statistical-problem-formulation

# 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: ```yaml 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 ```markdown # 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.