npcdist {np} | R Documentation

## Kernel Conditional Distribution Estimation with Mixed Data Types

### Description

`npcdist` computes kernel cumulative conditional distribution estimates on p+q-variate evaluation data, given a set of training data (both explanatory and dependent) and a bandwidth specification (a `condbandwidth` object or a bandwidth vector, bandwidth type, and kernel type) using the method of Li and Racine (2008) and Li, Lin, and Racine (2013). The data may be continuous, discrete (unordered and ordered factors), or some combination thereof.

### Usage

    npcdist(bws, ...)

    ## S3 method for class 'formula'
    npcdist(bws, data = NULL, newdata = NULL, ...)

    ## S3 method for class 'call'
    npcdist(bws, ...)

    ## S3 method for class 'condbandwidth'
    npcdist(bws,
            txdat = stop("invoked without training data 'txdat'"),
            tydat = stop("invoked without training data 'tydat'"),
            exdat,
            eydat,
            gradients = FALSE,
            ...)

    ## Default S3 method:
    npcdist(bws, txdat, tydat, ...)

### Arguments

`bws`

a bandwidth specification. This can be set as a `condbandwidth` object returned from a previous invocation of `npcdistbw`, or as a p+q-vector of bandwidths, with each element i up to i=q corresponding to the bandwidth for column i in `tydat`, and each element i from i=q+1 to i=p+q corresponding to the bandwidth for column i-q in `txdat`. If specified as a vector, then additional arguments will need to be supplied as necessary to specify the bandwidth type, kernel types, training data, and so on.

`gradients` | a logical value specifying whether to return estimates of the gradients at the evaluation points. Defaults to `FALSE`.

`...`

additional arguments supplied to specify the bandwidth type, kernel types, and so on. This is necessary if you specify bws as a p+q-vector and not a `condbandwidth` object, and you do not desire the default behaviours. To do this, you may specify any of `bwmethod`, `bwscaling`, `bwtype`, `cxkertype`, `cxkerorder`, `cykertype`, `cykerorder`, `uxkertype`, `oxkertype`, `oykertype`, as described in `npcdistbw`.

`data` | an optional data frame, list or environment (or object coercible to a data frame by `as.data.frame`) containing the variables in the model. If not found in data, the variables are taken from `environment(bws)`, typically the environment from which `npcdistbw` was called.
`newdata` | An optional data frame in which to look for evaluation data. If omitted, the training data are used.
`txdat` | a p-variate data frame of sample realizations of explanatory data (training data). Defaults to the training data used to compute the bandwidth object.
`tydat` | a q-variate data frame of sample realizations of dependent data (training data). Defaults to the training data used to compute the bandwidth object.
`exdat` | a p-variate data frame of explanatory data on which cumulative conditional distributions will be evaluated. By default, evaluation takes place on the data provided by `txdat`.
`eydat` | a q-variate data frame of dependent data on which cumulative conditional distributions will be evaluated. By default, evaluation takes place on the data provided by `tydat`.

### Details

`npcdist` implements a variety of methods for estimating multivariate conditional cumulative distributions (p+q-variate) defined over a set of possibly continuous and/or discrete (unordered, ordered) data. The approach is based on Li and Racine (2004) who employ ‘generalized product kernels’ that admit a mix of continuous and discrete data types.

Three classes of kernel estimators for the continuous data types are available: fixed, adaptive nearest-neighbor, and generalized nearest-neighbor. Adaptive nearest-neighbor bandwidths change with each sample realization in the set, x[i], when estimating the cumulative conditional distribution at the point x. Generalized nearest-neighbor bandwidths change with the point at which the cumulative conditional distribution is estimated, x. Fixed bandwidths are constant over the support of x.

Training and evaluation input data may be a mix of continuous (default), unordered discrete (to be specified in the data frames using `factor`), and ordered discrete (to be specified in the data frames using `ordered`). Data can be entered in an arbitrary order and data types will be detected automatically by the routine (see `np` for details).

A variety of kernels may be specified by the user. Kernels implemented for continuous data types include the second, fourth, sixth, and eighth order Gaussian and Epanechnikov kernels, and the uniform kernel. Unordered discrete data types use a variation on Aitchison and Aitken's (1976) kernel, while ordered data types use a variation of the Wang and van Ryzin (1981) kernel.

### Value

`npcdist` returns a `condistribution` object. The generic accessor functions `fitted`, `se`, and `gradients`, extract estimated values, asymptotic standard errors on estimates, and gradients, respectively, from the returned object. Furthermore, the functions `predict`, `summary` and `plot` support objects of both classes. The returned objects have the following components:

`xbw` | bandwidth(s), scale factor(s) or nearest neighbours for the explanatory data, `txdat`
`ybw` | bandwidth(s), scale factor(s) or nearest neighbours for the dependent data, `tydat`
`xeval` | the evaluation points of the explanatory data
`yeval` | the evaluation points of the dependent data
`condist` | estimates of the conditional cumulative distribution at the evaluation points
`conderr` | standard errors of the cumulative conditional distribution estimates
`congrad` | if invoked with `gradients = TRUE`, estimates of the gradients at the evaluation points
`congerr` | if invoked with `gradients = TRUE`, standard errors of the gradients at the evaluation points
`log_likelihood` | log likelihood of the cumulative conditional distribution estimate

### Usage Issues

If you are using data of mixed types, then it is advisable to use the `data.frame` function to construct your input data and not `cbind`, since `cbind` will typically not work as intended on mixed data types and will coerce the data to the same type.

### Author(s)

Tristen Hayfield tristen.hayfield@gmail.com, Jeffrey S. Racine racinej@mcmaster.ca

### References

Aitchison, J. and C.G.G. Aitken (1976), “Multivariate binary discrimination by the kernel method,” Biometrika, 63, 413-420.

Hall, P. and J.S. Racine and Q. Li (2004), “Cross-validation and the estimation of conditional probability densities,” Journal of the American Statistical Association, 99, 1015-1026.

Li, Q. and J.S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press.

Li, Q. and J.S. Racine (2008), “Nonparametric estimation of conditional CDF and quantile functions with mixed categorical and continuous data,” Journal of Business and Economic Statistics, 26, 423-434.

Li, Q. and J. Lin and J.S. Racine (2013), “Optimal bandwidth selection for nonparametric conditional distribution and quantile functions”, Journal of Business and Economic Statistics, 31, 57-65.

Pagan, A. and A. Ullah (1999), Nonparametric Econometrics, Cambridge University Press.

Scott, D.W. (1992), Multivariate Density Estimation. Theory, Practice and Visualization, New York: Wiley.

Silverman, B.W. (1986), Density Estimation, London: Chapman and Hall.

Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,” Biometrika, 68, 301-309.

### See Also

`npudens`

### Examples

    ## Not run:
    # EXAMPLE 1 (INTERFACE=FORMULA): For this example, we load Giovanni
    # Baiocchi's Italian GDP panel (see Italy for details), and compute the
    # cross-validated bandwidths (default) using a second-order Gaussian
    # kernel (default). Note - this may take a minute or two depending on
    # the speed of your computer.

    data("Italy")
    attach(Italy)

    # First, compute the bandwidths.

    bw <- npcdistbw(formula=gdp~ordered(year))

    # Next, compute the condistribution object...

    Fhat <- npcdist(bws=bw)

    # The object Fhat now contains results such as the estimated cumulative
    # conditional distribution function (Fhat$condist) and so on...

    summary(Fhat)

    # Call the plot() function to visualize the results (<ctrl>-C will
    # interrupt on *NIX systems, <esc> will interrupt on MS Windows
    # systems).

    plot(bw)

    detach(Italy)

    # EXAMPLE 1 (INTERFACE=DATA FRAME): For this example, we load Giovanni
    # Baiocchi's Italian GDP panel (see Italy for details), and compute the
    # cross-validated bandwidths (default) using a second-order Gaussian
    # kernel (default). Note - this may take a minute or two depending on
    # the speed of your computer.

    data("Italy")
    attach(Italy)

    # First, compute the bandwidths.

    # Note - we cast `X' and `y' as data frames so that plot() can
    # automatically grab names (this looks like overkill, but in
    # multivariate settings you would do this anyway, so may as well get in
    # the habit).

    X <- data.frame(year=ordered(year))
    y <- data.frame(gdp)

    bw <- npcdistbw(xdat=X, ydat=y)

    # Next, compute the condistribution object...

    Fhat <- npcdist(bws=bw)

    # The object Fhat now contains results such as the estimated cumulative
    # conditional distribution function (Fhat$condist) and so on...

    summary(Fhat)

    # Call the plot() function to visualize the results (<ctrl>-C will
    # interrupt on *NIX systems, <esc> will interrupt on MS Windows systems).

    plot(bw)

    detach(Italy)

    # EXAMPLE 2 (INTERFACE=FORMULA): For this example, we load the old
    # faithful geyser data from the R `datasets' library and compute the
    # conditional distribution function.

    library("datasets")
    data("faithful")
    attach(faithful)

    # Note - this may take a few minutes depending on the speed of your
    # computer...

    bw <- npcdistbw(formula=eruptions~waiting)

    summary(bw)

    # Plot the conditional cumulative distribution function (<ctrl>-C will
    # interrupt on *NIX systems, <esc> will interrupt on MS Windows
    # systems).

    plot(bw)

    detach(faithful)

    # EXAMPLE 2 (INTERFACE=DATA FRAME): For this example, we load the old
    # faithful geyser data from the R `datasets' library and compute the
    # cumulative conditional distribution function.

    library("datasets")
    data("faithful")
    attach(faithful)

    # Note - this may take a few minutes depending on the speed of your
    # computer...

    # Note - we cast `X' and `y' as data frames so that plot() can
    # automatically grab names (this looks like overkill, but in
    # multivariate settings you would do this anyway, so may as well get in
    # the habit).

    X <- data.frame(waiting)
    y <- data.frame(eruptions)

    bw <- npcdistbw(xdat=X, ydat=y)

    summary(bw)

    # Plot the conditional cumulative distribution function (<ctrl>-C will
    # interrupt on *NIX systems, <esc> will interrupt on MS Windows systems)

    plot(bw)

    detach(faithful)

    ## End(Not run)

[Package np version 0.60-9 Index]