## Parametric vs. Nonparametric Approach to Estimations

Parametric statistics assume that the unknown CDF belongs to a family of CDFs characterized by a parameter (vector) . As the form of is assumed, the target of estimation is its parameters . Thus, all uncertainty about is comprised of uncertainty about its parameters. Parameters are estimated by , and estimates are be substituted into the assumed distribution to conduct inference for the quantities of interest. If the assumed distribution is incorrect, inference may also be inaccurate, or trends in the data may be missed.

To demonstrate the parametric approach, consider independent and identically distributed random variables generated from an exponential distribution with rate . Investigators wish to estimate the 75 percentile and erroneously assume that their data is normally distributed. Thus, is assumed to be the Normal CDF but and are unknown. The parameters and are estimated in their typical way by and , respectively. Since the normal distribution belongs to the location-scale family, an estimate of the percentile is provided by,

where is the standard normal quantile function, also known as the probit.

set.seed(12345)
library(tidyverse, quietly = T)
# Generate data from Exp(2)
x <- rexp(n = 100, rate = 2)

# True value of 75th percentile with rate = 2
true <- qexp(p = 0.75, rate = 2)
true
## [1] 0.6931472
# Estimate mu and sigma
xbar <- mean(x)
s    <- sd(x)

# Estimate 75th percentile assuming mu = xbar and sigma = s
param_est <- xbar + s * qnorm(p = 0.75)
param_est
## [1] 0.8792925

The true value of the 75 percentile of is 0.69 while the parametric estimate is 0.88.

Nonparametric statistics make fewer distributions about the unknown distribution , requiring only mild assumptions such as continuity or the existence of specific moments. Instead of estimating parameters of , itself is the target of estimation. is commonly estimated by the empirical cumulative distribution function (ECDF) ,

Any statistic that can be expressed as a function of the CDF, known as a statistical functional and denoted , can be estimated by substituting for . That is, plug-in estimators can be obtained as .

## Motivation

For observed pairs , , the relationship between and can be defined generally as

where and . If we are unsure about the form of , our objective may be to estimate without making too many assumptions about its shape. In other words, we aim to “let the data speak for itself”.

Simulated scatterplot of . Here, and . The true function is displayed in green.

Non-parametric approaches require only that be smooth and continuous. These assumptions are far less restrictive than alternative parametric approaches, thereby increasing the number of potential fits and providing additional flexibility. This makes non-parametric models particularly appealing when prior knowledge about ‘s functional form is limited.

## Estimating the Regression Function

If multiple values of were observed at each , could be estimated by averaging the value of the response at each . However, since is often continuous, it can take on a wide range of values making this quite rare. Instead, a neighbourhood of is considered.

Result of averaging at each . The fit is extremely rough due to gaps in and low frequency at each .

Define the neighbourhood around as for some bandwidth . Then, a simple non-parametric estimate of can be constructed as average of the ‘s corresponding to the within this neighbourhood. That is,

(1)

where

is the uniform kernel. This estimator, referred to as the Nadaraya-Watson estimator, can be generalized to any kernel function (see my previous blog bost). It is, however, convention to use kernel functions of degree (e.g. the Gaussian and Epanechnikov kernels).

The red line is the result of estimating with a Gaussian kernel and arbitrarily selected bandwidth of . The green line represents the true function .