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A new kernel estimator of hazard ratio and its asymptotic mean squared error

Taku Moriyama, Yoshihiko Maesono

Abstract

The hazard function is a ratio of a density and survival function, and it is a basic tool of the survival analysis. In this paper we propose a kernel estimator of the hazard ratio function, which are based on a modification of Ćwik and Mielniczuk's method. We study nonparametric estimators of the hazard function and compare those estimators by means of asymptotic mean squared error ($AMSE$). We obtain asymptotic bias and variance of the new estimator, and compare them with a naive estimator. The asymptotic variance of the new estimator is always smaller than the naive estimator's, so we also discuss an improvement of $AMSE$ using Terrell and Scott's bias reduction method. The new modified estimator ensures the non-negativity, and we demonstrate the numerical improvement.

A new kernel estimator of hazard ratio and its asymptotic mean squared error

Abstract

The hazard function is a ratio of a density and survival function, and it is a basic tool of the survival analysis. In this paper we propose a kernel estimator of the hazard ratio function, which are based on a modification of Ćwik and Mielniczuk's method. We study nonparametric estimators of the hazard function and compare those estimators by means of asymptotic mean squared error (). We obtain asymptotic bias and variance of the new estimator, and compare them with a naive estimator. The asymptotic variance of the new estimator is always smaller than the naive estimator's, so we also discuss an improvement of using Terrell and Scott's bias reduction method. The new modified estimator ensures the non-negativity, and we demonstrate the numerical improvement.

Paper Structure

This paper contains 7 sections, 3 theorems, 43 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Hazard ratio estimators and asymptotic properties
  3. Hazard ratio estimators
  4. Asymptotic properties
  5. Comparison of kernel hazard estimators
  6. Bias reduction and simulation study
  7. Appendices: Proofs of Theorems

Key Result

Theorem 1

Let us assume that ${\rm (i)}$$f(\cdot)$ is 4 times continuously differentiable at $x_0$, ${\rm(ii)}$$K$ is symmetric and the support is $[-d, d]$ for some positive number $d>0$ and ${\rm(iii)}$$A_{1,4}$ and $A_{2,0}$ are bounded. Then, the $MSE$ of $\widehat{H}(x_0)$ is given by

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3