Fast and stable algorithm for nonparametric estimation of regression
functions and their derivatives via **l**ocal **p**olynomials with
**Epa**nechnikov weight function.

1 2 |

`x` |
vector of design points, not necessarily ordered. |

`y` |
vector of observations of the same length as |

`bandwidth` |
bandwidth(s) for nonparametric estimation. Either a
number or a vector of the same length as |

`deriv` |
order of derivative of the regression function to be
estimated; defaults to |

`n.out` |
number of output design points where the function has to
be estimated. The default is |

`x.out` |
vector of output design points where the function has to
be estimated. The default value is an equidistant grid of |

`order` |
integer, order of the polynomial used for local
polynomials. Must be |

`mnew` |
integer forcing to restart the algorithm after |

`var` |
logical flag: if |

More details are described in the first reference SBE\&G (1994)
below. In S\&G, a bad finite sample behaviour of local polynomials
for random designs was found.
For practical use, we therefore propose local polynomial regression
fitting with ridging, as implemented in the function
`lpridge`

. In `lpepa`

, several parameters described
in SBE\&G are fixed either in the fortran
routine or in the **R**-function. There, you find comments how to change them.

For `var=TRUE`

, the variance of the estimator proportional to the
residual variance is computed, i.e., the exact finite sample variance of the
regression estimator is `var(est) = est.var * sigma^2`

.

a list including used parameters and estimator.

`x` |
vector of ordered design points. |

`y` |
vector of observations ordered according to x. |

`bandwidth` |
vector of bandwidths actually used for nonparametric estimation. |

`deriv` |
order of derivative of the regression function estimated. |

`x.out` |
vector of ordered output design points. |

`order` |
order of the polynomial used for local polynomials. |

`mnew` |
force to restart the algorithm after mnew updating steps. |

`var` |
logical flag: whether the variance of the estimator was computed. |

`est` |
estimator of the derivative of order deriv of the regression function. |

`est.var` |
estimator of the variance of est (proportional to residual variance). |

See also http://www.unizh.ch/biostat/ under ‘Manuscripts’ etc.

- Numerical stability and computational speed:

B. Seifert, M. Brockmann, J. Engel and T. Gasser (1994)
Fast algorithms for nonparametric curve estimation.
*J. Computational and Graphical Statistics* **3**, 192–213.

- Statistical properties:

Seifert, B. and Gasser, T. (1996)
Finite sample variance of local polynomials: Analysis and solutions.
*J. American Statistical Association* **91**(433), 267–275.

Seifert, B. and Gasser, T. (2000)
Data adaptive ridging in local polynomial
regression. *J. Computational and Graphical Statistics* **9**,
338–360.

Seifert, B. and Gasser, T. (1998)
Ridging Methods in Local Polynomial Regression.
in: S. Weisberg (ed), *Dimension Reduction, Computational Complexity,
and Information*, Vol.**30** of Computing Science \& Statistics,
Interface Foundation of North America, 467–476.

Seifert, B. and Gasser, T. (1998)
Local polynomial smoothing.
in: *Encyclopedia of Statistical Sciences*,
Update Vol.**2**, Wiley, 367–372.

Seifert, B., and Gasser, T. (1996)
Variance properties of local polynomials and ensuing
modifications. in: *Statistical Theory and Computational Aspects
of Smoothing*, W. Härdle, M. G. Schimek (eds), Physica, 50–127.

`lpridge`

, and also `lowess`

and
`loess`

which do local linear and quadratic regression
quite a bit differently.

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