单位指数分布的偏差校正方法及其应用.docx

东北石油大学本科生毕业设计（论文）

摘要

在诸多领域涉及比例变量建模时，单位指数分布具有重要应用价值，但小样本条件下，其最大似然估计量的偏差会引发显著估计误差。为获取更可靠的估计量，本文展开深入研究。首先，提出最大似然法与参数自助偏差校正方法，详细剖析推导实现偏差修正最大似然估计方法的具体过程，并给出获取偏差校正自助估计器的步骤。其次，通过蒙特卡罗模拟实验，对两种偏差校正方法进行全面性能评估，结果表明，所提方法对大部分参数组合，均可显著降低最大似然估计量的偏差与均方误差。再者，采用土壤水分数据集与数值示例进行实例分析，进一步验证方法在实际应用中的有效性。本研究在理论上完善了单位指数分布估计的理论架构，在实践中为相关领域提供了更精准可靠的分析工具，对推动单位指数分布在各领域的应用具有重要意义。

关键词：单位指数分布、最大似然估计、偏差校正、自助法、蒙特卡洛模拟

Abstract

Whenmodelingproportionalvariablesinmanyfields,unitexponentialdistributionhasimportantapplicationvalue,butundersmallsampleconditions,thedeviationofitsmaximumlikelihoodestimatorcancausesignificantestimationerrors.Toobtainmorereliableestimates,thisarticleconductsin-depthresearch.Firstly,themaximumlikelihoodmethodandparameterbootstrapbiascorrectionmethodareinnovativelyproposed,andthespecificprocessofimplementingthebiascorrectedmaximumlikelihoodestimationmethodisanalyzedandderivedindetail.Thestepsforobtainingthebiascorrectedbootstrapestimatorarealsoprovided.Secondly,throughMonteCarlosimulationexperiments,acomprehensiveperformanceevaluationwasconductedonthetwobiascorrectionmethods.Theresultsshowedthattheproposedmethodcansignificantlyreducethebiasandmeansquareerrorofthemaximumlikelihoodestimatorformostparametercombinations.Furthermore,asoilmoisturedatasetandnumericalexampleswereusedforcaseanalysistofurthervalidatetheeffectivenessofthemethodinpracticalapplications.Inaddition,correspondingoptimizationstrategiesareproposedforcomplexdatafeaturesthatmayoccurinpracticalapplications,suchasoutliers,datamissing,andseasonalfluctuations.Thisstudyhastheoreticallyimprovedthetheoreticalframeworkforestimatingunitindexdistributionsandprovidedmoreaccurateandreliableanalyticaltoolsforrelatedfieldsinpractice,whichisofgreatsignificanceforpromotingtheapplicationofunitindexdistributionsinvari