Sufficient statistics are powerful tools that condense complex datasets while preserving essential parameter information. They simplify data analysis by capturing all relevant sample information, allowing for efficient estimation and reducing computational burden.

The factorization theorem provides a formal method for identifying sufficient statistics. It links sufficiency to the structure of the likelihood function, offering a practical approach to determine which statistics are sufficient for various probability distributions.

Sufficient Statistics

Role of sufficient statistics

• Sufficient statistics condense complex datasets into simplified form while preserving all relevant parameter information
• Function of data captures essential information about parameter of interest enables efficient estimation
• Properties include capturing all relevant sample information and allowing data reduction without information loss
• Simplifies complex datasets reduces computational burden preserves essential information for parameter estimation
• Sample mean for normal distribution (unknown mean, known variance) summarizes data efficiently
• Sample variance for normal distribution (known mean, unknown variance) captures spread information

Factorization theorem proof

• Theorem states T(X) sufficient for θ if likelihood function factored as $L(θ; x) = g(T(x), θ) \cdot h(x)$
• g(T(x), θ) depends on x only through T(x) h(x) independent of θ
• Proof involves two directions:
  1. Forward: If T(X) sufficient, likelihood can be factored
  2. Reverse: If likelihood factored, T(X) sufficient
• Links sufficiency concept to likelihood function structure
• Provides method for identifying sufficient statistics in practice

Sufficient statistics for distributions

• Bernoulli: Sum of successes captures all information about success probability
• Poisson: Sum of observations summarizes rate parameter efficiently
• Exponential: Sum of observations sufficient for rate parameter estimation
• Uniform: Minimum and maximum observations capture distribution endpoints
• Normal (unknown mean, known variance): Sample mean summarizes location parameter
• Normal (unknown mean and variance): Sample mean and variance jointly sufficient

Applications of factorization theorem

• Applying theorem involves:
  1. Writing likelihood function
  2. Identifying parameter-dependent terms
  3. Factoring out parameter-independent terms
  4. Identifying sufficient statistic from factored form
• Bernoulli example:
  • Likelihood: $L(p; x) = p^{\sum x_i} (1-p)^{n-\sum x_i}$
  • Factored: $g(T(x), p) = p^T (1-p)^{n-T}$, T = $\sum x_i$
  • h(x) = 1 (constant)
  • Sufficient statistic: T = $\sum x_i$ (sum of successes)
• Validates sufficient statistic choices
• Helps identify minimal sufficient statistics for efficient inference