robustSFA (version 0.2.0)

bootstrap_test: Bootstrap Test for Comparing Estimates in Stochastic Frontier Models

Description

This function performs a bootstrap test to assess the closeness of two MDPD estimates at different values of \(\alpha\). This test is based on the observation that outliers particularly affect the estimation of \(\sigma^2\) and \(\lambda\). Accordingly, the bootstrap test is conducted using the following similarity measure:

$$ sim(\hat\theta_0,\hat\theta_1)= \frac{|\hat \sigma^2_1-\hat\sigma^2_0|}{\hat\sigma^2_0} +\frac{|\hat\lambda_1-\hat\lambda_0|}{\hat\lambda_0},$$

where \(\hat\theta_0\) and \(\hat\theta_1\) are the MDPD estimates corresponding to \(\alpha = \alpha_0\) and \(\alpha = \alpha_1\), respectively. If the two MDPD estimates are close, \(sim(\hat\theta_0,\hat\theta_1)\) will be close to zero. We note that this similarity measure differs from the one used in Song et al. (2017). Apart from this, the bootstrap procedure follows the same steps described in Song et al. (2017). A low p-value indicates that the two estimates are significantly different. Note that this test may require significant computational time, as it involves numerous estimation procedures.

Usage

bootstrap_test(formula, data = NULL, alpha0, alpha1, B = 99)

Value

A numeric. p-value of the bootstrap test.

Arguments

formula

A symbolic description of the model to be estimated, specified using the standard R formula syntax (e.g., y ~ x1 + x2).

data

A data frame containing the variables in the model.

alpha0

First value of \(\alpha\). The bootstrap samples are generated using the MDPD estimates with \(\alpha=\alpha_0\).

alpha1

Second value of \(\alpha\).

B

A numeric value specifying the number of bootstrap replications. The default is 99.

Examples

# \donttest{
## Example using the 'riceProdPhil' dataset from the `frontier` package
library(frontier)
data(riceProdPhil)

my.model <- log(PROD) ~ log(AREA) + log(LABOR) + log(NPK) + log(OTHER)


## Evaluate the closeness of ML estimates (alpha = 0) and
## MDPD estimates with alpha = 0.5.
bootstrap_test(my.model, data = riceProdPhil, alpha0=0.5, alpha1=0)


## Data with a single outlying observation
riceProdPhil2 <- riceProdPhil
riceProdPhil3 <- riceProdPhil

idx <- which.max(riceProdPhil$PROD)
riceProdPhil2$PROD[idx] <- riceProdPhil$PROD[idx]*10
riceProdPhil3$PROD[idx] <- riceProdPhil$PROD[idx]/100


## Evaluate the closeness of ML estimates (alpha = 0) and
## MDPD estimates with alpha = 0.5.
bootstrap_test( my.model, data = riceProdPhil2, alpha0=0.5, alpha1=0)
bootstrap_test( my.model, data = riceProdPhil3, alpha0=0.5, alpha1=0)
# }