for construction than wholesale trade, while logit estimates an approximately 3% effect that is not statistically significant.Source: §4.4 · p. 18
• 1618.73 posterior predictive deviance for GEV, versus 1653.40 for logit, 1643.04 for probit, and 1637.92 for Cloglog, gives GEV the best holdout prediction on the full EPS data.The holdout set contains 1501 observations; the authors report no indication of overfitting.Source: §4.4 · p. 18

5. Conclusion and discussion.

• The GEV link flexibly fits response-curve skewness and required about 2 minutes to fit in R for a two-covariate simulation, compared with about 8 hours for the GT link in FORTRAN 95.The authors describe the GEV computation burden as much lower.Source: §5 · p. 19
• The proposed GEV link is presented for binary response data with covariates in the EPS adoption setting.Source: §5 · p. 19
• The GEV shape parameter ξ affects both skewness-related flexibility and distribution tail heaviness, rather than controlling skewness alone.The authors identify separating skewness and tail-heaviness control as a way to further improve flexibility.Source: §5 · p. 19

Simulated data examples.

• In Simulation ♯1, GEV coefficient estimates are close to the true Cloglog model, while shape-parameter uncertainty decreases as sample size rises from 200 to 5000.At n = 200, the 95% HPD interval for ξ includes the true value.Source: Simulated data examples · p. 20
• GEV average covariate effects are closer to Cloglog than logit in Simulation ♯1, and holdout posterior deviance favors GEV and Cloglog over logit.DIC also favors GEV and Cloglog over logit; marginal likelihood and BIC align with those comparisons only at n = 5000.Source: Simulated data examples · pp. 20, 22, Table 7 · p. 21, Table 8 · p. 21
• In Simulation ♯2, GEV shape-parameter intervals include the value approximating the standard normal distribution, but precise estimation requires larger samples.At n = 5000, all model-comparison criteria except marginal likelihood show some GEV advantage over logit and Cloglog; marginal likelihood requires an even larger sample under flat priors.Source: Simulated data examples · pp. 22–23, Table 10 · p. 23