Exchangeable random variables: Difference between revisions

Revision as of 02:07, 7 November 2018

Revision as of 21:33, 14 November 2018

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* O'Neill, B. (2009) Exchangeability, Correlation and Bayes' Effect. ''International Statistical Review'' '''77(2)''', pp. 241-250.</ref>

'''The Representation Theorem:''' This statement is based on the presentation in O'Neill (2009) in references below. Given an infinite sequence of random variables <math>\boldmathbf{X}=(X_1,X_2,X_3,...)</math> we define the limiting [[empirical distribution function]] <math>F_\boldmathbf{X}</math> by:

:::::<math>F_\boldmathbf{X}(x) = \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n I(X_i \le x).</math>

(This is the [[Cesaro summation|Cesaro limit]] of the indicator functions. In cases where the Cesaro limit does not exist this function can actually be defined as the [[Banach limit]] of the indicator functions, which is an extension of this limit. This latter limit always exists for sums of indicator functions, so that the empirical distribution is always well-defined.) If the sequence <math>\boldmathbf{X}</math> is exchangeable then the elements of <math>\boldmathbf{X} | F_\boldmathbf{X}</math> are independent with distribution function <math>F_\boldmathbf{X}</math>. This means that for any vector of random variables in the sequence we have joint distribution function given by:

:::::<math>\Pr (X_1 \le x_1,X_2 \le x_2,...,X_n \le x_n) = \int \prod_{i=1}^n F_\boldmathbf{X}(x_i)\,dP(F_\boldmathbf{X}).</math>

If the distribution function <math>F_\boldmathbf{X}</math> is indexed by another parameter <math>\theta</math> then (with densities appropriately defined) we have:

:::::<math>p(X_1,X_2,...,X_n) = \int \prod_{i=1}^n p(X_i|\theta)\,dP(\theta).</math>

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<br>'''Covariance for exchangeable sequences (infinite):''' If the sequence <math>X_1,X_2,X_3,...</math> is exchangeable then:

::::: <math> \operatorname{cov} (X_i,X_j) = \operatorname{var} (\operatorname{E}(X_i|F_\boldmathbf{X})) = \operatorname{var} (\operatorname{E}(X_i|\theta)) \ge 0 \quad\text{for }i \ne j.</math>

<br>'''Covariance for exchangeable sequences (finite):''' If <math>X_1,X_2,...,X_n</math> is exchangeable with <math>\sigma^2 = \operatorname{var} (X_i)</math> then: