Fit of a $L_q$ Regression Model

Fits a regression model in the $L_q$ norm (also labeled as the $L_p$ norm). In more detail, the optimization function $ \sum_i | y_i - x_i \beta | ^p$ is optimized. The nondifferentiable function is approximated by a differentiable approximation, i.e., we use $|x| \approx \sqrt{x^2 + \varepsilon } $. The power $p$ can also be estimated by using est_pow=TRUE, see Giacalone, Panarello and Mattera (2018). The algorithm iterates between estimating regression coefficients and the estimation of power values. The estimation of the power based on a vector of residuals e can be conducted using the function lq_fit_estimate_power.

Using the $L_q$ norm in the regression is equivalent to assuming an expontial power function for residuals (Giacalone et al., 2018). The density function and a simulation function is provided by dexppow and rexppow, respectively. See also the normalp package.

Usage

lq_fit(y, X, w=NULL, pow=2, eps=0.001, beta_init=NULL, est_pow=FALSE, optimizer="optim",
    eps_vec=10^seq(0,-10, by=-.5), conv=1e-4, miter=20, lower_pow=.1, upper_pow=5)

lq_fit_estimate_power(e, pow_init=2, lower_pow=.1, upper_pow=10)

dexppow(x, mu=0, sigmap=1, pow=2, log=FALSE)

rexppow(n, mu=0, sigmap=1, pow=2, xbound=100, xdiff=.01)

Arguments

y: Dependent variable
X: Design matrix
w: Optional vector of weights
pow: Power $p$ in $L_q$ norm- The power $p=0$ is handled using the loss function $f(e)=e^2/(e^2+\varepsilon)$.
est_pow: Logical indicating whether power should be estimated
eps: Parameter governing the differentiable approximation
e: Vector of resiuals
pow_init: Initial value of power
beta_init: Initial vector
optimizer: Can be "optim" or "nlminb".
eps_vec: Vector with decreasing $\varepsilon$ values used in optimization
conv: Convergence criterion
miter: Maximum number of iterations
lower_pow: Lower bound for estimated power
upper_pow: Upper bound for estimated power
x: Vector
mu: Location parameter
sigmap: Scale parameter
log: Logical indicating whether the logarithm should be provided
n: Sample size
xbound: Lower and upper bound for density approximation
xdiff: Grid width for density approximation

Value

List with following several entries

coefficients: Vector of coefficients
res_optim: Results of optimization
...: More values

References

Giacalone, M., Panarello, D., & Mattera, R. (2018). Multicollinearity in regression: an efficiency comparison between $L_p$-norm and least squares estimators. Quality & Quantity, 52(4), 1831-1859. doi:10.1007/s11135-017-0571-y

Examples

#############################################################################
# EXAMPLE 1: Small simulated example with fixed power
#############################################################################

set.seed(98)
N <- 300
x1 <- stats::rnorm(N)
x2 <- stats::rnorm(N)
par1 <- c(1,.5,-.7)
y <- par1[1]+par1[2]*x1+par1[3]*x2 + stats::rnorm(N)
X <- cbind(1,x1,x2)

#- lm function in stats
mod1 <- stats::lm.fit(y=y, x=X)

#- use lq_fit function
mod2 <- sirt::lq_fit( y=y, X=X, pow=2, eps=1e-4)
mod1$coefficients
mod2$coefficients

if (FALSE) {
#############################################################################
# EXAMPLE 2: Example with estimated power values
#############################################################################

#*** simulate regression model with residuals from the exponential power distribution
#*** using a power of .30
set.seed(918)
N <- 2000
X <- cbind( 1, c(rep(1,N), rep(0,N)) )
e <- sirt::rexppow(n=2*N, pow=.3, xdiff=.01, xbound=200)
y <- X %*% c(1,.5) + e

#*** estimate model
mod <- sirt::lq_fit( y=y, X=X, est_pow=TRUE, lower_pow=.1)
mod1 <- stats::lm( y ~ 0 + X )
mod$coefficients
mod$pow
mod1$coefficients
}

Fit of a \(L_q\) Regression Model

Usage

Arguments

Value

References

Examples