STA305H1 Lecture Notes - Lecture 6: Maximum Likelihood Estimation, Pareto Distribution, Semiparametric Model
18 Jan 2018
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Document Summary
Suppose that x1, , xn are i. i. d. non-negative random variables whose distribution function. = 1 (2) for each t > 0. Simple examples of slowly varying functions are l(x) = ln(x) and l(x) = constant; more importantly, the condition (2) guarantees that for any > 0, l(x) x and. L(x) x for large values of x. The model (1) is often used as a semi-parametric model for heavy tailed data, that is, data that exhibit more extreme values than might be seen, for example, from a normal or exponential distribution. Some examples include stock returns (and other nancial data) as well as le transmission times over a network. Moreover, the value of also describes the distribution of the next record value given the current record value x(n) = y from the rst n observations x1, , xn. If xn+1, xn+2, are subsequent observations, we de ne z to be the next record, that is, the rst value of.
