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We discuss the estimation of hazard rates under random censoring with the kernel method. Two practically relevant problems that occur when applying unmodified kernel estimators are boundary effects near the endpoints of the support of the hazard rate, and a substantial increase in the variance from left to right over the range of abscissae where the hazard rate is estimated. A new class of boundary kernels is proposed for the first problem. Explicit formulas for these kernels are developed, and it is shown that this boundary correction works well in practice. A data-adaptive varying bandwidth selection procedure is proposed for the second problem. This procedure generally will lead to increasing bandwidths near the left endpoint and toward the right endpoint, and will lead to smaller integrated mean squared error of the hazard rate estimator as compared to a fixed bandwidth method. A practically feasible method incorporating the new boundary kernels and local bandwidth choices is implemented and illustrated with survival data from a leukemia study.