在R中获得伽玛分布参数（特别是位置参​​数）的MLE

Question

嗨，我想用手估算伽玛分布参数！我知道很多R函数可以估算形状和比例参数，但是似乎很难找到有关估算位置参数的代码。

x <- c(108,91,62,59,84,60,71,105,70,69,66,65,78,83,82,68,107,68,68,69,80,
       75,89,68,64,68,70,57,62,87,51,55,56,57,75,98,60,68,81,47,76,48,63,
       58,40,62,61,58,38,40,45,68,56,64,49,53,50,39,54,47,37,50,54,70,49,
       57,52,47,43,52,57,46,63,56,50,51,50,42,46,56,52,59,45,50,59,44,52,
       54,53,63,45,56,55,53,56,46,45,49,63,50,41,42,53,50,58,50,37,53,58,
       49,53,51,64,44,53,53,55,43,50,60,51,55,56,52,51,45,49,51,63,48,51,
       60,45,40,50,66,62,69,53,54,49,47,63,55,62,57,58,51,50,57,62,45,47,
       52,35,41,53,48,59,45,41,52,36,84,62,31,41,48,47,50,50,57,53,37,46,
       41,56,51,39,59,53,51,49,45,42,32,55,34,43,35,48,33,41,38,57,37,40,
       34,44,43,62,36,41,51,48,31,28,33,35,48,31)
# estimate shape and scale parameter
gamma_likelihood <- function(para){
  sum (  (para[2] -1)*log(x) - para[2]*log(para[1]) - log(gamma(para[2])) - x/para[1] + 1/para[1])
}

MLE = optim(c(10,10), 
            fn = gamma_likelihood, 
            method = "L-BFGS-B", 
            lower = 0.00001, 
            control = list(fnscale = -1), 
            hessian = T 
)
MLE$par


# estimate location, shape and scale parameter
gamma_likelihood <- function(para){
  x = x[x > para[1]]
  sum (  (para[3] -1)*log(x - para[1]) - para[3]*log(para[2]) - 
           log(gamma(para[3])) - x/para[2] + para[1]/para[2] )
}

MLE = optim(c(23,6,7), 
            fn = gamma_likelihood,
            method = 'L-BFGS-B',
            lower = 0.00000001,
            control = list(fnscale = -1)
)
MLE$par

这是我的代码，我可以估算形状和比例参数。但是，当将位置参数添加到对数似然中时。结果似乎不正确。TRUE参数为c（21.4，5.47，6.0）。

Answer 1

如果观测值小于或等于location参数，则该lambda值的整体可能性必须为0（请记住，它是参数的函数，而不是观测值）。

-Inf删除了对特定位置参数没有意义的观察结果，如果函数x中的任何一个应返回0，则函数将返回一个有效数字。 “无效”，因为您有# estimate location, shape and scale parameter gamma_likelihood <- function(para){ if(min(x) < para[1]) return(-Inf) sum ( (para[3] -1)*log(x - para[1]) - para[3]*log(para[2]) - log(gamma(para[3])) - x/para[2] + para[1]/para[2] ) } MLE = optim(c(23,6,7), fn = gamma_likelihood, method = 'L-BFGS-B', lower = 0.00000001, control = list(fnscale = -1) ) MLE$par 的可能性。

这是对数似然函数的更正版本：

[1] 21.161109  5.394343  6.136862

导致：{{1}}