问一道题：NO.PZ2020010304000052

问题如下：

If a p-VaR model is well specified, HITs should be iid Bernoulli(1 - p). What is the probability of observing two HITs in a row? Can you think of how this could be used to perform a test that the model is correct?

选项：

解释：

The probability of a HIT should be 1 - p if the model is correct. They should also be independent, and so the probability of observing two HITs in a row should be $(1 - p)^2$. They can be formulated as the null that $H_0: E[HIT_i * HIT_{i + 1}] = (1 - p)^2$ and tested against the alternative $H_1: E[HIT_i * HIT_{i + 1}] ≠ (1 - p)^2$. This can be implemented as a simple test of a mean by defining the random variable $X_i = HIT_i * HIT_{i + 1}$ and then testing the null $H_0: \mu_X = (1 - p)^2$ using a standard test of a mean.