Linear Conditional Expectation for Discretized Distributions.

Area of Science:

• Statistics
• Probability Theory
• Econometrics

Background:

• Many statistical models for continuous data assume linear conditional expectations.
• Multivariate data components are frequently measured using discrete ordinal scales, derived from underlying continuous latent variables.
• Understanding the properties of these discretized distributions is crucial for accurate statistical inference.

Purpose of the Study:

• To investigate whether common discretized multivariate distributions maintain a linear conditional expectation.
• To provide theoretical results and empirical evidence for the behavior of these distributions.
• To inform the application of statistical methods to data with mixed continuous and discrete components.

Main Methods:

• Derivation of theoretical properties for discretized bivariate and trivariate distributions.
• Conducting simulation studies to empirically verify the findings.
• Analyzing the conditional expectation under discretization.

Main Results:

• Common examples of discretized bivariate and trivariate distributions demonstrate a linear conditional expectation.
• The theoretical results are supported by simulation outcomes.
• The findings hold for typical discretization methods of latent continuous variables.

Conclusions:

• The assumption of linear conditional expectation is often preserved in common discretized multivariate distributions.
• This research validates the use of methods assuming linear conditional expectation in certain discrete data contexts.
• Provides a foundation for more robust statistical modeling with ordinal or discretized data.