Observed data can depart from ideal statistical assumptions because of systematic distortion, random measurement error, and irregular reporting. This dissertation develops robust estimation and inference methods for three such settings. The first study considers latent correlation when both variables are affected by multiplicative functions of an observed confounder. A log-adjusted residual estimator is combined with empirical likelihood, jackknife empirical likelihood, and adjusted jackknife empirical likelihood. Theoretical derivations and simulations examine linear, nonlinear, and periodic distortions under multiple confounder distributions. The numerical results generally show improved coverage and shorter intervals after log adjustment, particularly for complex distortion functions, while the real-data analyses illustrate practical implementation and empirical-likelihood diagnostics. The second study extends the correlation framework to a hybrid model containing multiplicative distortion, additive distortion, and classical measurement error. Log-adjusted kernel calibration removes systematic confounding, and variance deattenuation accounts for measurement-error inflation. The resulting measurement-error corrected log-adjusted residual estimator is used with jackknife and adjusted jackknife empirical likelihood. The stated regularity conditions establish consistency and asymptotic normality of the ME-LAR estimator, together with chi-square limits for the likelihood-ratio statistics. Simulations evaluate bias, root mean squared error, coverage, and interval length across measurement-error strengths. An air-quality sensor analysis illustrates replicate-free sensitivity analysis and the interpretation of estimates near the correlation boundary. The third study evaluates least absolute deviations (LAD), equivalently minimization of the sum of absolute deviations, as a robust sensitivity-analysis criterion for time-varying susceptible–exposed–infectious–removed models fitted to reporting-interval incidence. Three positive transmission families are considered: cosine turning-point, exponential, and logistic decline. In a phase-aware Monte Carlo study with 1,000 paired replications for each of 60 conditions, paired 95% Monte Carlo intervals favored LAD in 136 of 240 comparisons for mean absolute error and 139 for weighted interval score, while LSQ was favored in 46 and 47 comparisons, respectively. LAD was most often favored under isolated spikes, backlog release, and temporary underreporting followed by delayed release. In rolling-origin forecasts for four synthetic RAPIDD Ebola scenarios, simple baselines achieved the lowest weighted interval score in all 20 scenario–horizon cells, and logistic-decline LSQ was the strongest SEIR specification in all 20. Predictive-uncertainty and sensitivity diagnostics distinguish forecast accuracy, interval calibration, and biological-parameter identifiability. The findings support LAD as a targeted robustness check alongside LSQ. Together, the studies provide a unified perspective on robust inference when the observation process is distorted or irregular.