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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.

Asymptotic consistency of MLE. Fisher's information number.


roposition

Let MATH be an iid sample from the population with the distribution MATH . Assume that the distribution MATH is a smooth function of $\theta$ for all possible $y$ , MATH and let $\theta ^{\ast}$ be the MLE of $\theta_{0}$ MATH then MATH in distribution as the sample size $N$ approaches infinity. Here $\xi$ is a standard normal variable and the MATH is the Fisher's information number MATH

Proof

We introduce the log-likelihood function MATH and consider the Taylor expansion MATH Since $\theta^{\ast}$ is the MLE we have MATH , hence MATH The MATH is a sum of iid random variables. The mean of each MATH is zero. Indeed, MATH MATH Hence, according to the CLT ( Central Limit theorem ) MATH in distribution. Similarly, MATH with some number $\sigma$ . Since the expression MATH is not zero, we obtain MATH Observe that MATH Indeed, MATH MATH MATH Hence, MATH





Notation. Index. Contents.


















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