Parametric modal regression for right-censored positive responses

Abstract

We present a unified parametric framework for modal regression applicable to continuous positive distributions, with explicit support for right-censored observations. The key contribution is a systematic analytical reparameterization of density parameters as direct functions of the conditional mode. This closed-form mapping is derived for the Gamma, Beta, Weibull, Lognormal, and Inverse Gaussian distributions, directly linking the mode to a linear predictor. Maximum likelihood estimation is performed using the censored log-likelihood, with asymptotic inference based on the observed Fisher information matrix. A Monte Carlo simulation study across multiple distributions, sample sizes, and censoring levels confirms consistent parameter recovery. Empirical bias and RMSE decrease as expected, and Wald confidence intervals achieve nominal coverage. Finally, the proposed methodology is illustrated through an application to real-world reliability data. All methodology is implemented in the open-source R package ModalCens.

Figures (8)

• Figure 1: Root mean squared error (RMSE) of the regression coefficient estimates across $B = 1000$ replications. Rows: regression parameters ($\gamma_0$, $\gamma_1$, $\gamma_2$). Columns: distribution families. Line types and shades distinguish the censoring levels.
• Figure 2: Conditional densities and fitted modal regression lines (Weibull, Gamma, and Lognormal). Inverse Gaussian is excluded due to poor fit from the $\lambda > 3M_i$ restriction. Circles: observed failures; triangles: right-censored observations.
• Figure 3: Diagnostic plots for the Weibull modal regression on the motorette data. Left: randomized quantile residuals versus fitted modes. Right: normal Q-Q plot.
• Figure 4: Gamma family: boxplots of the estimation error $\hat{\gamma}_j - \gamma_j^0$ across $B = 1{,}000$ replications.
• Figure 5: Weibull family: boxplots of the estimation error $\hat{\gamma}_j - \gamma_j^0$ across $B = 1{,}000$ replications.

Theorems & Definitions (1)

• Remark 1