Maximum Likelihood Estimation under Right-Censored Exponential Data

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Survival Analysis / MLE.

Your Task
Let $T_1, \dots, T_n$ be independent failure times distributed exponentially with rate $\lambda$ ($f(t) = \lambda e^{-\lambda t}, S(t) = e^{-\lambda t}$). The study terminates at a fixed time $C > 0$ (Type-I right censoring). For each unit $i$, we observe $Y_i = \min(T_i, C)$ and the event indicator $\delta_i = \mathbb{I}_{\{T_i \le C\}}$.
(1) Derive the likelihood function and log-likelihood function.
(2) Derive the Maximum Likelihood Estimator $\hat{\lambda}$ and its asymptotic variance.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.