Maximal monotone lag reduction in exponentially weighted smoothers: the two-rate discounted-regression family and its critical reductions

Every causal smoother trades noise rejection for lag, and with exponential weights every standard attempt to compensate for this lag runs into the same obstacle: overshoot. We ask how far the evaluation point of a smoothing filter can be advanced before its unit-step response stops being monotone, and answer the question exactly for both classical weighting classes. For uniform (FIR) weights the answer is a familiar object in an unfamiliar role: in the continuous limit the weighted moving average (WMA) is precisely the furthest monotone advance of the least-squares regression line, at lag reduction r = 1/3 (at finite period the boundary is the ramp over the newest p - 1 samples, at r = (p+1)/(3(p-1))), a characterisation that appears never to have been stated. For exponential (IIR) weights the construction commonly treated as the analogue fails completely: we prove that advancing along the discounted regression line, the family (1+r) EMA - r EMA^[2] (Tillson's GDEMA, the building block of T3), overshoots for every advance r > 0, contradicting the standard practitioner belief that partial lag reduction cures overshoot. A two-rate construction repairs this: taking the trend correction from a lag-matched cascade one order higher restores a positive monotone maximum, with closed form r_crit^M(s) = (s^(s+1)/(s+1)^(s+2))·e^((2s+1)/(s+1)) at smoothness order s, characterised intrinsically, in the continuous limit, as the unique reduction whose step response has a stationary inflection. The resulting filters, FastEma (nominal period: strictly faster than the EMA, and at s = 1 pointwise no larger in step error) and LeadEma (inflated period: the EMA's lag with reduced undershoot), are linear, causal, O(s) per sample, and overshoot-free; the guarantee extends exactly to fractional smoothness orders through output-level interpolation between adjacent integer orders. At full advance the family reaches its zero-lag endpoint XEPMA, zero time lag at every order and, at s = 1, quadratic-exact (m1 = m2 = 0, discrete time included): there its trend correction is the genuine slope of a lag-matched smoother, a property neither Brown's DEMA nor Mulloy's TEMA shares, although TEMA is itself quadratic-exact. Between the plain cascade and this endpoint the family carries a three-boundary interior structure, r_crit^C < r_crit^M < r_crit^O, separating EMA-like (never re-accelerating), monotone and no-overshoot convergence. Benchmarks on ECG and image denoising against parameter-optimised classical baselines show the family is practically competitive in zero-phase denoising use, with an advantage at heavy broadband noise.