The Asymptotic Properties of the Second-Round Estimator in the Newton-Raphson Method

Ordinarily, iteration (4.4.2) is to be repeated until convergence takes place. However, if 0t is a consistent estimator of 60 such that ‘ІТфі — 0O) has a proper

image305 image306 Подпись: (A ~ Є0), Подпись: (4.4.6)

limit distribution, the second-round estimator в2 has the same asymptotic distribution as a consistent root of Eq. (4.1.9). In this case further iteration does not bring any improvement, at least asymptotically. To show this, con­sider

where в* lies between б, and в0. Inserting (4.4.6) into (4.4.2) yields

m – во)- {і – [Щ-ИаШJ} m – e°> <4A7)

Подпись: -[■1 PQ I

Тдж6} Vr дв],;

image310 image311 Подпись: (4.4.8)

But, because under the condition of Theorem 4.1.3

we have

(4.4.9)

which proves the desired result.

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