statistical-problem-formulation

Research

Formulate statistical research problems with formal notation, target parameters, assumptions, hypotheses, evaluation criteria, and theory targets.

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Statistical Problem Formulation

Overview

Use this skill before any method design, theory, experiment, or report writing. The goal is to transform a broad topic into a precise statistical problem.

Required Formulation Elements

Element	Questions
Observed data	What is observed? What is the sample size? Are samples iid, dependent, clustered, censored, or selected?
Data model	What family of distributions or data-generating processes is considered?
Target	What parameter, decision, prediction, or risk is the object of study?
Assumptions	What must hold for the target to be identifiable or the method to work?
Hypotheses	What claims should be supported, refuted, or made inconclusive?
Criteria	What metrics define success or failure?
Theory target	What property should be derived: bias, variance, consistency, rate, coverage, error bound, robustness, or impossibility?

Handoff Schema

The problem formulation should be precise enough to support this structured handoff:

topic_id: TXX
title: ""
research_question: ""
observed_data:
  notation: ""
  sampling: iid | dependent | clustered | time_series | selected | unknown
data_model:
  notation: ""
  family: ""
target:
  name: ""
  notation: ""
  type: estimand | decision | prediction | risk | descriptive_quantity
  truth_source: analytic | simulation | oracle | empirical_reference | not_applicable
assumptions:
  structural: []
  sampling: []
  regularity: []
  identifiability: []
claims:
  - id: C1
    statement: ""
    formal_statement: ""
evaluation_criteria:
  - name: ""
    direction: ""
theory_targets:
  - identifiability
  - bias
  - consistency
blocking_ambiguities: []

Template

# Problem Formulation

## Research Question
...

## Observed Data
Let ...

## Data-Generating Model
Assume ...

## Target / Estimand
Define ...

## Candidate Procedure Class
We consider procedures ...

## Assumptions
1. ...

## Claims / Hypotheses
- ...

## Evaluation Criteria
- ...

## Theoretical Questions
- ...

## Experimental Questions
- ...

Quality Bar

A formulation passes only if another researcher could implement or analyze the problem without guessing the target, assumptions, or success criteria.