ARtCensReg {ARCensReg} R Documentation

## Censored autoregressive regression model with Student-t innovations

### Description

It fits a univariate left, right, or interval censored linear regression model with autoregressive errors considering Student-t innovations, through the SAEM algorithm. It provides estimates and standard errors of the parameters, supporting missing values on the dependent variable.

### Usage

ARtCensReg(cc, lcl = NULL, ucl = NULL, y, x, p = 1, M = 10,
perc = 0.25, MaxIter = 400, pc = 0.18, nufix = NULL, tol = 1e-04,
show_se = TRUE, quiet = FALSE)


### Arguments

 cc Vector of censoring indicators of length n, where n is the total observations. For each observation: 0 if non-censored, 1 if censored/missing. lcl, ucl Vectors of length n that represent the lower and upper bounds of the interval, which contains the observade value of the censored observation. Default=NULL, indicating no-censored data. See details for more information. y Vector of responses of length n. x Matrix of covariates of dimension n \times l, where l is the number of fixed effects including the intercept, if considered (in models which include an intercept, x should contain a column of ones). p Order of the autoregressive process. It must be a positive integer value. M Size of the Monte Carlo sample generated in each step of the SAEM algorithm. Default=10. perc Percentage of burn-in on the Monte Carlo sample. Default=0.25. MaxIter The maximum number of iterations of the SAEM algorithm. Default=400. pc Percentage of initial iterations of the SAEM algorithm with no memory. It is recommended that 50

### Details

The linear regression model with autocorrelated errors, defined as a discrete-time autoregressive (AR) process of order p, at time t is given by

Y_t = x_t^T \beta + \xi_t,

\xi_t = \phi_1 \xi_{t-1} + ... + \phi_p \xi_{t-p} + \eta_t, t=1,..., n,

where Y_t is the response variable, \beta = (\beta_1,..., \beta_l)^T is a vector of regression parameters of dimension l, x_t = (x_{t1},..., x_{tl})^T is a vector of non-stochastic regressor variables values, and \xi_t is the AR error with \eta_t being a shock of disturbance following the Student-t distribution with \nu degrees of freedom, \phi = (\phi_1,..., \phi_p)^T being the vector of AR coefficients, and n denoting the sample size.

It is assumed that Y_t is not fully observed for all t. For left censored observations, we have lcl=-Inf and ucl=V_t, such that the true value Y_t \leq V_t. For right censoring, lcl=V_t and ucl=Inf, such that Y_t \geq V_t. For interval censoring, lcl and ucl must be finite values, such that V_{1t} \leq Y_t \leq V_{2t}. Missing data can be defined by setting lcl=-Inf and ucl=Inf.

The initial values are obtained by ignoring censoring and applying maximum likelihood estimation with the censored data replaced by their censoring limits. Moreover, just set cc as a vector of zeros to fit a regression model with autoregressive errors for non-censored data.

### Value

An object of class "ARtpCRM" representing the AR(p) censored regression Student-t fit. Generic functions such as print and summary have methods to show the results of the fit. The function plot provides convergence graphics for the parameter estimates.

Specifically, the following components are returned:

 beta Estimate of the regression parameters. sigma2 Estimated scale parameter of the innovation. phi Estimate of the autoregressive parameters. nu Estimated degrees of freedom. theta Vector of parameters estimate (\beta, \sigma^2, \phi, \nu). SE Vector of the standard errors of (\beta, \sigma^2, \phi, \nu). yest Augmented response variable based on the fitted model. uest Final estimated weight variables. x Matrix of covariates of dimension n \times l. iter Number of iterations until convergence. criteria Attained criteria value. call The ARtCensReg call that produced the object. tab Table of estimates. cens "left", "right", or "interval" for left, right, or interval censoring, respectively. nmiss Number of missing observations. ncens Number of censored observations. converge Logical indicating convergence of the estimation algorithm. MaxIter The maximum number of iterations used for the SAEM algorithm. M Size of the Monte Carlo sample generated in each step of the SAEM algorithm. pc Percentage of initial iterations of the SAEM algorithm with no memory. time Time elapsed in processing. plot A list containing convergence information.

### Warning

This algorithm assumes that the first p values in the response vector are completely observed.

### Author(s)

Katherine L. Valeriano, Fernanda L. Schumacher, and Larissa A. Matos

### References

Delyon B, Lavielle M, Moulines E (1999). “Convergence of a stochastic approximation version of the EM algorithm.” Annals of statistics, 94–128.

Valeriano KL, Schumacher FL, Galarza CE, Matos LA (2021). “Censored autoregressive regression models with Student- t  innovations.” arXiv preprint arXiv:2110.00224.

arima, ARCensReg

### Examples

## Example 1: (p = l = 1)
# Generating a sample
set.seed(1234)
n = 80
x = rep(1, n)
dat = rARCens(n=n, beta=2, phi=.6, sig2=.3, x=x, cens='right', pcens=.05,
innov='t', nu=4)

# Fitting the model (quick convergence)
fit0 = ARtCensReg(dat$data$cc, dat$data$lcl, dat$data$ucl, dat$data$y, x,
M=5, pc=.12, tol=0.001)
fit0

## Example 2: (p = l = 2)
# Generating a sample
set.seed(783796)
n = 200
x = cbind(1, runif(n))
dat = rARCens(n=n, beta=c(2,1), phi=c(.48,-.2), sig2=.5, x=x, cens='left',
pcens=.05, innov='t', nu=5)

# Fitting the model with nu known
fit1 = ARtCensReg(dat$data$cc, dat$data$lcl, dat$data$ucl, dat$data$y, x,
p=2, M=15, pc=.20, nufix=5)
summary(fit1)
plot(fit1)

# Fitting the model with nu unknown
fit2 = ARtCensReg(dat$data$cc, dat$data$lcl, dat$data$ucl, dat$data$y, x,
p=2, M=15, pc=.20)
summary(fit2)
plot(fit2)
`

[Package ARCensReg version 3.0.1 Index]