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Loại 2 ML ước tính phải có ít nhất là tiệm cận hiệu quả như bất kỳ ước tính root-n phù hợp khác đó là tiệm unbiased.4 Vì vậy, ít nhất là trong các mẫu lớn, ước tính khả năng tối đa sở hữu một tài sản tối ưu mà nói chung là không được chia sẻ bởi các phương pháp ước lượng khác . | 482 Discrete and Limited Dependent Variables Suppose that how long a state endures is measured by T a nonnegative continuous random variable with PDF f t and CDF F t where t is a realization of T. Then the survivor function is defined as S t - 1 - F t . This is the probability that a state which started at time t 0 is still going on at time t. The probability that it will end in any short period of time say the period from time t to time t At is Pr t T t At F t At - F t . 11.79 This probability is unconditional. For many purposes we may be interested in the probability that a state will end between time t and time t At conditional on having reached time t in the first place. This probability is Pr t T t At I T t F t At F t S t 11.80 Since we are dealing with continuous time it is natural to divide 11.79 and 11.80 by At and consider what happens as At 0. The limit of 1 At times 11.79 as At 0 is simply the PDF f t and the limit of 1 At times 11.80 is 11.81 IMSfA f7L. vl - S t 1 - F t The function h t defined in 11.81 is called the hazard function. For many purposes it is more interesting to model the hazard function than to model the survivor function directly. Functional Forms For a parametric model of duration we need to specify a functional form for one of the functions F t S t f t or h t which then implies functional forms for the others. One of the simplest possible choices is the exponential distribution which was discussed in Section 10.2. For this distribution f t e Qe et and F t e 1 - e et e 0. Therefore the hazard function is h t f t ee et 0. h t S t e-et e Thus if duration follows an exponential distribution the hazard function is simply a constant. Copyright 1999 Russell Davidson and James G. MacKinnon 11.8 Duration Models 483 Since the restriction that the hazard function is a constant is a very strong one the exponential distribution is rarely used in applied work. A much more flexible functional form is provided by the Weibull distribution which has .
