Limit theorems for the maximum likelihood estimate under general multiply Type II censoring

Abstract

Assume n items are put on a life-time test, however for various reasons we have only observed the r ₁-th,..., r _k-th failure times % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiEamaaBaaaleaamiaadkhadaWgaaqaaSGaaGymaiaacYcaaWqa% baGaamOBaiaacYcacaGGUaGaaiOlaiaac6caaSqabaGccaGGSaGaam% iEamaaBaaaleaamiaadkhadaWgaaqaaSGaam4AaiaacYcaaWqabaGa% amOBaaWcbeaaaaa!48BB!\[x_{r_{1,} n,...} ,x_{r_{k,} n} \]with % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaGimaiabgsMiJkaadIhadaWgaaWcbaadcaWGYbWaaSbaaeaaliaa% igdacaGGSaaameqaaiaad6gaaSqabaGccqGHKjYOcqWIVlctcqGHKj% YOcaWG4bWaaSbaaSqaaWGaamOCamaaBaaabaWccaWGRbGaaiilaaad% beaacaWGUbaaleqaaeXatLxBI9gBaGqbaOGae8hpaWJaeyOhIukaaa!521B!\[0 \le x_{r_{1,} n} \le \cdots \le x_{r_{k,} n} > \infty \]. This is a multiply Type II censored sample. A special case where each x _ri ,n goes to a particular percentile of the population has been studied by various authors. But for the general situation where the number of gaps as well as the number of unobserved values in some gaps goes to ∞, the asymptotic properties of MLE are still not clear. In this paper, we derive the conditions under which the maximum likelihood estimate of θ is consistent, asymptotically normal and efficient. As examples, we show that Weibull distribution, Gamma and Logistic distributions all satisfy these conditions.