On parameter estimation for the truncated skew-normal distribution

Abstract

Parameter estimation for the truncated skew-normal distribution is challenging, as truncation introduces additional nonlinearity into the likelihood function and often leads to numerical instability in existing estimation procedures. In this paper, we propose a grid-based estimation method, referred to as GRID-MOM, for parameter estimation in the truncated skew-normal distribution. The proposed approach fixes the shape parameter on a pre-specified grid and, for each grid point, estimates the location and scale parameters using the method of moments. The optimal value of the shape parameter is then selected via likelihood-based comparison, yielding the final parameter estimates. By decoupling the estimation of the shape parameter from that of the location and scale parameters, the proposed method reduces the complexity of the optimization problem and improves numerical stability. We evaluate the finite-sample performance of the proposed estimator through an extensive numerical study, comparing it with existing methods under a variety of scenarios. The results demonstrate that the proposed method provides stable and accurate estimation, particularly for the shape parameter, suggesting that the proposed method offers a practical alternative for inference in truncated skew-normal models. We further demonstrate the practical applicability of the proposed method using phosphoproteomics data and hospital admission data.

Figures (6)

• Figure 1: Comparison of GRID-MOM (red boxplots) and GRID-MLE (green boxplots) under truncation rate $\tau = 0.1$. Rows correspond to truncation direction (Left, Right, Double), and columns correspond to the true shape parameter $\alpha_0 \in \{1,2,4\}$. Across all simulation settings, the true location and scale parameters are fixed at $\xi_0 = 0$ and $\omega_0 = 1$, and the sample size is set to $n = 500$. Within each panel, the parameter estimates $(\hat{\xi}, \hat{\omega}, \hat{\alpha})$ based on 1,000 Monte Carlo replications are summarized using boxplots. The horizontal dashed lines are reference lines at 0, 1, 2, and 4.
• Figure 2: Comparison of GRID-MOM (red boxplots) and GRID-MLE (green boxplots) under truncation rate $\tau = 0.2$. The layout and all other simulation settings are identical to those in Figure \ref{['fig:GRID_MOM_MLE_tau=0.1']}.
• Figure 3: Computational cost of GRID-MOM and GRID-MLE as functions of sample size (left panel) and the number of grid points (right panel). In the left panel, the reported average computing times are based on 50 Monte Carlo replications with the number of grid points fixed at 401. In the right panel, the averages are based on 100 Monte Carlo replications with the sample size fixed at $n = 500$.
• Figure 4: Histogram of the test statistics from the phosphoproteomics data, overlaid with the estimated truncated densities obtained by MLE (red solid line), MOM (green dashed line), MWM (blue dotted line), GRID-MLE (purple dot-dashed line), and GRID-MOM (pink long-dashed line). The fitted density curves from MLE, MOM, GRID-MLE, and GRID-MOM are nearly indistinguishable and overlap almost completely.
• Figure 5: Histogram of the hospital admission days, overlaid with the estimated truncated densities obtained by MLE (red solid line), MOM (green dashed line), MWM (blue dotted line), GRID-MLE (purple dot-dashed line), and GRID-MOM (pink long-dashed line). The fitted density curves from MWM and GRID-MLE are nearly indistinguishable and overlap almost completely.

Theorems & Definitions (1)

• Remark 1