Any rejection from Lawford's test, however close to the threshold, commits the return tail to index 2 and leaves the fitted distribution without variance. Every admissible non-Gaussian member has that same survival tail index. The fitted parameter changes only the tail's scale; the index stays fixed. On daily S&P 500 log returns, c = 1.6358 gives a tail coefficient of 0.135. Small as that coefficient looks, E[Z^2] is infinite. Variance-based sizing cannot use such a tail estimate.
The paper fits the index itself, which cannot be held. Our tradable substitute would be SPY, and neither the fitted c of 1.6358 nor the -6.57 quantile transfers to the ETF. We have not backtested the diagnostic.
The source of the boundary
The normal CDF has an exact confluent hypergeometric representation. Lawford releases the denominator parameter of that 1F1, then imposes the endpoint conditions required of a distribution function. Those conditions set the numerator parameter at 1/2 and determine the normalising constant. Monotonicity remains valid for c >= 3/2. The resulting object is an admissible CDF by construction, avoiding the positivity repairs required by Gram-Charlier and Edgeworth expansions.
At c = 3/2, the result is exactly Phi. Above the boundary, the density falls as (c-3/2)|z|^-3, while the survival function falls as (c-3/2)/2 times z^-2. The probabilistic interpretation is clean: Z = eps/sqrt(Lambda), where eps is standard normal and Lambda is drawn from Beta(1, c-3/2). Under the Gaussian case, precision degenerates at unity. Only moments with r < 2 exist, so every c above the boundary has infinite variance.
Lawford estimates c by minimum distance on the CDF, using a tail-weighted measure proportional to [Phi(1-Phi)]^{-1/2} dPhi. Maximum likelihood is avoided for a stated reason. For every c > 3/2, the characteristic function lacks a second derivative at the origin, leaving moment, MGF and cumulant methods unavailable on any right neighbourhood of the boundary.
Calibration is the paper's strongest result
The statistic measures the improvement from estimating c rather than fixing it at 3/2. It is standardised with null constants H = 0.036295, Omega = 0.001191 and factor 60.953.
No nuisance parameter needs estimation.
Its limit places half the mass at zero and half on a chi-squared distribution with one degree of freedom. The boundary critical value is 2.706, below the naive 3.841, which therefore under-rejects.
For the fixed-scale version, simulated size ranges from 0.050 to 0.052 at nominal 5% across n = 25 to 1600. Boundary mass runs from 0.502 to 0.511, close to the theoretical 0.5. The local power calculation also tracks simulation. At c = 1.55, predicted power is 0.132, 0.184, 0.277 and 0.438 for n = 100, 200, 400 and 800. The simulated figures are 0.135, 0.186, 0.275 and 0.413.
Lawford acknowledges the weak power
The introduction concedes that omnibus procedures often have greater power against familiar non-Gaussian alternatives. Its defence is narrower: Jarque-Bera supplies neither a direction nor a fitted parameter carrying tail-risk information. At c = 1.60 and n = 400, JB rejects 0.931 of the time. The median/IQR plug-in, which Lawford recommends when location and scale are unknown, rejects 0.129. Against Student t3, the fixed-scale test has exactly zero power at every n from 25 to 800 because the t3 survival index is 3 and this family targets 2. Against chi-squared(3), rejection falls from 0.008 to 0.000; a symmetric family has nothing to say about skew.
The median/IQR plug-in does find symmetric heavy tails. Against Laplace, power rises from 0.272 to 0.999 as n moves from 25 to 800. Against t3, it rises from 0.195 to 0.977.
Lawford presents two failures and fixes both. The mean/SD plug-in has the right size yet almost no power against alternatives drawn from the family's own distribution: 0.004 at n = 25 and 0.000 by n = 100 against c = 3.0. Infinite variance makes the sample standard deviation diverge, wiping out the tail signal. Lawford replaces mean/SD with median/IQR, the version used in both empirical exercises.
Joint estimation of location, scale and shape degenerates for structural reasons. At the 10% level, rejection rises from 0.089 at n = 200 to 0.309 at n = 1600. Omega_cc.lambda, the score variance for shape after projecting out location and scale, collapses to 0.000013 against curvature of 0.002282. The statistic then divides by that vanishing quantity.
Oversizing remains throughout the simulations for the median/IQR version. Rejection is 0.095 at n = 25, 0.068 at n = 100 and 0.057 at n = 1600 against nominal 5% (0.107 against nominal 10%). The paper describes the distortion as declining in n, and the figures support that description.
Can fitted c support position sizing?
The empirical application uses 1255 daily log returns from FRED, covering 2 June 2021 to 1 June 2026. Excess kurtosis is 6.54 and skewness is 0.02. The statistic reaches 26.79, with c = 1.6358. After fitting GARCH(1,1) (alpha 0.1048, beta 0.8628), the statistic drops to 6.32 with p = 0.006 and c = 1.5625. Residual excess kurtosis is 1.56. Lawford's own summary is direct: "GARCH filtering substantially reduces but does not eliminate the symmetric heavy-tailed departure detected by the test."
Mean log scores on raw returns rank Student t at -1.40734, Laplace at -1.40947, this family at -1.42062 and Gaussian at -1.48007. On filtered residuals, the order is t at -1.39649, hypergeometric at -1.40096, Laplace at -1.40955 and Gaussian at -1.41557. The family therefore finishes third on raw returns and second on residuals, losing to a two-parameter t in both cases. The fitted t degrees of freedom adapt to the data, moving from 3.8667 on raw returns to 6.9703 on residuals. The hypergeometric index cannot move.
For raw returns, the 0.1% lower-tail quantiles are Gaussian -3.24, Laplace -4.60, empirical -4.82, Student t -5.51 and hypergeometric -6.57. Only the hypergeometric estimate extends beyond the sample minimum of -6.16, as Lawford notes. His VaR ratio table assumes a tail coefficient of 0.30. The raw-return fit reaches only 0.135, so the reported ratios of 1.73 at 99%, 2.17 at 99.5% and 3.98 at 99.9% overstate the implication of the fitted c.
Neither empirical exercise is backtested. The paper labels both descriptive and states that its i.i.d. null theory does not cover serially dependent returns or generated GARCH residuals.
Evidence that would change the verdict
The fitted c provides a single monotone number with a known null law and no nuisance estimation. As a tail model, it fixes the index at 2 before observing the data and estimates only scale. The residual series has excess kurtosis of 1.56 and skewness of -0.49. This symmetric family cannot capture that skew. We have examined the asymmetric GARCH side of the issue in /articles/asymmetric-long-memory-garch-sign-dependent-kernel-injection-in-a-two-dimensiona.
An exceedance backtest could settle the question: compare the fitted 99% and 99.5% quantiles with filtered historical simulation and GARCH-t, using a window longer than five years and more than one asset. Until that evidence exists, the paper's useful output is the rejection flag and its direction, which is also the claim Lawford makes for it.