混合對(duì)數(shù)正態(tài)分布最大值的極限分布

黃建文, 羅遠(yuǎn)峰

(遵義師范學(xué)院數(shù)學(xué)與計(jì)算科學(xué)學(xué)院,貴州遵義563002)

1 預(yù)備知識(shí)

對(duì)于極值理論中指定分布的最大值分布的極限分布以及相應(yīng)的賦范常數(shù)的研究是一個(gè)重點(diǎn)問(wèn)題和難點(diǎn)問(wèn)題.這一問(wèn)題,有部分學(xué)者對(duì)此做了比較深入的研究和應(yīng)用[1-8],對(duì)數(shù)正態(tài)分布的概率密度函數(shù)定義如下:對(duì)于x＞0

混合對(duì)數(shù)正態(tài)分布在現(xiàn)代統(tǒng)計(jì)的發(fā)展過(guò)程中作為一個(gè)模型得到廣泛深入的研究和應(yīng)用.關(guān)于混合對(duì)數(shù)正態(tài)分布的應(yīng)用已有很多研究成果:M.Yang[10]對(duì)于模擬峰度和偏度研究標(biāo)準(zhǔn)的對(duì)數(shù)正態(tài)混合模型；H.Chrisfina等[11]考慮在一個(gè)保險(xiǎn)政策中,假定索賠金額具有混合對(duì)數(shù)正態(tài)分布,研究得出給定參數(shù)值的中位數(shù)的分布服從對(duì)數(shù)正態(tài)分布；Y.Tian等[12]在均方損失函數(shù)下通過(guò)抽樣算法研究對(duì)數(shù)正態(tài)分布混合模型的參數(shù)估計(jì),而且通過(guò)一些模擬表明貝葉斯算法對(duì)他們的模型是有效的.S.Jamaludin等[13]引入幾種類型的混合分布在描述和擬合馬來(lái)西亞半島每天降雨量的數(shù)據(jù)時(shí),通過(guò)分析和檢驗(yàn),結(jié)果發(fā)現(xiàn)混合對(duì)數(shù)正態(tài)模型被認(rèn)為是最佳的模型.L.Dietmar[14]構(gòu)造出在Merton模型(Black-Scholes模型帶跳)和隨機(jī)波動(dòng)模型中近似終端分布的混合對(duì)數(shù)正態(tài)分布,并將其應(yīng)用于衍生品的定價(jià)和風(fēng)險(xiǎn)管理等方面.

本文主要研究同服從混合對(duì)數(shù)正態(tài)分布的獨(dú)立隨機(jī)變量、最大值的極限分布及其相應(yīng)的規(guī)范化常數(shù).主要結(jié)果給出了這一重要分布最大值的漸近形式.這2個(gè)命題的結(jié)論可以用于多元數(shù)據(jù)處理和參數(shù)估計(jì)中理論分析研究.

為了得到本文的主要結(jié)論,下面給出幾個(gè)引理.這2個(gè)引理來(lái)自于參考文獻(xiàn)[15].

2 輔助引理

3 主要結(jié)果

4 證明

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