Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
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Asymptotics and Simulation · Chapter II-05

Stochastic Orders and Competing Sample Sizes

Asymptotic negligibility depends on the normalization used by the estimator. Several sample sizes can grow at different rates, so convergence of a raw error term does not determine its contribution after root-n scaling. Triangular arrays and counterexample sequences make those comparisons explicit.

Conceptual map

  1. II-05.01Convergence notation
  2. II-05.02Statistical scales
  3. II-05.03Triangular arrays
  4. II-05.04Competing rates
  5. II-05.05Counterexample sequences

1. The estimator chooses the relevant normalization

An approximation error can converge to zero and still enter a root-sample-size limit. Raw convergence and normalized negligibility are separate statements.

Let the statistical sample size be \(n\), the number of independently simulated inequalities or units be \(n_{I}\), and the number of simulation draws per unit be \(n_{s}\). Suppose a directional simulation bias has order \(n_{s}^{-1}\) and a centered independent simulation average has standard deviation of order \((n_{I}n_{s})^{-1/2}\).

Proposition 1 · Root-n negligibility conditions

At root-\(n\) scale, the squared bias and centered variance vanish when

\[n/n_{s}^{2}\to 0, \quad n/(n_{I}n_{s})\to 0.\](1)

Proof. Multiply the bias by \(\sqrt{n}\) and square, giving order \(n/n_{s}^{2}\). Multiplying the centered error by \(\sqrt{n}\) gives variance of order \(n/(n_{I}n_{s})\). Convergence of these nonnegative orders to zero yields the claims. ∎

2. Three growing sample sizes can lie on different boundaries

Set \(n=m^{4}\), \(n_{I}=m\), and \(n_{s}=m^{3}\). Then

\[n/n_{s}^{2}=m^{-2}\to 0, \quad n/(n_{I}n_{s})=1.\](2)

The directional bias is negligible at root-\(n\) scale, while centered simulation noise remains first order. Increasing raw simulation sizes therefore does not make every computational term asymptotically negligible.

3. Exponent inequalities classify the regimes

Let \(n=m^{a}\), \(n_{I}=m^{b}\), and \(n_{s}=m^{c}\) for positive exponents. The bias condition is \(2c>a\); the centered-noise condition is \(b+c>a\). Equality places a term on the first-order boundary, and reversing an inequality makes that term dominate the root-\(n\) scale.

Slowly varying factors matter on equality boundaries. For example, \(n_{s}=\sqrt{n}\) log \(n\) makes \(\sqrt{n}/n_{s}\) vanish, while \(n_{s}=\sqrt{n}\)/log \(n\) makes it diverge. Exponent labels alone classify neither sequence because both have exponent one half relative to \(n\).

4. Raw convergence supplies no normalized conclusion

Failure case · Consistency rate substituted for inference rate

Let \(e_{n}=n^{-1/4}\). Then \(e_{n}\to0\) while \(\sqrt{n}e_{n}=n^{1/4}\to \infty\). The approximation is consistent and dominates a root-\(n\) expansion.

Stochastic orders require the same comparison: \(R_{n}=o_{p}(1)\) does not imply \(\sqrt{n}R_{n}=o_{p}(1)\). Dependence can also change the nominal product \(n_{I}n_{s}\); that variance calculation is examined in II-07.

5. Implementation, exercises, and sources

Create one row per error component with raw order, estimator normalization, normalized order, source of dependence, and boundary regime. Evaluate sequences numerically over increasing \(m\), while retaining the analytic exponent calculation and any slowly varying factors.

Download the volume verification script →

Exercises

  1. Verify both ratios in (2) for \(m=10\),100,1000.
  2. Classify the (\(b,c\)) plane for fixed \(a=1\).
  3. Construct a logarithmic boundary sequence for the centered-noise condition.
Partial solutions

1. The bias ratios are \(10^{-2},10^{-4},10^{-6}\); the centered-noise ratio stays one. 2. Bias is negligible above \(c=1/2\), and centered noise is negligible above \(b+c=1\). 3. Multiply or divide one boundary sample size by log \(m\) and evaluate the resulting squared factor.

  1. A. W. van der Vaart, Asymptotic Statistics, Chapter 2.Stochastic orders and convergence modes.
  2. Patrick Billingsley, Convergence of Probability Measures, chapters on weak convergence.Normalization and triangular arrays.
  3. Whitney K. Newey and Daniel McFadden (1994), “Large Sample Estimation and Hypothesis Testing,” in Handbook of Econometrics, Volume IV.Remainder rates in estimator expansions.

6. Audit checkpoint

State every sample size, their joint sequence, statistical normalization, raw bias and variance orders, dependence structure, exponent equality cases, slowly varying factors, and the normalized order of each remainder.

7. Scope boundary

The chapter covers scalar rate comparisons for finite-dimensional triangular arrays. Process-level weak convergence and high-dimensional rate theory require additional results.

Prerequisites