M_split(q) estimation is a development of M-estimation which is based on the assumption that a functional model of observations can be split into q competitive ones. The main idea behind such an assumption is that the observation set might be a mixture of realizations of different random variables which differ from each other in location parameters that are estimated. The paper is focused on the robustness of M_split(q) estimates against outlying observations. The paper presents derivatives of the general expressions of the respective influence functions and weight functions which are the main basis for theoretical analysis. To recognize the properties of M_split(q) estimates in a better way, we propose considering robustness from two points of view, namely local and global ones. Such an approach is a new one, but it reflects the nature of the estimation method in question very well. Thus, we consider the local breakdown point (LBdP) and the global one (GBdP) that are both based on the maximum sensitivities of the estimates. LBdP describes the mutual relationship between the “neighboring” M_split(q) estimates, whereas GBdP concerns the whole set of the estimates and describes the robustness of the method itself (in more traditional sense). The paper also presents GBdP with an extension, which shows how an outlier might influence M_split(q) estimates. The general theory proposed in the paper is applied to investigate the squared M_split(q) estimation, the variant which is used in some practical problems in geodesy, surveying, remote sensing or geostatistics, and which can also be applied in other geosciences.

中文翻译：

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M split（q）估计的稳健性：一种理论方法

M _split（q）估计是M估计的发展，它基于这样的假设：观测的功能模型可以分为q个竞争模型。这种假设背后的主要思想是，观测集可能是不同随机变量的实现的混合，这些随机变量的估计位置参数彼此不同。本文着重于M _split（q）估计相对于外围观测值的鲁棒性。本文介绍了各个影响函数和权函数的一般表达式的导数，这是理论分析的主要基础。识别M _split（q）的性质为了以更好的方式进行估算，我们建议从本地和全局两个角度考虑稳健性。这种方法是一种新方法，但它很好地反映了所讨论估计方法的性质。因此，我们考虑基于估计的最大敏感度的本地分解点（LBdP）和全局分解点（GBdP）。LBdP描述了“相邻的” M _split（q）估计之间的相互关系，而GBdP涉及整个估计集合并描述了方法本身的鲁棒性（在更传统的意义上）。本文还介绍了带有扩展名的GBdP，它显示了异常值如何影响M _split（q）估计。本文提出的一般理论用于研究平方M_split（q）估计值，它是在大地测量，勘测，遥感或地统计学中的一些实际问题中使用的变体，也可以在其他地球科学中应用。