Non-parametric Maximum Likelihood (NPML)
- Estimates model parameters while avoiding assumptions about the data’s underlying distribution.
- Implementations include kernel density estimation for probability density functions and Kaplan–Meier for survival functions.
- Useful when data are complex or the true distribution is unknown, but can be computationally intensive.
Definition
Section titled “Definition”Nonparametric maximum likelihood (NPML) is a statistical method used to estimate the parameters of a model without making any assumptions about the underlying distribution of the data.
Explanation
Section titled “Explanation”NPML seeks the parameter estimates (or function estimates) that maximize the likelihood of the observed data while imposing no parametric form on the underlying distribution. This approach is applied when the data are complex or when the underlying distribution is unknown or difficult to model, allowing estimation without specifying a fixed distributional family. Implementations of NPML often rely on nonparametric estimators—such as kernel density estimators for PDFs or Kaplan–Meier estimators for survival functions—to represent the estimated distribution or function whose likelihood is maximized.
Examples
Section titled “Examples”Estimating a probability density function (PDF)
Section titled “Estimating a probability density function (PDF)”An example of NPML is estimating the PDF of a dataset by maximizing the likelihood of the data given an estimated PDF. This can be carried out using a kernel density estimator, a function that estimates the PDF by smoothing the data with a kernel function. For instance, if a dataset consists of observations of a continuous variable (such as heights of individuals), NPML can be used to estimate the PDF by finding the kernel-smoothed PDF that best fits the observed data.
Estimating a survival function
Section titled “Estimating a survival function”Another example of NPML is estimating the survival function of a population, i.e., the probability that an individual survives a given time period based on characteristics and other factors. This can be done by maximizing the likelihood of the data given an estimated survival function. One nonparametric method cited for this purpose is Kaplan–Meier estimation: divide the population into groups based on characteristics, calculate the probability of survival for each group, and estimate the overall survival function by taking the product of the probabilities of survival for each group.
Use cases
Section titled “Use cases”- Applying NPML when the data are complex or the underlying distribution is unknown or difficult to model.
- Estimating model parameters when parametric methods are inappropriate because the data are too complex to fit a specified distributional family.
Notes or pitfalls
Section titled “Notes or pitfalls”- NPML can be computationally intensive and may not always be the most efficient method for parameter estimation.
Related terms
Section titled “Related terms”- Probability density function (PDF)
- Kernel density estimation / Kernel density estimator
- Survival function
- Kaplan–Meier estimation
- Parametric methods