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not be representative of the population defined by the later period because it was drawn from a different population! This is, of course, an extreme case. More often, structural market variations are due to subtle influences that are sometimes impossible to identify, especially before the fact. In some cases, the market may still be fundamentally the same, but it may be going through different phases; each sample drawn might inadvertently be taken from a different phase and be representative of that phase alone, not of the market as a whole. How can it be determined that the population from which a sample is drawn for the purpose of system development is the same as the population on which the system will be traded Short of hopping into a time machine and sampling the future, there is no reliable way to tell if tomorrow will be the day the market undergoes a systemkilling metamorphosis! Multiple out-of-sample tests, conducted over a long period of time, may provide some assurance that a system will hold up, since they may show that the market has not changed substantially across several sampling periods. Given a representative sample, statistics can help make accurate inferences about the population from which the sample was drawn. Statistics cannot, however, reveal whether tomorrow s market will have changed in some fundamental manner.

Another issue found in trading system development is optimization, i.e., improving the performance of a system by adjusting its parameters until the system performs its best on what the developer hopes is a representative sample. When the system fails to hold up in the future (or on out-of-sample data), the optimization process is pejoratively called curve-fitting. However, there is good curve-fitting and bad curve-fitting. Good curve-fitting is when a model can be fit to the entire relevant population (or, at least, to a sufficiently large sample thereof), suggesting that valid characteristics of the entire population have been captured in the model. Bad curve-@zing occurs when the system only fits chance characteristics, those that are not necessarily representative of the population from which the sample was drawn. Developers are correct to fear bad curve-fitting, i.e., the situation in which parameter values are adapted to the particular sample on which the system was optimized, not to the population as a whole. If the sample was small or was not representative of the population from which it was drawn, it is likely that the system will look good on that one sample but fail miserably on another, or worse, lose money in real-time trading. However, as the sample gets larger, the chance of this happening becomes smaller: Bad curve-fitting declines and good curve-fitting increases. All the statistics discussed reflect this, even the ones that specifically concern optimization. It is true that the more combinations of things optimized, the greater the likelihood good performance may be obtained by chance alone.

The Web is an excellent ordering medium You never get put on hold Each year more people learn to use the Web Some catalogers estimate that by 2005 they may have 35 percent of their orders placed through the Web The Web, however, does not seem to be an effective sales medium Very few of the orders that arrive at a catalog Web site come from people who were simply browsing the Web What brings the visitors is the arrival of the paper catalog Once they had built their catalog Web site, many catalogers tried to use advertising, mailed promotions, and af liate programs to bring people to the Web site to shop However, the increase in sales seldom paid for the cost of the promotions The catalogers discovered that the paper catalog itself is the only really effective way to drive customers to their Web sites.

However, if the statistical result was sufficiently good, or the sample on which it was based large enough to reduce the probability that the outcome was due to chance, the result might still be very real and significant, even if many parameters were optimized Some have argued that size does not matter, ie, that sample size and the number of trades studied have little or nothing to do with the risk of overoptimization, and that a large sample does not mitigate curve-fitting This is patently untrue, both intuitively and mathematically Anyone would have less confidence in a system that took only three or four trades over a lo-year period than in one that took over 1,000 reasonably profitable trades Think of a linear regression model in which a straight line is being fit to a number of points.