Old Statistical Methods for New Tricks in Analysis
McNicholl, Dennis, Modern Trader
EQUITY TRADING TECHNIQUES
The frequent occurrence of an approximately normal distribution for stock closing prices opens the way for market technicians to apply traditional, well-founded, statistical decision methods instead of the more specialized methods of current technical analysis, at least for short-term trading.
The number and complexity of technical analysis indica tors have increased rapidly over the last few years, made possible by the personal computer and the easy availability of market price time series data. These indicators are nearly always heuristic in origin and are the outcome of the hard work and ingenuity of today's technical analysis experts.
However, this article is based on the central idea that short-term indicator design can be based on familiar sample-data, statistical-inference methods. That is, traders can apply the same standard distribution parameter and shape estimators used routinely in business, engineering and medical research applications. For example, one classic measure of the skew of a sample distribution, called the skewness coefficient, can be adapted to technical analysis as an indicator for the onset and probable duration of a trend.
Moreover, many potential newcomers to short-term trading are probably already practicing standard statistical methods as a normal part of their daily work and would welcome a shortcut way to learn trading indicators based on knowledge they already possess.
(Note: In this article, the term "short-term" refers to average round turn trades of five days or less.)
Central limit theorem One reason such an approach to technical analysis is feasible is the central limit theorem of mathematical statistics. In practical terms, the central limit theorem states that whenever an observed random variable in an experiment is the sum of many small component independent random variables, then a sample of the observed random variable will have an approximately normal distribution. Moreover, the observed random variable will have an approximately normal distribution no matter what the distribution of each component random variable. It is only necessary that the component random variables be independent and that no one or a few of the component random variables makes up a dominant share of the observed variable.
Thus, because any stock's daily closing price is the sum of all the random upticks and downticks of the price during the day, you can expect that a sample of any one stock's closing prices will have an approximately normal distribution.
The easiest way to understand and appreciate the action of the central limit theorem is to study a series of examples. First is a rudimentary example where an observed random variable is the sum of only two component independent random variables (see "Two variables equal one bell-shaped curve," below).
The central limit theorem, as the term "limit" implies, in its general form requires a very large number of component random variables in the sum for the observed random variable. In practice, however, the number of component random variables in the observed random variable sum is not required to be that large, as the simple example illustrates. To construct this example, suppose that two independent samples of 150 sample points each are drawn, one after the other, from a population that has the uniform distribution with the probability density function defined as:
f(X)=0, X < 0
= 1, 0= < X < = 1
=0, X >1
The mean for f(x) is 0.5 and the variance is 1/12.
Let the symbol X1 represent any one member of the first-drawn 150-- point sample and the symbol X2 represent any one member of the second-drawn sample. The shapes of the histograms for the random sample of X1 and the random sample of X2 resemble the uniform distribution defined in the above equation and, therefore, do not resemble the normal distribution. …