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Quoting Weerahandi, Generalized Confidence Intervals (1993):

  • Confidence interval (Property 1) --- Consider a particular situation of interval estimation of a parameter $\theta$. If the same experiment is repeated a large number of times to obtain new sets of observations, then the $95\%$-confidence intervals will correctly include the true value of the parameter $\theta$ $95\%$ of the time.

  • Generalized confidence interval (Property 2) --- After a large number of independent situations of setting (generalized) $95\%$-confidence intervals for certain parameters of interest, the investigator will have correctly included the true values of the parameters in the corresponding intervals $95\%$ of the time.

  • Property 1 implies Property 2.

This definition of a generalized confidence interval is not clear to me. Is there a mathematical, rigorous phrasing of Property 2? This is my main question.

I'm also not able to figure out what this property means in rigorous terms with the help of the mathematical construction of generalized confidence intervals (using generalized pivotal quantities). How can we "see" Property 2 from this construction?

It is also not clear to me why Property 2 does not imply Property 1. If the same expriment is repeated, these are not two "independent situations" in the sense of Property 2?

Note: It is known that Property 2 does not imply Property 1; a GCI is not a CI in general.


Edit

Theorem 2.1 of this paper gives more information. Its statement is the following one.

For every integer $k \geq 1$, let ${\cal M}_k$ be a statistical model with unknown parameter $\theta_k$. The ${\cal M}_k$'s have independent sample spaces.

Assume there are some observations for each ${\cal M}_k$ and denote by $I_k$ a corresponding generalized $95\%$-confidence interval.

Set $\delta_k=1$ if $\theta_k \in I_k$ and $\bar\delta_n = \frac{\sum_{k=1}^n\delta_k}{n}$. Then $\Pr(\lim_{n\to\infty}\bar\delta_n = 95\%)=1$. That is, as $n\to\infty$, the mean number of the GCIs $I_k$ containing $\theta_k$ tends to $95\%$ almost surely.

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The two properties imply each other. Indeed, the implication is nearly trivial provided we formulate them mathematically, as you have requested: let's begin there.

I would like to remark that the language is confusing because it is attempting to make statements about probabilities by referring to "a large number of": in other words, it is appealing implicitly to laws of large numbers to equate probabilities with asymptotic frequencies in sequences of independent trials. I will avoid such solecisms by translating these statements into what I think they are trying to say about probabilities.

For background, let us understand a "situation" to consist of collecting $n$ independent observations $\mathbf{x}=(x_1, x_2, \ldots, x_n)$ from some distribution $F$ assumed to lie within a specific family $\Omega$ of distributions.

A "parameter" is a function $\delta:\Omega\to\mathbb{R}$.

An "interval" can be represented as an indexed pair of functions $l_n,u_n:\mathbb{R}^n\to\mathbb{R}$ with the restriction that $l_n(\mathbf{x})\le u_n(\mathbf{x})$ for all $n\ge 1$ and all $\mathbf{x}\in\mathbb{R}^n$. For any $\mathbf{x}$, these functions determine the interval $[l_n(\mathbf{x}), u_n(\mathbf{x})]$.

  1. A $1-\alpha$ confidence interval for $\delta$ is a pair $l_n, u_n$ for which $${\Pr}_F(\delta(F)\in [l_n(\mathbf{x}), u_n(\mathbf{x})])=1-\alpha$$ for all $F\in\Omega$ and all $n\ge 1$.

  2. Consider a nonempty set of distributional families and parameters $\mathcal{S}=\{\delta_i:\Omega_i\to\mathbb{R}\}$. This models "a large number of independent situations." A $1-\alpha$ generalized confidence interval for $\mathcal{S}$ is a collection of functions $l_{i;n}, u_{i;n}$ indexed by $i\in\mathcal{S}$ and integers $n\ge 1$ such that for any sequence of length $s$ of samples $\mathbf{x}_j$ of sizes $n_j$ taken from $\Omega_j$, $$\frac{1}{s}\sum_{j=1}^s {\Pr}_F(\delta_{i_j}(\mathbf{x}_j)\in [l_{i_j;n_j}(\mathbf{x}_j), u_{i_j;n_j}(\mathbf{x}_j)])=1-\alpha.\tag{2}$$for all sequences $(F_j)$ with $F_j\in\Omega_j$.

By taking $\mathcal{S}=\{\delta\}$, $(2)$ trivially implies $(1)$. To go the other way, let $l_{i_j;n},u_{i_j;n}$ be a $1-\alpha$ confidence interval for each situation $j$: since all the probabilities appearing in formula $(2)$ exactly equal $1-\alpha$, the mean (on the left) also is $1-\alpha$.

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  • $\begingroup$ Hello whuber. It is not true that a generalized confidence interval is a confidence interval. Hence Property 2 should not imply Property 1. I should have mentioned this point in my post. $\endgroup$ – Stéphane Laurent May 14 '17 at 18:17
  • $\begingroup$ Thank you. In that case, your quotation does not provide a sufficiently clear definition. Could you provide any information to make it less ambiguous? $\endgroup$ – whuber May 14 '17 at 22:05
  • $\begingroup$ Hi whuber. I can provide the definition of the construction of a GCI, but I'm not sure this will help. I'll do it later. $\endgroup$ – Stéphane Laurent May 26 '17 at 10:56
  • $\begingroup$ Although a construction could help illustrate the GCI, what we need is a sufficiently clear definition. $\endgroup$ – whuber May 26 '17 at 12:23
  • $\begingroup$ Sure. But if I had a clear definition I would not have opened this question ^^ $\endgroup$ – Stéphane Laurent May 26 '17 at 13:02

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