We discuss a new kernel type estimator for density function \(f_X(x)\) with nonnegative support. Here, we use a type of gamma density as a kernel function and modify it with expansions of exponential and logarithmic functions. Our modified gamma kernel density estimator is not only free of the boundary bias, but the variance is also in smaller orders, which are \(O(n^{-1}h^{-1/4})\) in the interior and \(O(n^{-1}h^{-3/4})\) in the boundary region. Furthermore, the optimal orders of its mean squared error are \(O(n^{-8/9})\) in the interior and \(O(n^{-8/11})\) in the boundary region. Simulation results that demonstrate the proposed method’s performances are also presented.

中文翻译：

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新型伽玛核密度估计器

我们讨论了具有非负支持的密度函数\（f_X（x）\）的新内核类型估计器。在这里，我们使用一种伽玛密度作为核函数，并用指数和对数函数的扩展对其进行修改。我们的改性伽马核密度估计器是不仅自由边界偏置的，但方差也更小的订单，这是\（O（N ^ { - 1} H ^ { - 1/4}）\）在内部和\（O（n ^ {-1} h ^ {-3/4}）\）在边界区域中。此外，它的均方误差的最佳顺序是\（O（N ^ { - 8/9}）\）在内部和\（O（N ^ { - 8/11}）\）在边界区域中。仿真结果也证明了该方法的性能。