Strong consistency of the maximum spacing estimate. (English) Zbl 0921.62025

Ekström, Magnus

Theory Probab. Math. Stat. 55, 55-73 (1997) and Teor. Jmovirn. Mat. Stat. 55, 55-72 (1996).

Let \(\xi_1\),…,\(\xi_n\) be an i.i.d. sample with distribution \(F_{\theta^0}(x)\), \(x\in R\), where \(\theta^0\in\Theta\subset R^k\) is an unknown parameter. Let \(-\infty=\xi_{(0)}<\xi_{(1)}<\dots<\xi_{(n)}<\xi_{(n+1)}=\infty\) be the order statistics. The maximum spacing estimate (MSP) \(\hat\theta\) is defined as the value of a parameter that maximizes \[ S_n(\theta)=(n+1)^{-1}\sum_{j=1}^{n+1} \log\left( (n+1)(F_\theta(\xi_{(j)})-F_\theta(\xi_{(j-1)})\right). \] Convergence of \(\hat\theta_n\) to \(\theta_0\) a.s. as \(n\to\infty\) is proved under some mild conditions on \(F_\theta\). It is demonstrated that MSP estimates are consistent for some nonregular problems when MLE fails. This is the case of estimation of the mixture parameters \( F_\theta(x)=0.5\Phi(x)+0.5\Phi((x-\mu)/\sigma) \), \(\theta=(\mu,\sigma)\in R\times R^{+}\), and the case of estimation of \(\gamma\)-distribution parameters.

Reviewer: R.E.Maiboroda (Kyiv)

MSC:

62F12	Asymptotic properties of parametric estimators
62F10	Point estimation

Keywords:

consistency; maximum spacing estimates; nonregular estimation problems