waldtypetest function

Robust Wald-type Tests

waldtypetest provides Wald-type tests for both simple and composite hypotheses for beta regression based on the robust estimators (LSMLE, LMDPDE, SMLE, and MDPDE).

waldtypetest(object, FUN, ...)

Arguments

• object: fitted model object of class robustbetareg (see robustbetareg).
• FUN: function representing the null hypothesis to be tested.
• ...: further arguments to be passed to the FUN function.

Returns

waldtypetest returns an output for the Wald-type test containing the value of the test statistic, degrees-of-freedom and p-value.

Details

The function provides a robust Wald-type test for a general hypothesis $m(\theta) = \eta_0$, for a fixed $\eta_0 \in R^d$, against a two sided alternative; see Maluf et al. (2022) for details. The argument FUN specifies the function $m(\theta) - \eta_0$, which defines the null hypothesis. For instance, consider a model with $\theta = (\beta_1, \beta_2, \beta_3, \gamma_1)^\top$ and let the null hypothesis be $\beta_2 + \beta_3 = 4$. The FUN argument can be FUN = function(theta) {theta[2] + theta[3] - 4}. It is also possible to define the function as FUN = function(theta, B) {theta[2] + theta[3] - B}, and pass the B argument through the waldtypetest function. If no function is specified, that is, FUN=NULL, the waldtypetest

returns a test in which the null hypothesis is that all the coefficients are zero.

Examples

# generating a dataset
set.seed(2022)
n <- 40
beta.coef <- c(-1, -2)
gamma.coef <- c(5)
X <- cbind(rep(1, n), x <- runif(n))
mu <- exp(X%*%beta.coef)/(1 + exp(X%*%beta.coef))
phi <- exp(gamma.coef) #Inverse Log Link Function
y <- rbeta(n, mu*phi, (1 - mu)*phi)
y[26] <- rbeta(1, ((1 + mu[26])/2)*phi, (1 - ((1 + mu[26])/2))*phi)
SimData <- as.data.frame(cbind(y, x))
colnames(SimData) <- c("y", "x")

# Fitting the MLE and the LSMLE
fit.mle <- robustbetareg(y ~ x | 1, data = SimData, alpha = 0)
fit.lsmle <- robustbetareg(y ~ x | 1, data = SimData)

# Hypothesis to be tested: (beta_1, beta_2) = c(-1, -2) against a two
# sided alternative
h0 <- function(theta){theta[1:2] - c(-1, -2)}
waldtypetest(fit.mle, h0)
waldtypetest(fit.lsmle, h0)
# Alternative way:
h0 <- function(theta, B){theta[1:2] - B}
waldtypetest(fit.mle, h0, B = c(-1, -2))
waldtypetest(fit.lsmle, h0, B = c(-1, -2))

References

Maluf, Y. S., Ferrari, S. L. P., and Queiroz, F. F. (2022). Robust beta regression through the logit transformation. arXiv:2209.11315.

Basu, A., Ghosh, A., Martin, N., and Pardo, L. (2018). Robust Wald-type tests for non-homogeneous observations based on the minimum density power divergence estimator. Metrika, 81:493–522.

Ribeiro, K. A. T. and Ferrari, S. L. P. (2022). Robust estimation in beta regression via maximum Lq-likelihood. Statistical Papers.

See Also

robustbetareg

Author(s)

Yuri S. Maluf (yurimaluf@gmail.com ), Francisco F. Queiroz (ffelipeq@outlook.com ) and Silvia L. P. Ferrari.