26. Limit Distribution and Scale-Free Reparameterizationb

Settings

1. Background: In econometric theory proofs, eliminating a nuisance macroscopic parameter (like ) can sometimes be achieved through a mathematical trick called "Scale-Free" or "Reparameterization".

2. Condition: This trick works perfectly when (single variable, in [Ketz(2019)]), which allows us to utilize the ratio property of the statistic.

3. Objective: Explain why and how the nuisance parameter can be completely absorbed (normalized to 1) under the condition, specifically in the context of the limit distribution involving a truncated maximum function.

Derivation

1. The Physical Essence of the Statistic as a Ratio: The basic structure of a statistic (used for confidence intervals or hypothesis testing) is always: In the limit state (e.g., based on Equation 14):
• Numerator (Estimated Error): Follows a truncated distribution . Here, is a normal random variable with mean 0 and variance .
• Denominator (Standard Error): In the limit, this is the square root of the variance, .

2. Algebraic Transformation (Absorbing the Denominator): Dividing the limit state numerator and denominator yields the statistic in the limit: Since the denominator is a positive number, we can use the property of the maximum function () and directly pull it inside the bracket:

3. Perfect Absorption of (The "Scale-Free" Magic):
• Right Term : A normal variable with variance divided by its standard deviation. This is standard Z-score normalization. After division, it becomes a variable that always follows the standard normal distribution , which we call . The parameter has vanished, and the variance is normalized to 1.
• Left Term : This still contains . However, our ultimate goal is to find the infimum for the worst-case coverage probability (AsySz), which requires to traverse from to . Since is a variable that takes any value in , regardless of whether the constant denominator is 2 or 200, the entire term remains a new variable that takes any value in . We can reparameterize this entire term as a new relative scaling drift parameter, . The final limit statistic becomes:

Conclusion

1. Why is Eliminated: In the new formula, is always a standard normal distribution (variance 1, independent of ). To find the worst-case coverage, we simply let run from to . The parameter has not truly disappeared; it has been completely absorbed by . In the univariate case, changing only scales the coordinate axis, which does not affect the result of finding the infimum (the worst-case scenario).

2. Contrast: Why This Fails for : If (two variance parameters), the underlying is a covariance matrix. Although matrix multiplication can normalize the two diagonal elements to 1, the correlation coefficients (off-diagonal elements) will still remain and be governed by . We cannot use a simple scalar to absorb multi-dimensional correlation. This is why the perfect dimensionality reduction (normalizing the variance-covariance structure to completely rid of ) is a special property unique to the case.