Suppose, $n$ units are placed on a life test. The time-to-failure follows a continuous probability distribution with non-existing finite moments(like a lower-truncated Cauchy or inverse Lomax). Let, $X_1, X_2,\cdots, X_n$ be a random sample of size $n$ from a continuous probability distribution with cumulative distribution function (CDF) $F(x)$ and probability density function (PDF) $f(x)$. Let $X_{(1:n)}\leq X_{(2:n)}\leq\cdots\leq X_{(n:n)}$ denote the corresponding order statistics. Obviously, the consecutive failures will be the order statistics $X_{(1:n)}\leq X_{(2:n)}\leq\cdots\leq X_{(n:n)}$ coming from the said distribution.
If we want to find the expected time to complete the test, finding $E(X_{n:n})$ will suffice.
the corresponding $1^{st}$ order moment of maximum order statistics is given by; $$E(X_{n:n})=\mu_{n:n}=n\int_{-\infty}^{\infty}x[F(x)]^{n-1}f(x)dx$$ It has been shown that $\mu_{n:n}$ exists provided that $E(X)$ exists. Even if we can't find the explicit expression for the expectation, the Monte Carlo technique can help to approximate that (courtesy of SLLN). But when the expectation is itself non-existent, even Monte Carlo will suffer. In such cases, what are the alternative ways to measure the average test completion time? Can we obtain other measures like median or mode and will it make sense in the context of the life-testing?
Note: This question seeks the answer within a specific context originated from the discussion in this post.
