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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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 $T\left( X\right) $ is sufficient if the distribution $X$ conditionally on given value of $T\left( X\right) $ does not depend on $\theta$ .

In other words the sufficient statistic $T\left( X\right) $ captures all the relevant information.

Theorem

(Factorization). Let MATH MATH is a sufficient statistic for $\theta$ if and only if there exist some functions MATH and $h\left( x\right) $ such that MATH

Proof

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

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





Notation. Index. Contents.


















Copyright 2007