Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.

Asymptotic efficiency of the MLE. Cramer-Rao low bound.

n this section we obtain a low bound for variance of an estimator under some reasonable regularity assumptions.

We use the Cauchy inequality MATH where the MATH is a joint distribution of the sample MATH We wish to apply the above relationship with MATH and MATH hence, we set MATH and MATH We obtain MATH MATH MATH MATH Therefore, MATH The last result is called the Cramer-Rao low bound.

If the variables MATH are iid then MATH and the denominator may be simplified MATH MATH MATH MATH MATH MATH MATH Hence, MATH Therefore, according to the last section the MLE asymptotically achieves the low bound.

Notation. Index. Contents.

Copyright 2007