npbr (version 1.8)

poly_degree: AIC and BIC criteria for choosing the optimal degree of the polynomial frontier estimator

Description

Computes the optimal degree of the unconstrained polynomial frontier estimator proposed by Hall, Park and Stern (1998).

Usage

poly_degree(xtab, ytab, prange=0:20, type="AIC", 
 control = list("tm_limit" = 700))

Value

Returns an integer.

Arguments

xtab

a numeric vector containing the observed inputs \(x_1,\ldots,x_n\).

ytab

a numeric vector of the same length as xtab containing the observed outputs \(y_1,\ldots,y_n\).

prange

a vector of integers specifying the range in which the optimal degree of the polynomial frontier estimator is to be selected.

type

a character equal to "AIC" or "BIC".

control

a list of parameters to the GLPK solver. See *Details* of help(Rglpk_solve_LP).

Author

Hohsuk Noh.

Details

As the degree \(p\) of the polynomial estimator \(\hat \varphi_{n,p}\) (see poly_est) determines the dimensionality of the approximating function, we may view the problem of choosing p as model selection. By analogy to the information criteria proposed by Daouia et al. (2016) in the boundary regression context, we obtain the optimal polynomial degree by minimizing $$ AIC(p) = \log \left( \sum_{i=1}^{n} (\hat \varphi_{n,p}(x_i)-y_i)\right) + (p+1)/n ,$$ $$BIC(p) = \log \left( \sum_{i=1}^{n} (\hat \varphi_{n,p}(x_i)-y_i)\right) + \log n (p+1)/(2n).$$ The first one (option type = "AIC") is similar to the famous Akaike information criterion Akaike (1973) and the second one (option type = "BIC") to the Bayesian information criterion Schwartz (1978).

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, in Second International Symposium of Information Theory, eds. B. N. Petrov and F. Csaki, Budapest: Akademia Kiado, 267--281.

Daouia, A., Noh, H. and Park, B.U. (2016). Data Envelope fitting with constrained polynomial splines. Journal of the Royal Statistical Society: Series B, 78(1), 3-30. doi:10.1111/rssb.12098.

Hall, P., Park, B.U. and Stern, S.E. (1998). On polynomial estimators of frontiers and boundaries. Journal of Multivariate Analysis, 66, 71-98.

Schwartz, G. (1978). Estimating the dimension of a model, Annals of Statistics, 6, 461--464.

See Also

poly_est

Examples

data("air")
x.air <- seq(min(air$xtab), max(air$xtab), 
 length.out = 101)
# Optimal polynomial degrees via the AIC criterion
(p.aic.air <- poly_degree(air$xtab, air$ytab, 
 type = "AIC"))
# Optimal polynomial degrees via the BIC criterion  
(p.bic.air <- poly_degree(air$xtab, air$ytab, 
 type = "BIC"))