EstemPMM: Polynomial Maximization Method for Non-Gaussian Regression

Overview

EstemPMM implements the Polynomial Maximization Method (PMM) for parameter estimation in linear regression and time series models when the error distribution deviates from normality. PMM achieves lower variance estimates than OLS by exploiting higher-order moments of the residual distribution.

The package provides two complementary estimators:

Method	Targets	Key quantity	Typical gain
PMM2 (S=2)	Asymmetric errors (\(\\|\gamma_3\\| > 0.3\))	\(g_2 = 1 - \gamma_3^2/(2+\gamma_4)\)	10–60% variance reduction
PMM3 (S=3)	Symmetric platykurtic errors (\(\gamma_4 < -0.7\))	\(g_3 = 1 - \gamma_4^2/(6+9\gamma_4+\gamma_6)\)	10–40% variance reduction

Use pmm_dispatch() to automatically choose between OLS, PMM2, and PMM3 based on residual cumulants.

Installation

# Released version from CRAN
install.packages("EstemPMM")

# Development version from GitHub
devtools::install_github("SZabolotnii/EstemPMM")

Quick Start

Automatic Method Selection

library(EstemPMM)

# Fit OLS first, then let pmm_dispatch() choose the best method
fit_ols <- lm(mpg ~ acceleration, data = auto_mpg)
pmm_dispatch(residuals(fit_ols))
# -> Recommends PMM2 (skewed residuals, gamma3 ≈ 0.5)

fit_ols2 <- lm(mpg ~ horsepower + I(horsepower^2), data = na.omit(auto_mpg))
pmm_dispatch(residuals(fit_ols2))
# -> Recommends PMM3 (symmetric platykurtic residuals, gamma4 ≈ -1.3)

PMM2 — Asymmetric Errors

library(EstemPMM)
set.seed(42)

# Simulate regression with skewed (gamma-distributed) errors
n <- 200
x <- rnorm(n)
y <- 2 + 1.5 * x + (rgamma(n, shape = 2, rate = 2) - 1)

# Fit PMM2 and compare with OLS
fit_pmm2 <- lm_pmm2(y ~ x, data = data.frame(y, x))
summary(fit_pmm2)
# g2 < 1 indicates variance reduction over OLS

PMM3 — Symmetric Platykurtic Errors

# Simulate regression with uniform (platykurtic) errors
y3 <- 2 + 1.5 * x + runif(n, -sqrt(3), sqrt(3))

# Fit PMM3 and compare with OLS
fit_pmm3 <- lm_pmm3(y3 ~ x, data = data.frame(y3, x))
summary(fit_pmm3)
# g3 ≈ 0.64 -> 36% variance reduction for uniform errors

Time Series

# AR(1) with asymmetric innovations -> PMM2
y_ar <- arima.sim(n = 300, list(ar = 0.7), innov = rgamma(300, 2, 2) - 1)
fit_ar_pmm2 <- ar_pmm2(y_ar, order = 1)

# AR(1) with uniform innovations -> PMM3
y_ar3 <- arima.sim(n = 300, list(ar = 0.7), innov = runif(300, -sqrt(3), sqrt(3)))
fit_ar_pmm3 <- ar_pmm3(y_ar3, order = 1)

# ARIMA(1,1,1) with PMM3 (uses nonlinear solver)
y_int <- cumsum(arima.sim(n = 300, list(ar = 0.5, ma = 0.3),
                           innov = runif(300, -sqrt(3), sqrt(3))))
fit_arima_pmm3 <- arima_pmm3(y_int, order = c(1, 1, 1))

Functions

Linear Regression

Function	Description
`lm_pmm2()`	PMM2 linear regression (asymmetric errors)
`lm_pmm3()`	PMM3 linear regression (symmetric platykurtic errors)
`pmm_dispatch()`	Automatic OLS / PMM2 / PMM3 selection
`compare_with_ols()`	Side-by-side comparison with OLS
`pmm2_inference()`	Bootstrap inference for linear PMM2 models

PMM2 Time Series

Function	Model
`ar_pmm2()`	AR(p)
`ma_pmm2()`	MA(q)
`arma_pmm2()`	ARMA(p, q)
`arima_pmm2()`	ARIMA(p, d, q)
`sar_pmm2()`	Seasonal AR — SAR(p, P)_s
`sma_pmm2()`	Seasonal MA — SMA(Q)_s
`sarma_pmm2()`	SARMA(p,q)×(P,Q)_s
`sarima_pmm2()`	SARIMA(p,d,q)×(P,D,Q)_s
`ts_pmm2()`	Universal wrapper
`ts_pmm2_inference()`	Bootstrap inference for TS models

PMM3 Time Series

Function	Model
`ar_pmm3()`	AR(p)
`ma_pmm3()`	MA(q)
`arma_pmm3()`	ARMA(p, q)
`arima_pmm3()`	ARIMA(p, d, q) — nonlinear solver
`ts_pmm3()`	Universal wrapper

Utilities

Function	Description
`pmm_skewness()`	Skewness coefficient \(\gamma_3\)
`pmm_kurtosis()`	Excess kurtosis \(\gamma_4\)
`pmm_gamma6()`	Sixth-order cumulant \(\gamma_6\)
`test_symmetry()`	Formal symmetry test for residuals
`compute_moments()`	Moments m2, m3, m4
`compute_moments_pmm3()`	Moments m2, m4, m6 and PMM3 quantities
`pmm2_variance_factor()`	Theoretical \(g_2\) factor
`pmm3_variance_factor()`	Theoretical \(g_3\) factor

Datasets

Dataset	Description	N	Use case
`DCOILWTICO`	WTI crude oil daily prices (FRED)	~9000	PMM2 time series
`auto_mpg`	UCI Auto MPG vehicle data	398	PMM2 and PMM3 linear regression
`djia2002`	DJIA daily data, Jul–Dec 2002	127	PMM2 AR(1), published example

Vignettes

Vignette	Topic
`vignette("pmm2-introduction")`	PMM2 linear regression: theory, examples, comparison with OLS
`vignette("pmm2-time-series")`	PMM2 for AR/MA/ARMA/ARIMA/SAR/SMA models
`vignette("bootstrap-inference")`	Bootstrap confidence intervals for PMM2
`vignette("pmm3-symmetric-errors")`	PMM3 linear regression: uniform/beta/truncated-normal errors
`vignette("pmm3-time-series")`	PMM3 for AR/MA/ARMA/ARIMA time series

S4 Class Hierarchy

PMM2fit                      PMM3fit
BasePMM2                     TS3fit
  └─ TS2fit                    ├─ ARPMM3
       ├─ ARPMM2               ├─ MAPMM3
       ├─ MAPMM2               ├─ ARMAPMM3
       ├─ ARMAPMM2             └─ ARIMAPMM3
       ├─ ARIMAPMM2
       ├─ SARPMM2
       ├─ SMAPMM2
       ├─ SARMAPMM2
       └─ SARIMAPMM2

All classes provide: coef(), residuals(), fitted(), predict(), summary(), plot(), AIC().

Efficiency Summary

PMM2 — Monte Carlo Results (asymmetric innovations)

Model	Innovation	Variance Reduction
Linear regression	Gamma (shape=2)	35–57%
AR(1)	Exponential	15–20%
MA(1)	Chi-squared (df=3)	15–23%
ARMA(1,1)	Student-t (df=4)	30–45%
SAR(1,1)_12	Exponential	up to 62%
SARMA(1,0,1,1)_12	Exponential	up to 59%

PMM3 — Monte Carlo Results (platykurtic innovations)

Model	Innovation	\(g_3\)	Variance Reduction
Linear regression	Uniform	0.64	36%
AR(1)	Uniform	0.55–0.70	30–45%
ARIMA(1,1,0)	Uniform	0.32–0.70	30–68%

PMM2 Variants

Three implementation variants are available for all PMM2 time series functions via the pmm2_variant parameter:

Variant	Speed	Best for
`"unified_global"` (default)	Fast	All models, best speed/accuracy tradeoff
`"unified_iterative"`	Slower	Maximum accuracy, SARIMA
`"linearized"`	Fast	Pure MA/SMA models

arima_pmm2(y, order = c(1,1,1), pmm2_variant = "unified_iterative")
ma_pmm2(y, order = 1, pmm2_variant = "linearized")

References

Foundational theory: Kunchenko, Y.P., Lega, Y.G. (1992). Estimation of Random Variable Parameters by the Polynomial Maximization Method. Naukova Dumka, Kyiv.

Linear regression (PMM2): Zabolotnii S., Warsza Z.L., Tkachenko O. (2018). Polynomial Estimation of Linear Regression Parameters for the Asymmetric PDF of Errors. Automation 2018, AISC vol. 743. Springer. https://doi.org/10.1007/978-3-319-77179-3_75

Autoregressive models (PMM2): Zabolotnii S., Tkachenko O., Warsza Z.L. (2022). Application of the PMM for Estimation Parameters of Autoregressive Models with Asymmetric Innovations. Automation 2022, AISC vol. 1427. Springer. https://doi.org/10.1007/978-3-031-03502-9_37

Moving average models (PMM2): Zabolotnii S., Tkachenko O., Warsza Z.L. (2023). PMM for Estimation Parameters of Asymmetric Non-Gaussian Moving Average Models. Automation 2023, LNNS vol. 630. Springer. https://doi.org/10.1007/978-3-031-25844-2_21

Author

Serhii Zabolotnii — Cherkasy State Business College ORCID: 0000-0003-0242-2234

Bug reports and feature requests: https://github.com/SZabolotnii/EstemPMM/issues