• Statistical Inference
  • Transformation and Expectations
    • Transformation for distributions
    • Expectation
    • Moments
      • Moment generating function
    • Differentiating
  • Distribution
    • Discrete
    • Continuous
    • Exponential family
    • Inequality
      • Chebyshev’s inequality
  • Multiple Random Variables
    • Joint distribution
    • Marginal distribution
    • Conditional distribution
    • Bivariate transformation
    • Covariance and correlation
    • Multivariate distribution
    • Inequality
      • Jensen’s inequality
  • Properties of a Random Sample
    • Concepts
      • Sampling distributions
      • Convolution formula
    • Sampling from normal distribution
      • \(\chi^2\) distribution
      • t distribution
      • F distribution
    • Order statistics
    • Convergence
      • Converge in probability
      • Weak law of large numbers
      • Converge almost surely
      • Converge in distribution
      • Central limit theorem
      • Slutsky’s theorem
    • Delta method
  • Data Reduction
    • Sufficiency
    • Likelihood Principle
    • Equivariance Principle
  • Point Estimation
    • Find estimators
      • Methods of moments estimator (MME)
      • Maximum likelihood estimator (MLE)
        • Invariant property of MLEs
    • Evaluation
      • MSE
      • Best unbiased estimator
        • Cramer-Rao inequality
  • Hypothesis Testing
    • Definition
    • Likelihood ratio test
    • Evaluation
      • Type 1 & 2 error
      • Power
      • Unbiased test
    • Most powerful test
  • Interval Estimation
  • Asymptotic Evaluations
    • Efficiency

Statistical Inference

Textbook: Statistical Inference (second edition) by George Casella and Roger L. Berger.

Transformation and Expectations

Transformation for distributions

Expectation

Moments

Moment generating function

Differentiating

Distribution

Discrete

Continuous

Exponential family

Inequality

Chebyshev’s inequality

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Multiple Random Variables

Joint distribution

Marginal distribution

Conditional distribution

Bivariate transformation

Covariance and correlation

Multivariate distribution

Inequality

Jensen’s inequality

Properties of a Random Sample

Concepts

Sampling distributions

Convolution formula

Sampling from normal distribution

\(\chi^2\) distribution

t distribution

F distribution

Order statistics

Convergence

Converge in probability

Weak law of large numbers

Converge almost surely

Converge in distribution

Central limit theorem

Slutsky’s theorem

Delta method

Data Reduction

Sufficiency

Likelihood Principle

Equivariance Principle

Point Estimation

Find estimators

Methods of moments estimator (MME)

Maximum likelihood estimator (MLE)

Invariant property of MLEs

Evaluation

MSE

Best unbiased estimator

Cramer-Rao inequality

Hypothesis Testing

Definition

Likelihood ratio test

Evaluation

Type 1 & 2 error

Power

Unbiased test

Most powerful test

Interval Estimation

Asymptotic Evaluations

Efficiency