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1、如何求解参数的矩估与极大似然估计一、矩估计若统计量作为总体参数（或 g()）的估计时，就称为（或 g()）的估计量。定义6.1 矩估计量设nXXX,21是总体的样本，的分布函数),:(1kxF依赖于参数k,1，假定 X 的 r 阶矩为),(1krrEX,1kr（或 r 阶中心矩）相应的样本矩记为),(1nrXXA如下的 k 个议程kraXXAkrnr,1),(),(11（6.1）的解，称为未知参数k,:1的矩估计。二、最（极）大似然估计设总体的密度函数),(xf是参数或参数向量，nXXX,21是该总体的样本，对给定的一组观测值nxxx,21，其联合密度是的函数，又称似然函数，记为：nkknxf
2、xxLL11),(),()(其中为参数集，若存在,),(?1nxx使),()?(LL就称),(?1nxx是的最大似然估计值，而),(?1nXX是的最大似然估计量。注：）对给定的观测值，)(L是的函数，最大似然估计的原理是选择使观测值nxxx,21出现的“概率”达到最大的?作为的估计。）最大似然估计具有不变性，即若?是的最大似然估计，则)(g的最大似然估计为)?(g。但是，矩估计不具有不变性，例如假定是X的矩估计，一般情形下，2的矩估计不是2X。1.设总体服从指数分布，其概率密度函数为0001)(1xxexfx，（0）试求参数的矩估计和极大似然估计.解：的概率密度为1,0;,00,0 xexfx
3、x似然函数为：11ixniLe11111niiixnxnniee而11lnlnniiLnx令21ln110niidLnxd得到：11?niixn=X因此得到参数的极大似然估计量为：11?niiXn矩估计求法如下：因为1E令111niiAxn则11?niixn从而的矩估计量为11?niiXn=X2.设母体具有指数分布，密度函数为000),(xxexfx，（0）试求参数的矩估计和极大似然估计.解：参数的矩估计求法为：因为文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:C
4、R10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S
5、4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1
6、HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S
7、6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4
8、 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2
9、D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3
10、文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q311E令：1111niiAxn则的矩估计量为：111?niinAX极大似然估计求法如下：的概率密度为,0,0,0 xexfxx似然函数为：1,0inxiLex而1lnlnniiLnx令1ln0niidLnxd解得的极大似然估计量为：1?niinx3.设总体 X N(,1),),(1nXX为来自 X 的一个样本，试求参数的矩估计和最大似然估计.解：
11、矩估计求法为：1E X令111niiAxn则11?niixn极大似然估计求法为：文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR1
12、0Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V
13、9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY
14、5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N
15、10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 Z
16、L7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3
17、O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3X 的概率密度为：221;2xfxe似然函数为：22112ixniLe211222nii
18、xne而211lnln 222niinLx令1ln1202niidLxd即10niix解得的极大似然估计量为：11?niixn文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N1
19、0S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL
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22、码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10Y3S4V9X1 HY5V6S6N10S4 ZL7M2D3O8Q3文档编码:CR10
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