Simple Statistical Hypothesis

A Simple Statistical Hypothesis is a statistical hypothesis that completely specifies the probability distribution by assigning exact values to all distribution parameters.

• AKA: Simple Hypothesis, Point Hypothesis, Exact Hypothesis.
• Context:
  • It can typically specify exact values for all parameters like μ = 0 and σ² = 1.
  • It can typically enable exact calculation of test statistic distributions under the hypothesis.
  • It can often serve as a null hypothesis in Neyman-Pearson hypothesis testing.
  • It can often allow for most powerful test construction via the Neyman-Pearson lemma.
  • It can range from being a Univariate Simple Statistical Hypothesis to being a Multivariate Simple Statistical Hypothesis, depending on its parameter count.
  • It can range from being a Location Simple Statistical Hypothesis to being a Scale Simple Statistical Hypothesis, depending on its parameter type.
  • It can range from being a Null Simple Statistical Hypothesis to being a Alternative Simple Statistical Hypothesis, depending on its hypothesis role.
  • It can range from being a Testable Simple Statistical Hypothesis to being a Theoretical Simple Statistical Hypothesis, depending on its practical feasibility.
  • ...
• Example(s):
  • Simple Mean Hypotheses, such as:
    • H₀: μ = 100 (population mean equals exactly 100).
    • H₀: μ₁ = μ₂ (two population means are exactly equal).
  • Simple Distribution Hypotheses, such as:
    • H₀: X ~ N(0, 1) (standard normal distribution).
    • H₀: X ~ Poisson(λ = 5) (Poisson with rate 5).
  • Simple Parameter Hypotheses, such as:
    • H₀: p = 0.5 (probability equals exactly 0.5).
    • H₀: σ² = 25 (variance equals exactly 25).
  • ...
• Counter-Example(s):
  • A Composite Statistical Hypothesis, which specifies a range of parameter values.
  • A One-Sided Hypothesis, which specifies inequality constraints.
  • A Non-Parametric Hypothesis, which doesn't specify exact distributions.
• See: Composite Statistical Hypothesis, Statistical Hypothesis, Hypothesis Specification Type, Null Hypothesis, Neyman-Pearson Lemma, Likelihood Ratio Test, Most Powerful Test, Statistical Hypothesis Testing Task.

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