A new class of distributions as a finite functional mixture using functional weights.

Area of Science:

• Statistics
• Probability Theory
• Reliability Engineering

Background:

• Existing probability distributions may not fully capture complex data patterns.
• The need for flexible statistical models in analyzing time-to-event data is crucial.
• Warped probability density functions offer a way to create new distributions.

Purpose of the Study:

• Introduce a new family of probability distributions.
• Develop a specific three-parameter life-time distribution based on the exponential model.
• Provide analytical expressions for key statistical properties and parameter estimation methods.

Main Methods:

• Defining a probability density function as a weighted sum of a base and a warped version.
• Utilizing a dilation transformation and polynomial decay weight for a specific case.
• Deriving the moment generating function, moments, and quantile function.
• Employing the maximum likelihood estimation method for parameter estimation.

Main Results:

• A new flexible life-time distribution is proposed.
• Explicit formulas for the moment generating function, moments, and quantile function are derived.
• The maximum likelihood estimation method is shown to be applicable for parameter estimation.

Conclusions:

• The proposed distribution family offers a valuable addition to the statistical toolkit.
• The specific life-time distribution demonstrates practical applicability through data set examples.
• The methodology provides a framework for constructing other novel distributions.