Exchangeability

Exchangeable random variables

Prediction without covariates

Lemma 1

Suppose $Z_1, \cdots Z_n$ are exchangeable random variables. For any $\alpha \in \{0, 1\}$, $P[Z_n \leq \hat{Q}_n(\alpha)] \geq \alpha$

Moreover, if $Z_1, \cdots Z_{n}$ are a.s distinct, $P[Z_{n} \leq \hat{Q}_{n+1}(\alpha)] \leq \alpha + \frac{1}{n}$

• key proof: using the uniform distribution or these order statistics?

Lemma 2

Suppose $Z_1, \cdots Z_{n+1}$ are exchangeable random variables. For any $\alpha \in \{0, 1\}$, define $\alpha_n$ as $\alpha_n = (1 + \frac{1}{n})\alpha$. Then, $P[Z_{n+1} \leq \hat{Q}_{n+1}(\alpha_n)] \geq \alpha$

Moreover, if $Z_1, \cdots Z_{n+1}$ are a.s distinct, $P[Z_{n+1} \leq \hat{Q}_{n+1}(\alpha_n)] \leq \alpha + \frac{1}{n}$.

• Key of the proof:

  • use lemma 1 and change n points to (n+1) points of these order statistics

  • and you cannot jump for two points since you only added one point into the empirical distribution

One-sided prediction interval without covariates