Nonparametric maximum likelihood (NPML) is a statistical method that is used to estimate the parameters of a model without making any assumptions about the underlying distribution of the data. This method is often used when the data is complex or when the underlying distribution is unknown or difficult to model.

An example of NPML is the estimation of the probability density function (PDF) of a dataset. In this case, the goal is to estimate the PDF of the data by maximizing the likelihood of the data given the estimated PDF. This is done by finding the PDF that best fits the data, using a process called kernel density estimation.

For example, suppose we have a dataset that consists of a series of observations of a continuous variable, such as the heights of individuals in a population. We can use NPML to estimate the PDF of the data by maximizing the likelihood of the data given an estimated PDF. We can do this by using a kernel density estimator, which is a function that is used to estimate the PDF of a dataset by smoothing the data using a kernel function.

Another example of NPML is the estimation of the survival function of a population. In this case, the goal is to estimate the probability that an individual in the population will survive a given time period, based on their characteristics and other factors. This can be done by maximizing the likelihood of the data given an estimated survival function.

To do this, we can use a method called Kaplan-Meier estimation, which is a nonparametric method for estimating the survival function of a population. In this method, we divide the population into groups based on their characteristics and calculate the probability of survival for each group. We then estimate the overall survival function by taking the product of the probabilities of survival for each group.

NPML is a useful tool for estimating the parameters of complex or unknown distributions, as it allows us to estimate these parameters without making any assumptions about the underlying distribution. It is also useful for estimating the parameters of models when the data is too complex to be modeled using traditional parametric methods. However, NPML can be computationally intensive and may not always be the most efficient method for estimating the parameters of a model.