第六讲极大似然估计.docVIP

第六讲 极大似然估计 The Likelihood Function and Identification of the Parameters (极大似然函数及参数识别) 1、似然函数的表示 在具有n个观察值的随机样本中，每个观察值的密度函数为。由于n个随机观察值是独立的，其联合密度函数为 函数被称为似然函数，通常记为，或者。 The probability density function, or pdf for a random variable y, conditioned on a set of parameters, , is denoted . This function identifies the data generating process that underlies an observed sample of data and, at the same time, provides a mathematical description of the data that the process will produce. The joint density of n independent and identically distributed (iid) observations from this process is the product of the individual densities; (17-1) This joint density is the likelihood function, defined as a function of the unknown parameter vector, , where is used to indicate the collection of sample data. Note that we write the joint density as a function of the data conditioned on the parameters whereas when we form the likelihood function, we write this function in reverse, as a function of the parameters, conditioned on the data. Though the two functions are the same, it is to be emphasized that the likelihood function is written in this fashion to highlight our interest in the parameters and the information about them that is contained in the observed data. However, it is understood that the likelihood function is not meant to represent a probability density for the parameters as it is in Section 16.2.2. In this classical estimation framework, the parameters are assumed to be fixed constants which we hope to learn about from the data. It is usually simpler to work with the log of the likelihood function: . (17-2) Again, to emphasize our interest in the parameters, given the observed data, we denote this function . The likelihood function and its logarithm, evaluated at , are sometimes denoted simply and , respectively or, where no ambiguity can arise, just or . It will usually be necessary to generalize the concept of the likelihood function to allow the density to de