1. (BH 10.25) An important concept in frequentist statistics is that of a confidence interval (CI)....

 

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1. (BH 10.25) An important concept in frequentist statistics is that of a confidence interval (CI). Suppose we observe data X from a distribution with parameter 0. Unlike in Bayesian statistics, is treated as a fixed but unknown constant; it is not given a prior distribution. A 95% confidence interval consists of a lower bound L(X) and upper bound U(X) such that P(L(X) < 0 < U(X)) = 0.95 for all possible values of 0. In the above statement, L(X) and U(X) are random variables, as they are functions of the r.v. X, whereas is a constant. The definition says that the random interval (L(X), U(X)) has a 95% chance of containing the true value of 0. Imagine an army of frequentists all over the world, independently generating 95% CIs. The jth frequentist observes data X, and makes a confidence interval for the parameter 0₁. Show that if there are n of these frequentists, then the fraction of their intervals which contain the corresponding parameter approaches 0.95 as n→∞0. Hint: Consider the indicator r.v. I; = I(L(X;) < 0; < U(X;)). j j 1. (BH 10.25) An important concept in frequentist statistics is that of a confidence interval (CI). Suppose we observe data X from a distribution with parameter 0. Unlike in Bayesian statistics, is treated as a fixed but unknown constant; it is not given a prior distribution. A 95% confidence interval consists of a lower bound L(X) and upper bound U(X) such that P(L(X) < 0 < U(X)) = 0.95 for all possible values of 0. In the above statement, L(X) and U(X) are random variables, as they are functions of the r.v. X, whereas is a constant. The definition says that the random interval (L(X), U(X)) has a 95% chance of containing the true value of 0. Imagine an army of frequentists all over the world, independently generating 95% CIs. The jth frequentist observes data X, and makes a confidence interval for the parameter 0₁. Show that if there are n of these frequentists, then the fraction of their intervals which contain the corresponding parameter approaches 0.95 as n→∞0. Hint: Consider the indicator r.v. I; = I(L(X;) < 0; < U(X;)). j j

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ANS WER Let I j denote the indicator random variable for the j th confidence interval cont... View the full answer