Estimators when mean equals variance

Let $X_1,X_2,…,X_n$, where $n>1$, be a random sample from a $N(\theta,\theta)$ distribution, where $\theta>0$ is an unknown parameter. Suppose $$T_n=\frac1n\sum_{i=1}^n X_i\quad\text{and}\quad S_n=\frac1n\sum_{i=1}^n(X_i-T_n)^2.$$ Then which of the following statements is/are correct?