Composite Statistical Hypothesis
A Composite Statistical Hypothesis is a statistical hypothesis that incompletely specifies the probability distribution by allowing distribution parameters to take values from a set or range rather than a single point.
- AKA: Composite Hypothesis, Interval Hypothesis, Range Hypothesis.
- Context:
- It can typically specify parameter ranges like μ > 0 or σ² ∈ [1, 10].
- It can typically require consideration of multiple possible parameter values when computing test power.
- It can often serve as an alternative hypothesis in most hypothesis testing scenarios.
- It can often necessitate uniformly most powerful tests when they exist.
- It can range from being a One-Sided Composite Statistical Hypothesis to being a Two-Sided Composite Statistical Hypothesis, depending on its directional constraint.
- It can range from being a Bounded Composite Statistical Hypothesis to being a Unbounded Composite Statistical Hypothesis, depending on its parameter range.
- It can range from being a Simple Composite Statistical Hypothesis to being a Complex Composite Statistical Hypothesis, depending on its constraint complexity.
- It can range from being a Univariate Composite Statistical Hypothesis to being a Multivariate Composite Statistical Hypothesis, depending on its parameter dimensionality.
- ...
- Example(s):
- One-Sided Hypotheses, such as:
- H₁: μ > 0 (mean is positive).
- H₁: σ² < 10 (variance is less than 10).
- Two-Sided Hypotheses, such as:
- H₁: μ ≠ 0 (mean is not zero).
- H₁: p ≠ 0.5 (probability differs from 0.5).
- Range Hypotheses, such as:
- H₁: μ ∈ [10, 20] (mean between 10 and 20).
- H₁: λ > 5 (Poisson rate exceeds 5).
- ...
- One-Sided Hypotheses, such as:
- Counter-Example(s):
- A Simple Statistical Hypothesis, which specifies exact parameter values.
- A Point Null Hypothesis, which assigns a single value to parameters.
- A Non-Statistical Hypothesis, which isn't about probability distributions.
- See: Simple Statistical Hypothesis, Statistical Hypothesis, Hypothesis Specification Type, Alternative Hypothesis, Uniformly Most Powerful Test, Likelihood Ratio Test, Statistical Hypothesis Testing Task, Parameter Space.