simestgcv: Single Index Model Estimation: Objective Function Approach.
In simest: Constrained Single Index Model Estimation

Description Usage Arguments Details Value Author(s) Source Examples

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.

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)

`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 β vector.

The function minimizes

∑_{i=1}^n w_i(y_i - f(x_i^{\top}β))^2 + λ\int\{f''(x)\}^2dx

with no constraints on f. The penalty parameter λ 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.

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 xβ.
`deriv`	corresponding values of the derivative of same length as that of xβ.
`residuals`	residuals obtained from the fit.
`minvalue`	minimum value of the objective function attained.

Arun Kumar Kuchibhotla, arunku@wharton.upenn.edu

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

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))

Loading required package: nnls
Loading required package: cobs
function (x, y, w = NULL, beta.init = NULL, nmulti = NULL, L = NULL, 
    lambda = NULL, maxit = 100, bin.tol = 1e-05, beta.tol = 1e-05, 
    method = c("cvx.pen", "cvx.lip", "cvx.lse", "smooth.pen"), 
    progress = TRUE, force = FALSE) 
NULL

multistart  1  of  1  done!
Call:
simestgcv.default(x = x, y = y, nmulti = 1, lambda = c(20^{
    1/6
}, 20^{
    1/4
}), maxit = 10, beta.tol = 0.001, agcv.iter = 10)
Estimate of beta is:
[1] 0.1660906 0.9861105
Number of Starting Vectors:
[1] 1
Initial vector leading to the minimum:
[1] 0.1660906 0.9861105
Minimum Criterion Value Obtained:
[1] 1.070559
Total Number of Iterations:
[1] 2
Convergence Status:
[1] "1 out of 1 converged"
[1] 0.3739226