Confidence intervals for means of the normal distribution

1

One sample confidence intervals.
   Two sample confidence intervals :
      * Paired samples
      * Independent samples (variances known, unknown but         equal, unknown and not equal).

Approximate confidence intervals on means

2

Asymptotic interval  (no demonstration).
   Welch's approximation.

Tutorial

1

MSE of a parameter estimator.
   Best estimate of a random variable X.
   Best estimate of X when a second r.v. Y is available.
   Properties of Minimum Mean Square Error estimators.

Animation

MSE of a model :
       * As a function of the position of the measurement
          point.
       * As a function of the model complexity (bias-
          variance tradeoff).

Tutorial

1

Estimating the variance of the normal distribution.
   MSE of the corrected (unbiased) sample variance.
   MSE of the uncorrected (biased) sample variance.
   An even better estimator of the variance.
   Comparing the properties of the three estimators.

Animation

MSE of a model :
       * As a function of the position of the measurement
          point.
       * As a function of the model complexity (bias-
          variance tradeoff).

Examples of sufficient statistic

1

Sufficient statistics for :
          * The Bernoulli distribution b(p),
          * The Binomial distribution B(n, p),
          * The Poisson distribution P(l),
          * The uniform distribution U[0, q],
          * The truncated exponential exp(q - x),
       from the definition only.

Factorization Theorem

2

A necessary and sufficient condition for a statistic to be    sufficient.
   Functions of sufficient statistics.
    Examples : Bernoulli, uniform, Poisson, mean of normal (two
    methods), variance of normal, Gamma, exponential.

Tutorials

1

Expectation and variance of the score.
   "Basic" Cramér-Rao inequality.
   The two operational forms of the Cramér-Rao inequality.
   Regularity conditions.

2

A necessary and sufficient condition for the existence
   of an efficient estimator.
   Variance of the estimator.
   Efficient estimator and Maximum Likelihood.
   Efficient estimator and Sufficient statistic. 

Tutorial

3

Examples of applications of the Cramér-Rao lower bound :
      * Mean and variance of the normal distribution.
      * Mean of the exponential distribution.
      * Parameter of the Bernoulli distribution.
      * Mean of the Poisson distribution.
      -----
    Parameter of the uniform distribution :
       * Cramér-Rao does not apply.
       * An unbiased estimator "better" that the CR lower bound.

Tutorial

1

Reducing the variance of a statistic
   while preserving its expectation.
   The Rao-Blackwell theorem :
      * Creating the new statistic.
      * Preserving the expectation.
      * Reducing the variance.

Tutorials

2

First example of blackwellization :
      * Estimating the probability for X = 0 for a Poisson
         distribution with parameter unknown.

3

Second example of blackwellization :
      * Estimating the probability for X > t for an exponential
         distribution with parameter unknown.

Tutorial

1

Demonstration of the Neyman-Pearson lemma..
   First consequences on :
      * Power and significance level.
      * Probabilities of Type I and Type II errors.

Tutorial

2

Mean of normal distributions.
    Location parameter of the Cauchy distribution.
    Example in which no parameters are involved.
    Neyman-Pearson and sufficient statistic :
         * New expression of the likelihood ratio.
         * Mean of normal distribution revisited.

Tutorials

1

Reminder : the goal of ANOVA
   The principle of ANOVA

2

Variance decomposition
   Total Sum of Squares (SS_T )
      * Decomposition of SS_T
       * Factorial Sum of Squares (SS_F)
      * Residual Sum of Squares (SS_R)
   The "variance decomposition" equation

Tutorials

3

Total Sum of Squares
      * Distribution
     Residual Sum of Squares
      * Distribution
    A premature attempt
    Factorial Sum of Squares  (no demonstration)

4

The ANOVA statistic
      * Fisher's F statistic
      * Mean Squares
   The F test
   ANOVA table

Tutorials

1

What does confidence depend on ?
   The T statistic
   The assumptions
   Variance known or unknown
   Student's t distribution
   Degrees of freedom

Tutorials

2

The "Reference value" t test.
   The "Paired samples" t test.
   The "Independent samples t test".

3

Reading the results of a t test :
      * Standard error
      * Degrees of freedom
      * Significance and p-value

Tutorials

1

          GOODNESS-OF-FIT CHI-SQUARE TEST
   The binomial case.
   The general multinomial case.
   Influence of sample size.
   Unknown parameters.
   Testing a continuous distribution.

2

               IDENTITY CHI-SQUARE TEST
   The problem.
   The identity Chi-square test.
   Generalization to p variables.

Tutorials

3

          INDEPENDENCE CHI-SQUARE TEST
    The problem.
    The independence Cgi-square test.
    Largest value.
    Special case : 2x2 tables.

Tutorial

1

The Kruskal-Wallis statistic.
   Rationale of the test.
   The two forms of the Kruskal-Wallis statistic.
   The Chi-square approximation.
   Two examples (small and large samples).

Tutorial

2

Ties.
    The influence of ties on the Kruskal-Wallis statistic.
   ----
    Multiple comparisons beween treatments.
    Multiple comparisons with a reference groups.