Impact of Dependence on Parameter Estimates of Autoregressive Process with Gumbel Distributed Innovation

IF 0.3 Q4 MATHEMATICS
S. Bako, M. Adam, A. Fitrianto
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引用次数: 1

Abstract

Recent studies have shown that independent identical distributed Gaussian random variables is not suitable for modelling extreme values observed during extremal events. However, many real life data on extreme values are dependent and stationary rather than the conventional independent identically distributed data. We propose a stationary autoregressive (AR) process with Gumbel distributed innovation and characterise the short-term dependence among maxima of an (AR) process over a range of sample sizes with varying degrees of dependence. We estimate the maximum likelihood of the parameters of the Gumbel AR process and its residuals, and evaluate the performance of the parameter estimates. The AR process is fitted to the Gumbel-generalised Pareto (GPD) distribution and we evaluate the performance of the parameter estimates fitted to the cluster maxima and the original series. Ignoring the effect of dependence leads to overestimation of the location parameter of the Gumbel-AR (1) process. The estimate of the location parameter of the AR process using the residuals gives a better estimate. Estimate of the scale parameter perform marginally better for the original series than the residual estimate. The degree of clustering increases as dependence is enhance for the AR process. The Gumbel-AR(1) fitted to the Gumbel-GPD shows that the estimates of the scale and shape parameters fitted to the cluster maxima perform better as sample size increases, however, ignoring the effect of dependence lead to an underestimation of the parameter estimates of the scale parameter. The shape parameter of the original series gives a superior estimate compare to the threshold excesses fitted to the Gumbel-GPD.
依赖性对Gumbel分布创新自回归过程参数估计的影响
最近的研究表明,独立的同分布高斯随机变量不适合模拟极端事件中观测到的极端值。然而,现实生活中许多关于极值的数据是相关的和平稳的,而不是传统的独立的同分布数据。我们提出了一个具有Gumbel分布式创新的平稳自回归(AR)过程,并描述了在不同依赖程度的样本量范围内(AR)过程最大值之间的短期依赖关系。我们估计了Gumbel AR过程参数及其残差的最大似然,并评估了参数估计的性能。AR过程拟合到gumbel - generalized Pareto (GPD)分布,我们评估了拟合到簇最大值和原始序列的参数估计的性能。忽略依赖性的影响会导致高估Gumbel-AR(1)过程的位置参数。利用残差对AR过程的位置参数进行估计,得到了较好的估计。原始序列的尺度参数估计值比残差估计值稍好。AR过程的聚类程度随着依赖性的增强而增加。拟合到Gumbel-GPD的Gumbel-AR(1)表明,随着样本量的增加,拟合到簇最大值的尺度和形状参数的估计表现得更好,然而,忽略依赖性的影响会导致尺度参数的参数估计被低估。与Gumbel-GPD拟合的阈值过量相比,原始序列的形状参数给出了更好的估计。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Matematika
Matematika MATHEMATICS-
自引率
25.00%
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0
审稿时长
24 weeks
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