simestgcv function

Single Index Model Estimation: Objective Function Approach.

This function provides an estimate of the non-parametric function and the index vector by minimizing an objective function specified by the method argument and also by choosing tuning parameter using GCV. data

simestgcv(x, y, w = NULL, beta.init = NULL, nmulti = NULL,
                lambda = NULL, maxit = 100, bin.tol = 1e-06, 
                beta.tol = 1e-05, agcv.iter = 100, progress = TRUE)

## Default S3 method:
simestgcv(x, y, w = NULL, beta.init = NULL, nmulti = NULL,
            lambda = NULL, maxit = 100, bin.tol = 1e-06, 
            beta.tol = 1e-05, agcv.iter = 100, progress = TRUE)

Arguments

• x: a numeric matrix giving the values of the predictor variables or covariates. For functions plot and print, x' is an object of class sim.est'.
• y: a numeric vector giving the values of the response variable.
• lambda: a numeric vector giving lower and upper bounds for penalty used in cvx.pen and cvx.lip.
• w: an optional numeric vector of the same length as $x$; Defaults to all 1.
• beta.init: An numeric vector giving the initial value for the index vector.
• nmulti: An integer giving the number of multiple starts to be used for iterative algorithm. If beta.init is provided then the nmulti is set to 1.
• agcv.iter: An integer providing the number of random numbers to be used in estimating GCV. See smooth.pen.reg for more details.
• progress: A logical denoting if progress of the algorithm to be printed. Defaults to TRUE.
• bin.tol: A tolerance level upto which the x values used in regression are recognized as distinct values.
• beta.tol: A tolerance level for stopping iterative algorithm for the index vector.
• maxit: An integer specifying the maximum number of iterations for each initial $\beta$ vector.

Details

The function minimizes

with no constraints on f. The penalty parameter $\lambda$ is choosen by the GCV criterion between the bounds given by lambda. Plot and predict function work as in the case of sim.est function.

Returns

An object of class `sim.est', basically a list including the elements - beta: A numeric vector storing the estimate of the index vector.

• nmulti: Number of multistarts used.

• x.mat: the input `x' matrix with possibly aggregated rows.

• BetaInit: a matrix storing the initial vectors taken or given for the index parameter.

• lambda: Given input lambda.

• K: an integer storing the row index of BetaInit which lead to the estimator beta.

• BetaPath: a list containing the paths taken by each initial index vector for nmulti times.

• ObjValPath: a matrix with nmulti rows storing the path of objective function value for multiple starts.

• convergence: a numeric storing convergence status for the index parameter.

• itervec: a vector of length nmulti storing the number of iterations taken by each of the multiple starts.

• iter: a numeric giving the total number of iterations taken.

• method: method is always set to "smooth.pen.reg".

• regress: An output of the regression function used needed for predict.

• x.values: sorted `x.betahat' values obtained by the algorithm.

• y.values: corresponding `y' values in input.

• fit.values: corresponding fit values of same length as that of $x\beta$.

• deriv: corresponding values of the derivative of same length as that of $x\beta$.

• residuals: residuals obtained from the fit.

• minvalue: minimum value of the objective function attained.

Source

Kuchibhotla, A. K., Patra, R. K. and Sen, B. (2015+). On Single Index Models with Convex Link.

Author(s)

Arun Kumar Kuchibhotla, arunku@wharton.upenn.edu

Examples

args(sim.est)
x <- matrix(runif(20*2,-1,1),ncol = 2)
b0 <- rep_len(1,2)/sqrt(2)
y <- (x%*%b0)^2 + rnorm(20,0,0.3)
tmp2 <- simestgcv(x, y, lambda = c(20^{1/6}, 20^{1/4}), nmulti = 1, 
                                        agcv.iter = 10, maxit = 10, beta.tol = 1e-03)
print(tmp2)
plot(tmp2)
predict(tmp2, newdata = c(0,0))