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
##  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
##  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 .