ON THE CONSISTENCY OF MAXIMUM LIKELIHOOD ESTIMATORS BASED ON RANDOMLY CENSORED DATA IN THE WEIBULL DISTRIBUTION CASE

Lel \left(x_\mathrmy, \delta \geqslant 1\right. ) be i.i.d. r. \mathrmv^\prime.s with P\left(x_1 \leqslant x\right)=F(x, a, \lambda), where a>0, \lambda>0, F(x, a, \lambda)-1-e^-2 e^*(x>0), F(\alpha, \alpha, \lambda)=0(x<0) in which both of a and \lambda are unknown. Let \left(y_6, 6 \geqslant 1\right. ) be a sequenee of independent random variables with P\left(y_s \leqslant\right. s) =G_4(x), G_4(0)=0(i>1). Sappoes that the two sequences are independent of each other. Set t_4=\min \left(\alpha_i, y_i\right), \delta_i=I\left(x_k \leqslant y_i\right)(\ \geqslant 1), I(A) being the indicator function of A. As well-known, maximum likelihood estimators \hat\alpha_n and \hat\lambda_n of \alpha, \lambda based on data \left(t_2, \delta_1\right), \cdots,\left(t_n, z_1\right) are the solution of the following equations; \left\\beginarrayl \frac1\alpha+\frac\sum_i^n \delta_1 \ln t_i\sum_i^n \delta_i-\frac\sum_i^n t_i^3 \ln f_i\sum_i^n t_i=0 \\ \lambda-\frac\sum_i^n \delta_i\sum_i^n t_i^n . \endarray\right. In the present paper, we have proved the following theorem: (1) If there exists \alpha>0 such that \lim _n \frac1n \sum_1^n G_i(x)<1, then \hat\alpha_n and \hat\lambda_2 are consistent estimators of \alpha and \lambda respectively. (2) If there exists x>0 such that \varlimsup_n \frac1n \sum_i^n G_i(x)<1 and there exist \operatornamelimital \lim _n \frac1n \sum_1^n. G_1(y) for almost all y w,r.t. Lebeggue measure, then both \hata_n and \hat\lambda_n are strongly consistent.