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The maximum spacing method. An estimation method related to the maximum likelihood method. (English) Zbl 0545.62006
Approximating the Kullback-Leibler information gives rise to the maximum likelihood method, which can break down when the contributions to the likelihood function have particular properties. It is proved in this paper that another type of approximation leads to the so-called maximum spacing method.
Let $$g(x)$$ be the true density function of random variables and $$F(x,\theta)$$, $$\theta \in \Theta$$, some assigned model distribution functions. The maximum spacing estimate is the parameter value $${\hat \theta}_ n$$ which maximizes the following quantity: $S_ n(\theta)=(n+1)^{-1}\sum^{n+1}_{j=1}\ln((F(\xi_ j,\theta)- F(\xi_{j-1},\theta))(n+1)),$ for the order statistics $$-\infty =\xi_ 0<\xi_ 1<...<\xi_{n+1}=+\infty$$. It is proved that $${\hat \theta}_ n$$ converges in probability to the parameter’s true value as $$n$$ tends to infinity. Some other properties of this method are exhibited, showing for instance that it can be used to check whether the assigned model is incorrect or not, and some applications are given, proving that this method may work when the ML method does not.
Reviewer: B. Bouchon

MSC:
 62B10 Statistical aspects of information-theoretic topics