In the first chapter, we study the problem of nonparametric inference for a hazard ratio function under the constraint of monotonicity. The ratio of the hazard functions of two populations or two strata of a single population plays an important role in time-to-event analysis. Cox regression is commonly used to estimate the hazard ratio under the assumption that it is constant in time, which is known as the proportional hazards assumption. However, this assumption is often violated in practice, and when it is violated, the parameter estimated by Cox regression is difficult to interpret. The hazard ratio can be estimated in a nonparametric manner using smoothing, but smoothing-based estimators are sensitive to the selection of tuning parameters, and it is often difficult to perform valid inference with such estimators. In some cases, it is known that the hazard ratio function is monotone. In this chapter, we demonstrate that monotonicity of the hazard ratio function defines an invariant stochastic order, and we study the properties of this order. Furthermore, we introduce an estimator of the hazard ratio function under a monotonicity constraint. We demonstrate that our estimator converges in distribution to a mean-zero limit, and we use this result to construct asymptotically valid confidence intervals. Finally, we conduct numerical studies to assess the finite-sample behavior of our estimator, and we use our methods to estimate the hazard ratio of progression-free survival in pulmonary adenocarcinoma patients treated with gefitinib or carboplatin-paclitaxel. In the second chapter, we explore a novel nonparametric inference approach for a debiased kernel density estimator. Kernel density estimation is one of the most popular nonparametric methods for estimating probability density functions. However, it is well-known that kernel density estimators are biased. The robust bias correction approach proposed by Calonico et al. (2018) can effectively reduce this bias, leading to substantial improvements in confidence interval coverage. However, bias correction can result in negative density estimates. In this section, we propose bias correction and inference for kernel density estimators on the log density scale, which ensures positive density estimates wherever the original kernel density estimator is positive. We demonstrate our estimator is within oP(n−1) of the bias corrected estimator of Calonico et al. (2018), and that the t-statistic constructed with the logarithm-transformed estimator exhibits higher coverage accuracy compared to the t-statistic for the bias corrected estimator. Finally, we use an Edgeworth expansion of our estimator to demonstrate that the proposed approach yields the same rate of coverage error as that of Calonico et al. (2018). We conduct numerical studies illustrating the practical performance of our methods compared to ordinary and bias-corrected kernel density estimators. In the third chapter, we consider improving the monotonicity-constrained nonparametric inference with debiased kernel smoothing. The property of monotonicity plays an important role when dealing with survival data or regression relationships, and it is desired to have one estimator that is both monotone and smooth. However, monotonicity-constrained estimators can suffer from issues such as significant boundary bias, slower convergence rates, and lack of smoothness. Simply combining a monotone estimator with kernel smoothing can exacerbate these problems, leading to increased bias, loss of smoothness, and loss of monotonicity. In this section, our new method projects a debiased local linear regression estimator onto a monotonicity-constrained spline smoother. This resulting estimator adheres to shape constraints, ensures smoothness, achieves uniform consistency, reduces bias, and maintains a satisfactory rate of convergence. In the numerical study, we use bootstrap to demonstrate the superior performance of our estimator compared to the local linear estimator.