Set $Y_i=X_i/i$ . Since $X_i\sim\operatorname{U}(0,i\theta)$ , $Y_i\sim\operatorname{U}(0,\theta)$ for every $i$ , and the $Y_i$ are independent: so $Y_1,\dots,Y_n$ are iid $\operatorname{U}(0,\theta)$ .

(A): $E(Y_i)=\theta/2$ , $\operatorname{Var}(Y_i)=\theta^2/12$ for every $i$ (the factor $1/i^2$ exactly cancels the growth of $\operatorname{Var}(X_i)=(i\theta)^2/12$ ), so $\frac2n\sum Y_i$ is unbiased for $\theta$ with variance $\frac{4}{n^2}\cdot n\cdot\frac{\theta^2}{12}=\frac{\theta^2}{3n}\to0$ : this estimator is consistent, so (A)'s claim of inconsistency is false.

(B): $E(\bar X)=\frac1n\sum\frac{i\theta}2=\frac\theta4(n+1)\to\infty$ as $n\to\infty$ — the sample mean's expectation diverges, so it cannot be consistent for the fixed value $\theta$ . False.

(D): $X_{(n)}\ge X_n\sim\operatorname{U}(0,n\theta)$ , whose scale grows without bound, so $X_{(n)}$ stochastically diverges rather than settling near $\theta$ . False.

(C): $\max\{Y_1,\dots,Y_n\}$ is the maximum of $n$ iid $\operatorname{U}(0,\theta)$ variables — the classical MLE of $\theta$ for the uniform location family, well known to converge (in probability, indeed almost surely) to $\theta$ . True.