Current Status Linear Regression

Description

Groeneboom and Hendrickx semiparametric binary response estimator (scalar case) score estimator based on NPMLE avoids any smoothing proposed by Groneboom and Hendrickx (2018).

Usage

GH(b, X, y, eps = 0.001)

Arguments

Value

A list with components:

• evaluation of a score function at parameter value

• estimated standard error

• sindex single index linear predictor

References

Groeneboom, P. and K. Hendrickx (2018) Current Status Linear Regression, Annals of Statistics, 46, 1415-1444,

Current Status Linear Regression Standard Errors

Description

Groeneboom and Hendrickx semiparametric binary response estimator (scalar case) score estimator based on NPMLE avoids any smoothing proposed by Groneboom and Hendrickx (2018).

Usage

GH.se(bstar, X, y, eps = 0.001, hc = 2)

Arguments

Value

A list with components:

• evaluation of a score function at parameter value

• estimated standard error

• sindex single index linear predictor

References

Groeneboom, P. and K. Hendrickx (2018) Current Status Linear Regression, Annals of Statistics, 46, 1415-1444,

Control parameters for Gautier-Kitamura bivariate random coefficient binary response

Description

These parameters can be passed via the ... argument of the rcbr function. defaults as suggested in Gautier and Kitamura matlab code

Usage

GK.control(n, u = -20:20/10, v = -20:20/10, T = 3, TX = 10, Mn = 1/log(n)^2)

Arguments

Value

updated list

Horowitz (1993) Modal Choice Data

Description

Modal choice data for journey to work in the Washington DC area from the late 1960's. The variables are: * 'DCOST': difference in cost of car versus transit (transit - car) * 'CARS': number of cars at home * 'DOVTT': difference in out of vehicle time (transit - car) * 'DIVTT': difference in in vehicle time (transit - car) * 'DEPEND': coded 1 if by car, 0 if by mass transit

Usage

Horowitz93

Format

A data frame with 842 observations on 5 variables:

Source

https://www.gams.com/latest/gamslib_ml/libhtml/gamslib_mws.html

References

Horowitz, J L, (1993) Semiparametric estimation of a work-trip mode choice model. Journal of Econometrics, 58, 49-70.

Control parameters for NPMLE of bivariate random coefficient binary response

Description

These parameters can be passed via the ... argument of the rcbr function. The first three arguments are only relevant if full cell enumeration is employed for bivariate version of the NPMLE.

Usage

KW.control(
  uv = NULL,
  u = NULL,
  v = NULL,
  initial = c(0, 0),
  epsbound = 1,
  epstol = 1e-07,
  presolve = 1,
  verb = 0
)

Arguments

Value

updated list

Dual optimization for Kiefer-Wolfowitz problems

Description

Interface function for calls to optimizer from various REBayes functions There is currently only one option for the optimization that based on Mosek. It relies on the Rmosek interface to R see installation instructions in the Readme file in the inst directory of this package. This version of the function is intended to work with versions of Mosek after 7.0. A more experimental option employing the pogs package available from https://github.com/foges/pogs and employing an ADMM (Alternating Direction Method of Multipliers) approach has been deprecated, those interested could try installing version 1.4 of REBayes, and following the instructions provided there.

Usage

KWDual(A, d, w, ...)

Arguments

Value

Returns a list with components:

. Mosek termination messages are treated as warnings from an R perspective since solutions producing, for example, MSK_RES_TRM_STALL: The optimizer is terminated due to slow progress, may still provide a satisfactory solution, especially when the return status variable is "optimal".

Author(s)

R. Koenker

References

Koenker, R and I. Mizera, (2013) “Convex Optimization, Shape Constraints, Compound Decisions, and Empirical Bayes Rules,” JASA, 109, 674–685.

Mosek Aps (2015) Users Guide to the R-to-Mosek Optimization Interface, https://docs.mosek.com/8.1/rmosek/index.html.

Koenker, R. and J. Gu, (2017) REBayes: An R Package for Empirical Bayes Mixture Methods, Journal of Statistical Software, 82, 1–26.

New Incremental Cell Enumeration (in) R

Description

Find interior points and cell counts of the polygons (cells) formed by a line arrangement.

Usage

NICER(A, b, initial = c(0, 0), verb = TRUE, epsbound = 1, epstol = 1e-07)

Arguments

Details

Modified version of the algorithm of Rada and Cerny (2018). The main modifications include preprocessing as hyperplanes are added to determine which new cells are created, thereby reducing the number of calls to the witness function to solve LPs, and treatment of degenerate configurations as well as those in "general position." When the hyperplanes are in general position the number of polytopes (cells) is determined by the elegant formula of Zazlavsky (1975)

m = {n \choose d} + n + 1

. In degenerate cases, i.e. when hyperplanes are not in general position, the number of cells is more complicated as considered by Alexanderson and Wetzel (1981). The function polycount is provided to check agreement with their results in an effort to aid in the selection of tolerances for the witness function. Current version is intended for use with d = 2, but the algorithm is adaptable to d > 2, and there is an experimental version called NICERd in the package.

Value

A list with components:

• SignVector a n by m matrix of signs determining position of cell relative to each hyperplane.

• w a d by m matrix of interior points for the m cells

References

Alexanderson, G.L and J.E. Wetzel, (1981) Arrangements of planes in space, Discrete Math, 34, 219–240. Gu, J. and R. Koenker (2020) Nonparametric Maximum Likelihood Methods for Binary Response Models with Random Coefficients, J. Am. Stat Assoc Rada, M. and M. Cerny (2018) A new algorithm for the enumeration of cells of hyperplane arrangements and a comparison with Avis and Fukada's reverse search, SIAM J. of Discrete Math, 32, 455-473. Zaslavsky, T. (1975) Facing up to arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes, Memoirs of the AMS, Number 154.

Examples

{
if(packageVersion("Rmosek") > "8.0.0"){
    A = cbind(c(1,-1,1,-2,2,1,3), c(1,1,1,1,1,-1,-2))
    B = matrix(c(3,1,7,-2,7,-1,1), ncol = 1)
    plot(NULL,xlim = c(-10,10),ylim = c(-10,10))
    for (i in 1:nrow(A))
	  abline(a = B[i,1]/A[i,2], b = -A[i,1]/A[i,2],col = i)
    f = NICER(A, B)
    for (j in 1:ncol(f$SignVector))
    	  points(f$w[1,j], f$w[2,j], cex = 0.5)
    }
}

New (Accelerated) Incremental Cell Enumeration (in) R

Description

Find interior points and cell counts of the polygons (polytopes) formed by a hyperplane arrangement.

Usage

NICERd(
  A,
  b,
  initial = rep(0, ncol(A)),
  verb = TRUE,
  accelerate = FALSE,
  epsbound = 1,
  epstol = 1e-07
)

Arguments

Details

Modified version of the algorithm of Rada and Cerny (2018). The main modifications include preprocessing as hyperplanes are added to determine which new cells are created, thereby reducing the number of calls to the witness function to solve LPs, and treatment of degenerate configurations as well as those in "general position." (for d=2 for now). When the hyperplanes are in general position the number of cells (polytopes) is determined by the elegant formula of Zaslavsky (1975)

m = {n \choose d} + n + 1

. In degenerate cases, i.e. when hyperplanes are not in general position, the number of cells is more complicated as considered by Alexanderson and Wetzel (1981). The function polycount is provided to check agreement with their results in an effort to aid in the selection of tolerances for the witness function for arrangement in d=2. The current version is intended mainly for use with d = 2, but the algorithm is adapted to the general position setting with d > 2, although it requires hyperplanes in general position and may require some patience when both the sample size is large. if hyperplanes not general position (i.e. all cross at origin), turn off accelerate

Value

A list with components:

• SignVector a n by m matrix of signs determining position of cell relative to each hyperplane.

• w a d by m matrix of interior points for the m cells

References

Alexanderson, G.L and J.E. Wetzel, (1981) Arrangements of planes in space, Discrete Math, 34, 219–240. Rada, M. and M. Cerny (2018) A new algorithm for the enumeration of cells of hyperplane arrangements and a comparison with Avis and Fukada's reverse search, SIAM J. of Discrete Math, 32, 455-473. Zaslavsky, T. (1975) Facing up to arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes, Memoirs of the AMS, Number 154.

Prediction of Bounds on Marginal Effects

Description

Given a fitted model by the exact NPMLE procedure prediction is made at a new design point with lower and upper bounds for the prediction due to ambiguity of the assignment of mass within the cell enumerated polygons.

Usage

bounds.KW2(object, ...)

Arguments

Value

a list consisting of the following components:

phat

Point prediction

lower

lower bound prediction

upper

upper bound prediction

xpoly

indices of crossed polygons

Author(s)

Jiaying Gu

See Also

predict.KW2 for a simpler prediction function without bounds

log likelihood for Gautier Kitamura procedure

Description

log likelihood for Gautier Kitamura procedure

Usage

## S3 method for class 'GK'
logLik(object, ...)

Arguments

Value

a scalar log likelihood

log likelihood for KW1 procedure

Description

log likelihood for KW1 procedure

Usage

## S3 method for class 'KW1'
logLik(object, ...)

Arguments

Value

a scalar log likelihood

Check Neighbouring Cell Counts

Description

Compare cell counts for each cell with its neighbours and return indices of the locally maximal cells.

Usage

neighbours(SignVector)

Arguments

Value

Column indices of the cells that are locally maximal, i.e. those whose neighbours have strictly fewer cell counts. The corresponding interior points of these cells can be used as potential mass points for the NPMLE function rcbr.fit.KW.

Plot a GK object

Description

Given a fitted model by the Guatier-Kitamura procedure plot the estimated density contours

Usage

## S3 method for class 'GK'
plot(x, ...)

Arguments

Value

nothing (invisibly)

Plot a KW2 object

Description

Given a fitted model by the rcbr NPMLE procedure plot the estimated mass points

Usage

## S3 method for class 'KW2'
plot(x, smooth = 0, pal = NULL, inches = 1/6, N = 25, tol = 0.001, ...)

Arguments

Value

nothing (invisibly)

Check Cell Count for degenerate hyperplane arrangements

Description

When the hyperplane arrangement is degenerate, i.e. not in general position, the number of distinct cells can be checked against the formula of Alexanderson and Wetzel (1981).

Usage

polycount(A, b, maxints = 10)

Arguments

Value

number of distinct cells

References

Alexanderson, G.L and J.E. Wetzel, (1981) Arrangements of planes in space, Discrete Math, 34, 219–240.

Identify crossed polygons from existing cells when adding a new line (works only for dim = 2)

Description

Given an existing cell configuration represented by the Signvector and associated interior points w, identify the polygons crossed by the next new line.

Usage

polyzone(SignVector, w, A, b)

Arguments

Value

vector of indices of crossed polygons

Author(s)

Jiaying Gu

Profiling estimation methods for RCBR models

Description

Profile likelihood and (GEE) score methods for estimation of random coefficient binary response models. This function is a wrapper for rcbr that uses the offset argument to implement estimation of additional fixed parameters. It may be useful to restrict the domain of the optimization over the profiled parameters, this can be accomplished, at least for box constraints by setting omethod = "L-BFGS-B" and specifying the lo and up accordingly.

Usage

prcbr(
  formula,
  b0,
  data,
  logL = TRUE,
  omethod = "BFGS",
  lo = -Inf,
  up = Inf,
  ...
)

Arguments

Value

a list comprising the components:

bopt

output of the optimizer for the profiled parameters beta

fopt

output of the optimizer for the random coefficients eta

Prediction of Marginal Effects

Description

Given a fitted model by the Gautier Kitamura procedure predictions are made at new design points given by the newdata argument.

Usage

## S3 method for class 'GK'
predict(object, ...)

Arguments

Value

a vector pf predicted probabilities

Prediction of Marginal Effects

Description

Given a fitted model by the rcbr NPMLE procedure predictions are made at new design points given by the newdata argument.

Usage

## S3 method for class 'KW2'
predict(object, ...)

Arguments

Value

a vector pf predicted probabilities

See Also

bound.KW2 for a prediction function with bounds

Estimation of Random Coefficient Binary Response Models

Description

Two methods are implemented for estimating binary response models with random coefficients: A nonparametric maximum likelihood method proposed by Cosslett (1986) and extended by Ichimura and Thompson (1998), and a (hemispherical) deconvolution method proposed by Gautier and and Kitamura (2013). The former is closely related to the NPMLE for mixture models of Kiefer and Wolfowitz (1956). The latter is an R translation of the matlab implementation of Gautier and Kitamura.

Usage

rcbr(formula, data, subset, offset, mode = "GK", ...)

Arguments

Details

The predict method produces estimates of the probability of a "success" (y = 1) for a particular vector, (z,v), when aggregated over the estimated distribution of random coefficients.

The logLik produces an evaluation of the log likelihood value associated with a fitted model.

Value

of object of class GK, KW1, with components described in further detail in the respective fitting functions.

Author(s)

Jiaying Gu and Roger Koenker

References

Kiefer, J. and J. Wolfowitz (1956) Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters, Ann. Math. Statist, 27, 887-906.

Cosslett, S. (1983) Distribution Free Maximum Likelihood Estimator of the Binary Choice Model, Econometrica, 51, 765-782.

Gautier, E. and Y. Kitamura (2013) Nonparametric estimation in random coefficients binary choice models, Ecoonmetrica, 81, 581-607.

Gu, J. and R. Koenker (2020) Nonparametric Maximum Likelihood Methods for Binary Response Models with Random Coefficients, J. Am. Stat Assoc

Groeneboom, P. and K. Hendrickx (2016) Current Status Linear Regression, preprint available from https://arxiv.org/abs/1601.00202.

Ichimura, H. and T. S. Thompson, (1998) Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution," Journal of Econometrics, 86, 269-295.

Examples

{
if(packageVersion("Rmosek") > "8.0.0"){
    # Simple Test Problem for rcbr
    n <- 60
    B0 = rbind(c(0.7,-0.7,1),c(-0.7,0.7,1))
    z <- rnorm(n)
    v <- rnorm(n)
    s <- sample(0:1, n, replace = TRUE)
    XB0 <- cbind(1,z,v) %*% t(B0)
    u <- s * XB0[,1] + (1-s) * XB0[,2]
    y <- (u > 0) - 0
    D <- data.frame(z = z, v = v, y = y)
    f <- rcbr(y ~ z + v, mode = "KW", data = D)
    plot(f)
    # Simple Test Problem for rcbr
    set.seed(15)
    n <- 100
    B0 = rbind(c(0.7,-0.7,1),c(-0.7,0.7,1))
    z <- rnorm(n)
    v <- rnorm(n)
    s <- sample(0:1, n, replace = TRUE)
    XB0 <- cbind(1,z,v) %*% t(B0)
    u <- s * XB0[,1] + (1-s) * XB0[,2]
    y <- (u > 0) - 0
    D <- data.frame(z = z, v = v, y = y)
    f <- rcbr(y ~ z + v, mode = "GK", data = D)
    contour(f$u, f$v, matrix(f$w, length(f$u)))
    points(x = 0.7, y = -0.7, col = 2)
    points(x = -0.7, y = 0.7, col = 2)
    f <- rcbr(y ~ z + v, mode = "GK", data = D, T = 7)
    contour(f$u, f$v, matrix(f$w, length(f$u)))
    points(x = 0.7, y = -0.7, col = 2)
    points(x = -0.7, y = 0.7, col = 2)
    }
}

Fitting of Random Coefficient Binary Response Models

Description

Two methods are implemented for estimating binary response models with random coefficients: A nonparametric maximum likelihood method proposed by Cosslett (1986) and extended by Ichimura and Thompson (1998), and a (hemispherical) deconvolution method proposed by Gautier and and Kitamura (2013). The former is closely related to the NPMLE for mixture models of Kiefer and Wolfowitz (1956). The latter is an R translation of the matlab implementation of Gautier and Kitamura.

Usage

rcbr.fit(x, y, offset = NULL, mode = "KW", control)

Arguments

Details

The predict method produces estimates of the probability of a "success" (y = 1) for a particular vector, (z,v), when aggregated over the estimated distribution of random coefficients.

Value

of object of class GK, KW1, with components described in further detail in the respective fitting functions.

Author(s)

Jiaying Gu and Roger Koenker

References

Kiefer, J. and J. Wolfowitz (1956) Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters, Ann. Math. Statist, 27, 887-906.

Cosslett, S. (1983) Distribution Free Maximum Likelihood Estimator of the Binary Choice Model, Econometrica, 51, 765-782. Gautier, E. and Y. Kitamura (2013) Nonparametric estimation in random coefficients binary choice models, Ecoonmetrica, 81, 581-607.

Groeneboom, P. and K. Hendrickx (2016) Current Status Linear Regression, preprint available from https://arxiv.org/abs/1601.00202.

Ichimuma, H. and T. S. Thompson, (1998) Maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution," Journal of Econometrics, 86, 269-295.

Gautier and Kitamura (2013) bivariate random coefficient binary response

Description

This is an implementation based on the matlab version of Gautier and Kitamura's deconvolution method for the bivariate random coefficient binary response model. Methods based on the fitted object are provided for predict, logLik and plot.requires orthopolynom package for Gegenbauer polynomials

Usage

rcbr.fit.GK(X, y, control)

Arguments

Value

a list with components:

u

grid values

v

grid values

w

estimated function values on 2d u x v grid

X

design matrix

y

response vector

Author(s)

Gautier and Kitamura for original matlab version, Jiaying Gu and Roger Koenker for the R translation.

References

Gautier, E. and Y. Kitamura (2013) Nonparametric estimation in random coefficients binary choice models, Ecoonmetrica, 81, 581-607.

NPMLE fitting for the Cosslett random coefficient binary response model

Description

This is the original one dimensional version of the Cosslett model, also known as the current status model:

P(y = 1 | v) = \int I (\eta > v)dF(\eta).

invoked with the formula y ~ v. By default the algorithm computes a vector of potential locations for the mass points of \hat F by finding interior points of the intervals between the ordered v, and then solving a convex optimization problem to determine these masses. Alternatively, a vector of predetermined locations can be passed via the control argument. Additional covariate effects can be accommodated by either specifying a fixed offset in the call to rcbr or by using the profile likelihood function prcbr.

Usage

rcbr.fit.KW1(X, y, control)

Arguments

Value

a list with components:

• x evaluation points for the fitted distribution

• y estimated mass associated with the v points

• logLik the loglikelihood value of the fit

• status mosek solution status

Author(s)

Jiaying Gu and Roger Koenker

References

Gu, J. and R. Koenker (2018) Nonparametric maximum likelihood estimation of the random coefficients binary choice model, preprint.

NPMLE fitting for random coefficient binary response model

Description

Exact NPMLE fitting requires that the uv argument contain a matrix whose rows represent points in the interior of the locally maximal polytopes determined by the hyperplane arrangement of the observations. If it is not provided it will be computed afresh here; since this can be somewhat time consuming, uv is included in the returned object so that it can be reused if desired. Approximate NPMLE fitting can be achieved by specifying an equally spaced grid of points at which the NPMLE can assign mass using the arguments u and v. If the design matrix X contains only 2 columns, so we have the Cosslett, aka current status, model then the polygons in the prior description collapse to intervals and the default method computes the locally maximal count intervals and passes their interior points to the optimizer of the log likelihood. Alternatively, as in the bivariate case one can specify a grid to obtain an approximate solution.

Usage

rcbr.fit.KW2(x, y, control)

Arguments

Value

a list with components:

• uv evaluation points for the fitted distribution

• W estimated mass associated with the uv points

• logLik the loglikelihood value of the fit

• status mosek solution status

Author(s)

Jiaying Gu and Roger Koenker

References

Gu, J. and R. Koenker (2018) Nonparametric maximum likelihood estimation of the random coefficients binary choice model, preprint.

Find witness point

Description

Find (if possible) an interior point of a polytope solving a linear program

Usage

witness(A, b, s, epsbound = 1, epstol = 1e-07, presolve = 1, verb = 0)

Arguments

Details

Solves LP: max over {w,eps} {eps | SAw - eps >= Sb, 0 < eps <= epsbound} S is diag(s), if at the solution eps > 0, then w is a valid interior point otherwise the LP fails to find an interior point, another s must be tried. Constructs a problem formulation that can be passed to Rmosek for solution.

Value

List with components:

• w proposed interior point at solution

• fail indicator of whether w is a valid interior point