Relationship between Derivatives of the Observed and Full Loglikelihoods and Application to Newton-Raphson Algorithm

IF 1.2 4区数学

International Journal of Biostatistics Pub Date : 1900-01-01 DOI:10.2202/1557-4679.1010

D. Commenges, V. Rondeau

引用次数: 0

Abstract

In the case of incomplete data we give general relationships between the first and second derivatives of the loglikelihood relative to the full and the incomplete observation set-ups. In the case where these quantities are easy to compute for the full observation set-up we propose to compute their analogue for the incomplete observation set-up using the above mentioned relationships: this involves numerical integrations. Once we are able to compute these quantities, Newton-Raphson type algorithms can be applied to find the maximum likelihood estimators, together with estimates of their variances. We detail the application of this approach to parametric multiplicative frailty models and we show that the method works well in practice using both a real data and a simulated example. The proposed algorithm outperforms a Newton-Raphson type algorithm using numerical derivatives.

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观测似然与全对数似然的导数关系及其在Newton-Raphson算法中的应用

在数据不完整的情况下，我们给出对数似然的一阶导数和二阶导数相对于完整和不完整观测设置之间的一般关系。在这些量很容易计算完整观测设置的情况下，我们建议使用上述关系计算不完整观测设置的模拟量:这涉及数值积分。一旦我们能够计算这些量，Newton-Raphson型算法就可以应用于找到最大似然估计量，以及它们的方差估计。我们详细介绍了该方法在参数乘法脆弱性模型中的应用，并通过实际数据和模拟示例表明该方法在实践中效果良好。该算法优于使用数值导数的Newton-Raphson型算法。

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

International Journal of Biostatistics Mathematics-Statistics and Probability

CiteScore

2.30

自引率

8.30%

发文量

期刊介绍： The International Journal of Biostatistics (IJB) seeks to publish new biostatistical models and methods, new statistical theory, as well as original applications of statistical methods, for important practical problems arising from the biological, medical, public health, and agricultural sciences with an emphasis on semiparametric methods. Given many alternatives to publish exist within biostatistics, IJB offers a place to publish for research in biostatistics focusing on modern methods, often based on machine-learning and other data-adaptive methodologies, as well as providing a unique reading experience that compels the author to be explicit about the statistical inference problem addressed by the paper. IJB is intended that the journal cover the entire range of biostatistics, from theoretical advances to relevant and sensible translations of a practical problem into a statistical framework. Electronic publication also allows for data and software code to be appended, and opens the door for reproducible research allowing readers to easily replicate analyses described in a paper. Both original research and review articles will be warmly received, as will articles applying sound statistical methods to practical problems.