Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 1 Time Series.
 2 Classical statistics.
 A. Basic concepts and common notation of classical statistics.
 B. Chi squared distribution.
 C. Student's t-distribution.
 D. Classical estimation theory.
 a. Sufficient statistics.
 b. Sufficient statistic for normal sample.
 c. Maximal likelihood estimation (MLE).
 d. Asymptotic consistency of MLE. Fisher's information number.
 e. Asymptotic efficiency of the MLE. Cramer-Rao low bound.
 E. Pattern recognition.
 3 Bayesian statistics.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Sufficient statistics.

efinition

(Principle of sufficient statistic). The statistic is sufficient if the distribution conditionally on given value of does not depend on .

In other words the sufficient statistic captures all the relevant information.

Theorem

(Factorization). Let is a sufficient statistic for if and only if there exist some functions and such that

Proof

For simplicity we assume that the sample space is discrete. The is a random variable, the is a deterministic variable and possible realized value of . We prove direct statement. We assume that is a sufficient statistic: By ( Bayes_formula ), According to the formula ( Bayes formula ) and the direct claim follows.

We now prove the inverse statement. Suppose that there is the factorization. Then We also have the property Hence, Therefore, the ratio does not depend on This completes the proof.

 Notation. Index. Contents.