Type: Package
Title: Polynomial Maximization Method for Non-Gaussian Regression
Version: 0.1.1
Date: 2025-10-22
Description: Implements the Polynomial Maximization Method ('PMM') for parameter estimation in linear and time series models when error distributions deviate from normality. The 'PMM2' variant achieves lower variance parameter estimates compared to ordinary least squares ('OLS') when errors exhibit significant skewness. Includes methods for linear regression, 'AR'/'MA'/'ARMA'/'ARIMA' models, and bootstrap inference. Methodology described in Zabolotnii, Warsza, and Tkachenko (2018) <doi:10.1007/978-3-319-77179-3_75>, Zabolotnii, Tkachenko, and Warsza (2022) <doi:10.1007/978-3-031-03502-9_37>, and Zabolotnii, Tkachenko, and Warsza (2023) <doi:10.1007/978-3-031-25844-2_21>.
License: GPL-3
Encoding: UTF-8
Depends: R (≥ 3.5.0)
Imports: methods, stats, graphics, utils
Suggests: dplyr, ggplot2, gridExtra, testthat (≥ 3.0.0), rmarkdown, knitr, MASS
LazyData: true
RoxygenNote: 7.3.3
Config/testthat/edition: 3
URL: https://github.com/SZabolotnii/EstemPMM
BugReports: https://github.com/SZabolotnii/EstemPMM/issues
VignetteBuilder: knitr
NeedsCompilation: no
Packaged: 2025-11-04 19:50:39 UTC; serhiizabolotnii
Author: Serhii Zabolotnii ORCID iD [aut, cre]
Maintainer: Serhii Zabolotnii <zabolotniua@gmail.com>
Repository: CRAN
Date/Publication: 2025-11-07 13:50:12 UTC

EstemPMM: Polynomial Maximization Method for Robust Regression and Time Series

Description

The EstemPMM package provides robust methods for estimating parameters of linear models and time series models that are robust to non-Gaussian errors.

Linear Regression Functions

lm_pmm2 - Fit linear models using PMM2

compare_with_ols - Compare PMM2 with OLS

Time Series Functions

ts_pmm2 - General function for fitting time series models using PMM2

ar_pmm2 - Fit AR models

ma_pmm2 - Fit MA models

arma_pmm2 - Fit ARMA models

arima_pmm2 - Fit ARIMA models

compare_ts_methods - Compare PMM2 with classical methods

Statistical Inference

pmm2_inference - Bootstrap inference for linear models

ts_pmm2_inference - Bootstrap inference for time series models

Utilities

pmm_skewness - Compute skewness

pmm_kurtosis - Compute kurtosis

compute_moments - Compute moments and cumulants

Author(s)

Maintainer: Serhii Zabolotnii zabolotniua@gmail.com (ORCID)

See Also

Useful links:

PMM2 fitting algorithm - unified implementation

Description

Core iterative algorithm for PMM2 parameter estimation

Usage

.pmm2_fit(
  b_init,
  X,
  y,
  m2,
  m3,
  m4,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

b_init

initial parameter estimates (typically from OLS)

X

design matrix

y

response vector

m2, m3, m4

central moments

max_iter

maximum number of iterations

tol

convergence tolerance

regularize

logical, add small value to diagonal for numerical stability

reg_lambda

regularization parameter (if regularize=TRUE)

verbose

logical, whether to print progress information

Value

A list with components:

b

estimated parameters

convergence

logical convergence status

iterations

number of iterations performed

PMM2 fitting algorithm for time series models

Description

Implementation of PMM2 algorithm for time series models with various approaches depending on the model structure (AR, MA, ARMA, ARIMA)

Usage

.ts_pmm2_fit(
  b_init,
  innovations_init,
  model_info,
  m2,
  m3,
  m4,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

b_init

initial parameter estimates (AR followed by MA)

innovations_init

initial innovations (errors)

model_info

list with model structure information

m2, m3, m4

central moments

max_iter

maximum number of iterations

tol

convergence tolerance

regularize

logical, add small value to diagonal for numerical stability

reg_lambda

regularization parameter (if regularize=TRUE)

verbose

logical, whether to print progress information

Value

A list with components:

b

estimated parameters

convergence

logical convergence status

iterations

number of iterations performed

innovations

final innovations (errors)

Calculate AIC for PMM2fit object

Description

Calculate AIC for PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
AIC(object, ..., k = 2)

Arguments

object

PMM2fit object

...

Additional arguments (not used)

k

Penalty per parameter to be used; default is 2

Value

AIC value

S4 class for storing PMM2 ARIMA model results

Description

S4 class for storing PMM2 ARIMA model results

S4 class for storing PMM2 ARMA model results

Description

S4 class for storing PMM2 ARMA model results

S4 class for storing PMM2 AR model results

Description

S4 class for storing PMM2 AR model results

Base S4 class for storing PMM2 model results

Description

Base S4 class for storing PMM2 model results

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals

m2

numeric second central moment of initial residuals

m3

numeric third central moment of initial residuals

m4

numeric fourth central moment of initial residuals

convergence

logical or integer code indicating whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

WTI Crude Oil Prices

Description

Daily spot prices for West Texas Intermediate (WTI) crude oil in U.S. dollars per barrel.

Usage

DCOILWTICO

Format

A data frame with observations for each trading day:

observation_date

Date of observation in YYYY-MM-DD format

DCOILWTICO

Crude Oil Price: West Texas Intermediate (WTI) in USD per barrel

Source

Federal Reserve Economic Data (FRED), Federal Reserve Bank of St. Louis https://fred.stlouisfed.org/series/DCOILWTICO

Examples

data(DCOILWTICO)
head(DCOILWTICO)
summary(DCOILWTICO$DCOILWTICO)

S4 class for storing PMM2 MA model results

Description

S4 class for storing PMM2 MA model results

S4 class for storing PMM2 regression model results

Description

Class for storing results of linear model estimation using PMM2

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals

m2

numeric second central moment of initial residuals

m3

numeric third central moment of initial residuals

m4

numeric fourth central moment of initial residuals

convergence

logical or integer code indicating whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

Slots

coefficients

Estimated coefficients

residuals

Final residuals

m2

Second central moment

m3

Third central moment

m4

Fourth central moment

convergence

Convergence status

iterations

Number of iterations performed

call

Original call

Base S4 class for storing PMM2 time series model results

Description

Base class for storing results of time series model estimation using PMM2

Slots

coefficients

numeric vector of estimated parameters

residuals

numeric vector of final residuals

m2

numeric second central moment of initial residuals

m3

numeric third central moment of initial residuals

m4

numeric fourth central moment of initial residuals

convergence

logical or integer code indicating whether algorithm converged

iterations

numeric number of iterations performed

call

original function call

model_type

character string indicating model type

intercept

numeric value of intercept

original_series

numeric vector of original time series

order

list of order parameters

Slots

coefficients

Estimated coefficients

residuals

Final residuals

m2

Second central moment

m3

Third central moment

m4

Fourth central moment

convergence

Convergence status

iterations

Number of iterations performed

call

Original call

model_type

Model type

intercept

Intercept

original_series

Original time series

order

Model orders

Fit an AR model using PMM2 (wrapper)

Description

Fit an AR model using PMM2 (wrapper)

Usage

ar_pmm2(
  x,
  order = 1,
  method = "pmm2",
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

x

Numeric vector of time series data

order

Model order specification: - For AR models: a single integer (AR order) - For MA models: a single integer (MA order) - For ARMA models: vector c(p, q) (AR and MA orders) - For ARIMA models: vector c(p, d, q) (AR, differencing, and MA orders)

method

String: estimation method, one of "pmm2" (default), "css", "ml", "yw", "ols"

max_iter

Integer: maximum number of iterations for the algorithm

tol

Numeric: tolerance for convergence

include.mean

Logical: whether to include a mean (intercept) term

initial

List or vector of initial parameter estimates (optional)

na.action

Function for handling missing values, default is na.fail

regularize

Logical, add small values to diagonal for numerical stability

reg_lambda

Regularization parameter (if regularize=TRUE)

verbose

Logical: whether to print progress information

Value

An S4 object of class ARPMM2 containing fitted autoregressive coefficients, residuals, central moment estimates (m2-m4), model order, intercept, original series, and convergence diagnostics.

Fit an ARIMA model using PMM2 (wrapper)

Description

Fit an ARIMA model using PMM2 (wrapper)

Usage

arima_pmm2(
  x,
  order = c(1, 1, 1),
  method = "pmm2",
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

x

Numeric vector of time series data

order

Model order specification: - For AR models: a single integer (AR order) - For MA models: a single integer (MA order) - For ARMA models: vector c(p, q) (AR and MA orders) - For ARIMA models: vector c(p, d, q) (AR, differencing, and MA orders)

method

String: estimation method, one of "pmm2" (default), "css", "ml", "yw", "ols"

max_iter

Integer: maximum number of iterations for the algorithm

tol

Numeric: tolerance for convergence

include.mean

Logical: whether to include a mean (intercept) term

initial

List or vector of initial parameter estimates (optional)

na.action

Function for handling missing values, default is na.fail

regularize

Logical, add small values to diagonal for numerical stability

reg_lambda

Regularization parameter (if regularize=TRUE)

verbose

Logical: whether to print progress information

Value

An S4 object of class ARIMAPMM2 containing fitted AR and MA coefficients, residual series, central moments, differencing order, intercept, original series, and convergence diagnostics.

Fit an ARMA model using PMM2 (wrapper)

Description

Fit an ARMA model using PMM2 (wrapper)

Usage

arma_pmm2(
  x,
  order = c(1, 1),
  method = "pmm2",
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

x

Numeric vector of time series data

order

Model order specification: - For AR models: a single integer (AR order) - For MA models: a single integer (MA order) - For ARMA models: vector c(p, q) (AR and MA orders) - For ARIMA models: vector c(p, d, q) (AR, differencing, and MA orders)

method

String: estimation method, one of "pmm2" (default), "css", "ml", "yw", "ols"

max_iter

Integer: maximum number of iterations for the algorithm

tol

Numeric: tolerance for convergence

include.mean

Logical: whether to include a mean (intercept) term

initial

List or vector of initial parameter estimates (optional)

na.action

Function for handling missing values, default is na.fail

regularize

Logical, add small values to diagonal for numerical stability

reg_lambda

Regularization parameter (if regularize=TRUE)

verbose

Logical: whether to print progress information

Value

An S4 object of class ARMAPMM2 containing fitted AR and MA coefficients, residuals, central moments, model specification, intercept, original series, and convergence diagnostics.

Extract coefficients from PMM2fit object

Description

Extract coefficients from PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
coef(object, ...)

Arguments

object

PMM2fit object

...

Additional arguments (not used)

Value

Vector of coefficients

Extract coefficients from TS2fit object

Description

Extract coefficients from TS2fit object

Usage

## S4 method for signature 'TS2fit'
coef(object, ...)

Arguments

object

TS2fit object

...

Additional arguments (not used)

Value

Named vector of coefficients

Compare AR methods

Description

Compare AR methods

Usage

compare_ar_methods(x, order = 1, include.mean = TRUE, pmm2_args = list())

Arguments

x

Numeric vector of time series data

order

Model order specification (see ts_pmm2 for format)

include.mean

Logical, whether to include intercept term

pmm2_args

List of additional arguments to pass to ts_pmm2()

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare ARIMA methods

Description

Compare ARIMA methods

Usage

compare_arima_methods(
  x,
  order = c(1, 1, 1),
  include.mean = TRUE,
  pmm2_args = list()
)

Arguments

x

Numeric vector of time series data

order

Model order specification (see ts_pmm2 for format)

include.mean

Logical, whether to include intercept term

pmm2_args

List of additional arguments to pass to ts_pmm2()

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare ARMA methods

Description

Compare ARMA methods

Usage

compare_arma_methods(
  x,
  order = c(1, 1),
  include.mean = TRUE,
  pmm2_args = list()
)

Arguments

x

Numeric vector of time series data

order

Model order specification (see ts_pmm2 for format)

include.mean

Logical, whether to include intercept term

pmm2_args

List of additional arguments to pass to ts_pmm2()

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare MA methods

Description

Compare MA methods

Usage

compare_ma_methods(x, order = 1, include.mean = TRUE, pmm2_args = list())

Arguments

x

Numeric vector of time series data

order

Model order specification (see ts_pmm2 for format)

include.mean

Logical, whether to include intercept term

pmm2_args

List of additional arguments to pass to ts_pmm2()

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare PMM2 with classical time series estimation methods

Description

Compare PMM2 with classical time series estimation methods

Usage

compare_ts_methods(
  x,
  order,
  model_type = c("ar", "ma", "arma", "arima"),
  include.mean = TRUE,
  pmm2_args = list()
)

Arguments

x

Numeric vector of time series data

order

Model order specification (see ts_pmm2 for format)

model_type

Model type: "ar", "ma", "arma", or "arima"

include.mean

Logical, whether to include intercept term

pmm2_args

List of additional arguments to pass to ts_pmm2()

Value

A named list containing the fitted objects for each estimation approach (e.g., YW/OLS/MLE or CSS/ML alongside PMM2) plus two data frames: coefficients (side-by-side parameter estimates) and residual_stats (residual RSS, MAE, skewness, and kurtosis).

Compare PMM2 with OLS

Description

Compare PMM2 with OLS

Usage

compare_with_ols(formula, data, pmm2_args = list())

Arguments

formula

Model formula

data

Data frame

pmm2_args

List of arguments to pass to lm_pmm2()

Value

List with OLS and PMM2 fit objects

Calculate moments and cumulants of error distribution

Description

Calculate moments and cumulants of error distribution

Usage

compute_moments(errors)

Arguments

errors

numeric vector of errors

Value

list with moments, cumulants and theoretical variance reduction coefficient

Compute final residuals for time series models

Description

Compute final residuals for time series models

Usage

compute_ts_residuals(coefs, model_info)

Arguments

coefs

Estimated coefficients

model_info

Model information

Value

Vector of residuals

Create design matrix for AR model

Description

Create design matrix for AR model

Usage

create_ar_matrix(x, p)

Arguments

x

Centered time series

p

AR order

Value

Design matrix with lagged values

Create design matrix for time series

Description

Create design matrix for time series

Usage

create_ts_design_matrix(x, model_info, innovations = NULL)

Arguments

x

Time series data

model_info

List with model parameters

innovations

Optional innovations/residuals for MA components

Value

List with design matrix, response variable, and other components

Extract fitted values from PMM2fit object

Description

Extract fitted values from PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
fitted(object, data = NULL, ...)

Arguments

object

PMM2fit object

data

Optional data source for model reconstruction, if object does not contain saved data

...

Additional arguments (not used)

Value

Vector of fitted values

Extract fitted values from TS2fit object

Description

Extract fitted values from TS2fit object

Usage

## S4 method for signature 'TS2fit'
fitted(object, ...)

Arguments

object

TS2fit object

...

Additional arguments (not used)

Value

Vector of fitted values

Helper function for extracting fitted values

Description

Helper function for extracting fitted values

Usage

fitted_values(object, data = NULL)

Arguments

object

PMM2fit object

Value

Vector of fitted values

Get fitted values for AR model

Description

Get fitted values for AR model

Usage

get_ar_fitted(object)

Arguments

object

TS2fit object with model_type="ar"

Value

Vector of fitted values

Get initial parameter estimates for time series models

Description

Get initial parameter estimates for time series models

Usage

get_initial_estimates(
  model_params,
  initial = NULL,
  method = "pmm2",
  verbose = FALSE
)

Arguments

model_params

Validated model parameters from validate_ts_parameters

initial

Optionally user-provided initial estimates

method

Estimation method

verbose

Output detailed information

Value

List containing:

b_init

Vector of initial AR/MA coefficients

x_mean

Estimated mean (if include.mean=TRUE)

innovations

Initial residuals/innovations

x_centered

Centered (or differenced + centered) series

m2

Second central moment of initial residuals

m3

Third central moment of initial residuals

m4

Fourth central moment of initial residuals

Get Yule-Walker estimates for AR(p)

Description

Get Yule-Walker estimates for AR(p)

Usage

get_yw_estimates(x, p)

Arguments

x

Numeric vector

p

Integer value of AR order

Value

Numeric vector of length p (AR coefficients)

PMM2: Main function for PMM2 (S=2)

Description

Fits a linear model using the Polynomial Maximization Method (order 2), which is robust to non-Gaussian errors.

Usage

lm_pmm2(
  formula,
  data,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  na.action = na.fail,
  weights = NULL,
  verbose = FALSE
)

Arguments

formula

R formula for the model

data

data.frame containing variables in the formula

max_iter

integer: maximum number of iterations for the algorithm

tol

numeric: tolerance for convergence

regularize

logical: add small value to diagonal for numerical stability

reg_lambda

numeric: regularization parameter (if regularize=TRUE)

na.action

function for handling missing values, default is na.fail

weights

optional weight vector (not yet implemented)

verbose

logical: whether to print progress information

Details

The PMM2 algorithm works as follows:

  1. Fits ordinary least squares (OLS) regression to obtain initial estimates

  2. Computes central moments (m2, m3, m4) from OLS residuals

  3. Iteratively improves parameter estimates using a gradient-based approach

PMM2 is especially useful when error terms are not Gaussian.

Value

S4 object of class PMM2fit

Examples

set.seed(123)
n <- 80
x <- rnorm(n)
y <- 2 + 3 * x + rt(n, df = 3)
dat <- data.frame(y = y, x = x)

fit <- lm_pmm2(y ~ x, data = dat)
summary(fit, formula = y ~ x, data = dat)

Fit an MA model using PMM2 (wrapper)

Description

Fit an MA model using PMM2 (wrapper)

Usage

ma_pmm2(
  x,
  order = 1,
  method = "pmm2",
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

x

Numeric vector of time series data

order

Model order specification: - For AR models: a single integer (AR order) - For MA models: a single integer (MA order) - For ARMA models: vector c(p, q) (AR and MA orders) - For ARIMA models: vector c(p, d, q) (AR, differencing, and MA orders)

method

String: estimation method, one of "pmm2" (default), "css", "ml", "yw", "ols"

max_iter

Integer: maximum number of iterations for the algorithm

tol

Numeric: tolerance for convergence

include.mean

Logical: whether to include a mean (intercept) term

initial

List or vector of initial parameter estimates (optional)

na.action

Function for handling missing values, default is na.fail

regularize

Logical, add small values to diagonal for numerical stability

reg_lambda

Regularization parameter (if regularize=TRUE)

verbose

Logical: whether to print progress information

Value

An S4 object of class MAPMM2 containing moving-average coefficients, residual innovations, central moments, model order, intercept, original series, and convergence diagnostics.

Plot diagnostic plots for PMM2fit object

Description

Plot diagnostic plots for PMM2fit object

Usage

## S4 method for signature 'PMM2fit,missing'
plot(x, y, which = 1:4, ...)

Arguments

x

PMM2fit object

y

Not used (compatibility with generic)

which

Set of plots to display (values 1-4)

...

Additional arguments passed to plotting functions

Value

Invisibly returns the input object

Build diagnostic plots for TS2fit objects

Description

Build diagnostic plots for TS2fit objects

Usage

## S4 method for signature 'TS2fit,missing'
plot(x, y, which = c(1:4), ...)

Arguments

x

TS2fit object

y

Not used (for S4 method compatibility)

which

Integer vector indicating which plots to produce

...

additional arguments passed to plot functions

Value

Invisibly returns x

Plot bootstrap distributions for PMM2 fit

Description

Plot bootstrap distributions for PMM2 fit

Usage

plot_pmm2_bootstrap(object, coefficients = NULL)

Arguments

object

Result from pmm2_inference

coefficients

Which coefficients to plot, defaults to all

Value

Invisibly returns histogram information

Universal PMM2 algorithm for all model types

Description

Universal PMM2 algorithm for all model types

Usage

pmm2_algorithm(
  b_init,
  X,
  y,
  m2,
  m3,
  m4,
  max_iter = 50,
  tol = 1e-06,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE,
  poly_terms = NULL
)

Arguments

b_init

Initial parameter estimates

X

Design matrix

y

Response vector

m2, m3, m4

Central moments

max_iter

Maximum number of iterations

tol

Tolerance for convergence

regularize

Whether to add regularization

reg_lambda

Regularization parameter

verbose

Whether to print progress information

poly_terms

Pre-computed polynomial coefficients (list with elements A, B, C); allows passing custom values for special scenarios, otherwise they are computed from moments

Value

List with estimation results

Bootstrap inference for PMM2 fit

Description

Bootstrap inference for PMM2 fit

Usage

pmm2_inference(
  object,
  formula,
  data,
  B = 200,
  seed = NULL,
  parallel = FALSE,
  cores = NULL
)

Arguments

object

object of class PMM2fit

formula

the same formula that was used initially

data

data frame that was used initially

B

number of bootstrap replications

seed

(optional) for reproducibility

parallel

logical, whether to use parallel computing

cores

number of cores to use for parallel computing, defaults to auto-detect

Value

data.frame with columns: Estimate, Std.Error, t.value, p.value

Monte Carlo comparison of PMM2 estimation methods

Description

Function generates time series for given models, repeatedly estimates parameters using different methods and compares their accuracy by MSE criterion. Additionally outputs theoretical and empirical characteristics of the innovation distribution (skewness, excess kurtosis, theoretical gain of PMM2).

Usage

pmm2_monte_carlo_compare(
  model_specs,
  methods = c("css", "pmm2"),
  n,
  n_sim,
  innovations = list(type = "gaussian"),
  seed = NULL,
  include.mean = TRUE,
  progress = interactive(),
  verbose = FALSE
)

Arguments

model_specs

List of model specifications. Each element must contain:

model

"ar", "ma" or "arma"

order

order (for AR/MA) or vector c(p, q) for ARMA

theta

numeric vector of true parameters; for ARMA a list list(ar = ..., ma = ...)

label

(optional) model name in report

innovations

(optional) description of innovation distribution: list(type = "gamma", shape = 2), list(type = "student_t", df = 5), etc. Can also pass an arbitrary generation function via generator.

methods

Vector of estimation methods (e.g., c("css","pmm2")). The first method is considered baseline for relative MSE calculation.

n

Sample size for simulation.

n_sim

Number of Monte Carlo experiments.

innovations

Function or distribution description, used by default for all models (if not specified in spec).

seed

Initial seed for random number generator (optional).

include.mean

Logical flag: whether to include intercept during estimation.

progress

Logical flag: print Monte Carlo progress.

verbose

Whether to print diagnostic messages on failures.

Value

List with three components:

parameter_results

MSE and relative MSE for each parameter

summary

Averaged MSE over parameters for each model/method

gain

Comparison of theoretical and empirical PMM2 gain

Calculate theoretical skewness, kurtosis coefficients and variance reduction factor

Description

Calculate theoretical skewness, kurtosis coefficients and variance reduction factor

Usage

pmm2_variance_factor(m2, m3, m4)

Arguments

m2, m3, m4

central moments of second, third and fourth orders

Value

List with fields c3, c4 and g

Calculate theoretical variance matrices for OLS and PMM2

Description

Calculate theoretical variance matrices for OLS and PMM2

Usage

pmm2_variance_matrices(X, m2, m3, m4)

Arguments

X

Design matrix with column of ones

m2, m3, m4

central moments of OLS residuals

Value

List with fields ols, pmm2, c3, c4, g

Calculate kurtosis from data

Description

Calculate kurtosis from data

Usage

pmm_kurtosis(x, excess = TRUE)

Arguments

x

numeric vector

excess

logical, whether to return excess kurtosis (kurtosis - 3)

Value

Kurtosis value

Calculate skewness from data

Description

Calculate skewness from data

Usage

pmm_skewness(x)

Arguments

x

numeric vector

Value

Skewness value

Prediction method for PMM2fit objects

Description

Prediction method for PMM2fit objects

Usage

## S4 method for signature 'PMM2fit'
predict(object, newdata = NULL, debug = FALSE, ...)

Arguments

object

PMM2fit object

newdata

New data frame for prediction

debug

Logical value, whether to output debug information

...

additional arguments (not used)

Value

Vector of predictions

Prediction method for TS2fit objects

Description

Prediction method for TS2fit objects

Usage

## S4 method for signature 'TS2fit'
predict(object, n.ahead = 1, ...)

Arguments

object

TS2fit object

n.ahead

Number of steps ahead for prediction

...

additional arguments (not used)

Value

Vector or list of predictions, depending on model type

Extract residuals from PMM2fit object

Description

Extract residuals from PMM2fit object

Usage

## S4 method for signature 'PMM2fit'
residuals(object, ...)

Arguments

object

PMM2fit object

...

Additional arguments (not used)

Value

Vector of residuals

Extract residuals from TS2fit object

Description

Extract residuals from TS2fit object

Usage

## S4 method for signature 'TS2fit'
residuals(object, ...)

Arguments

object

TS2fit object

...

Additional arguments (not used)

Value

Vector of residuals (innovations)

Universal solver for PMM2 system of equations

Description

Universal solver for PMM2 system of equations

Usage

solve_pmm2(
  b_init,
  X,
  y,
  m2,
  m3,
  m4,
  max_iter = 1000,
  tol = 1e-05,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

b_init

Initial parameter estimates (usually from OLS or MLE)

X

Design matrix (including intercept and all predictors)

y

Response vector

m2

Second central moment of residuals

m3

Third central moment of residuals

m4

Fourth central moment of residuals

max_iter

Maximum number of iterations

tol

Convergence tolerance

regularize

Whether to add regularization to Jacobian matrix

reg_lambda

Regularization parameter

verbose

Print progress information

Value

Vector of PMM2 parameter estimates

Generic summary method for PMM2fit objects

Description

Generic summary method for PMM2fit objects

Usage

## S4 method for signature 'PMM2fit'
summary(object, formula = NULL, data = NULL, B = 100, ...)

Arguments

object

object of class "PMM2fit"

formula

(optional) formula used for the model

data

(optional) data used

B

number of bootstrap replications for statistical inference

...

additional arguments (not used)

Value

Prints summary to console; returns object (invisibly).

Generic summary method for TS2fit objects

Description

Generic summary method for TS2fit objects

Usage

## S4 method for signature 'TS2fit'
summary(object, ...)

Arguments

object

object of class "TS2fit" or subclass

...

additional arguments (not used)

Value

Prints summary to console; returns object (invisibly).

Fit a time series model using the PMM2 method

Description

Fit a time series model using the PMM2 method

Usage

ts_pmm2(
  x,
  order,
  model_type = c("ar", "ma", "arma", "arima"),
  method = "pmm2",
  max_iter = 50,
  tol = 1e-06,
  include.mean = TRUE,
  initial = NULL,
  na.action = na.fail,
  regularize = TRUE,
  reg_lambda = 1e-08,
  verbose = FALSE
)

Arguments

x

Numeric vector of time series data

order

Model order specification: - For AR models: a single integer (AR order) - For MA models: a single integer (MA order) - For ARMA models: vector c(p, q) (AR and MA orders) - For ARIMA models: vector c(p, d, q) (AR, differencing, and MA orders)

model_type

String specifying the model type: "ar", "ma", "arma", or "arima"

method

String: estimation method, one of "pmm2" (default), "css", "ml", "yw", "ols"

max_iter

Integer: maximum number of iterations for the algorithm

tol

Numeric: tolerance for convergence

include.mean

Logical: whether to include a mean (intercept) term

initial

List or vector of initial parameter estimates (optional)

na.action

Function for handling missing values, default is na.fail

regularize

Logical, add small values to diagonal for numerical stability

reg_lambda

Regularization parameter (if regularize=TRUE)

verbose

Logical: whether to print progress information

Details

The PMM2 algorithm works as follows:

  1. Fits an initial model using a standard method (OLS, Yule-Walker, CSS or ML)

  2. Computes central moments (m2, m3, m4) from initial residuals/innovations

  3. Uses these moments with a specialized solver (pmm2_algorithm) to find robust parameter estimates

Value

An S4 object TS2fit of the corresponding subclass

Bootstrap inference for PMM2 time series models

Description

Bootstrap inference for PMM2 time series models

Usage

ts_pmm2_inference(
  object,
  x = NULL,
  B = 200,
  seed = NULL,
  block_length = NULL,
  method = c("residual", "block"),
  parallel = FALSE,
  cores = NULL,
  debug = FALSE
)

Arguments

object

object of class TS2fit

x

(optional) original time series; if NULL, uses object@original_series

B

number of bootstrap replications

seed

(optional) for reproducibility

block_length

block length for block bootstrap; if NULL, uses heuristic value

method

bootstrap type: "residual" or "block"

parallel

logical, whether to use parallel computing

cores

number of cores for parallel computing

debug

logical, whether to output additional diagnostic information

Value

data.frame with columns: Estimate, Std.Error, t.value, p.value

Update MA model innovations

Description

Update MA model innovations

Usage

update_ma_innovations(x, ma_coef)

Arguments

x

Centered time series

ma_coef

Vector of MA coefficients

Value

Vector of innovations

Validate and prepare time series parameters

Description

Validate and prepare time series parameters

Usage

validate_ts_parameters(x, order, model_type, include.mean)

Arguments

x

Time series data

order

Model order specification

model_type

Model type (ar, ma, arma, or arima)

include.mean

Whether to include mean/intercept

Value

List of validated parameters and model information