Nonparametric statistics uses models whose effective dimension can grow with the sample size instead of fixing a finite-dimensional parametric family.
Density estimation constructs an estimate of an unknown probability density function from sampled observations.
A kernel for density estimation is an integrable function with unit integral, usually nonnegative and centered, whose rescaling averages observations over bandwidth .
For a symmetric bandwidth- averaging kernel and a density with bounded first derivative, the interior pointwise bias is at most a constant times . Near a support boundary, an uncorrected symmetric kernel can instead have nonvanishing bias.
Nonparametric regression estimates a regression function without restricting it to a fixed finite-dimensional family.
A linear estimator at has the form , where the weights depend on the design and tuning parameters but not on the observed responses.
Local polynomial regression fits a low-degree polynomial by weighted least squares near each target point.
The local polynomial Gram matrix is the weighted matrix of inner products of the local monomial basis. Its invertibility makes the local fit unique.
Effective kernel weights are the coefficients for which a local polynomial estimate can be written .
Local polynomial weights of degree reproduce every multivariate polynomial of total degree at most : applying the estimator to its design values returns its value at the target point.
Articles by others on the same topic
Nonparametric statistics refers to a branch of statistics that does not assume a specific distribution for the population from which the samples are drawn. Unlike parametric methods, which rely on assumptions about the parameters (such as mean and variance) of a population's distribution (often assuming a normal distribution), nonparametric methods are more flexible as they can be used with data that do not meet these assumptions.