- There are three kinds of lies: innocent, brazen, and statistical © Mark Twain
- "Bleek's Paradox: we conduct several experiments and calculate the probability of the null hypothesis for each. Although all the statistical results of the individual experiments were "successful", i.e. the null hypothesis for each was rejected with probability p < n, after meta-analysis we get the opposite result: p > n.
- Before applying statistics in a particular area, it is necessary to make sure that we are dealing with ergodic environment. Otherwise, it will turn out to be a game of numbers with a clever face.
- "Bleck's Paradox: we conduct several experiments and calculate the probability of the null hypothesis for each of them. Despite the fact that all statistical results of individual experiments were "successful", i.e. the null hypothesis for each of them was rejected with probability p < n, after meta-analysis we get the opposite result: p > n.
This is an interesting paradox. Where can I learn more about this?
Interesting paradox. Where can I find out more about it?
Your article gives me a double impression.
Plus. In this forum, the very act of asking about hypothetical evaluation of results is very important. The forum is full of people who draw a mashka and assume that this is the case rather than a mashka in the interval.
Minus.
Totally agree with Reshetov. All that you have told - this refers to stationary series or close to them - i.e. series with little change of mo and variance over time. But there are no such series on financial markets and the whole application of statistics on financial markets revolves around the stationarity of time series. The most famous examples are ARIMA, ARCH and all the rest.
Your random series, the histogram of which is shown in Fig. 2, shows that the series has a weak relation to the stationary one, it is skewed and has significantly different tails. It is especially well seen in relation to the perfectly normal curve drawn by you. As such, your reasoning does not apply at all to your example. This one is just an illustration of Reshetov's thoughts.
PS. The most dangerous and despicable concept in statistics is correlation. It is better not to mention it at all.
...All that you have told - this refers to stationary series or close to them - i.e. series with little change of mo and variance over time. And there are no such series on financial markets, and the whole application of statistics on financial markets revolves around the stationarity of time series. The most famous examples are ARIMA, ARCH and all the rest.
Your random series, the histogram of which is shown in Fig. 2, shows that the series has a weak relation to the stationary one, it is skewed and has significantly different tails. It is especially well seen in relation to the perfectly normal curve drawn by you. As such, your reasoning does not apply at all to your example. This one is an illustration of Reshetov's thoughts.
Thank you for your opinion!
I will give my counterarguments.
Stationarity is a characteristic of a time series. Figure 2 is a variation series. The article doesn't talk about time series! Although I agree that time is a useful characteristic.....
As far as I understand, ergodicity means a certain stability of the system under study....
So, I would like to note an important point. If the system, let's talk about a financial time series, is not stationary, we can still use econometrics to find a stable model (e.g. GARCH) describing the behaviour of the model. And in this I see the constancy of the system - behaviour according to the model.... but with the condition that there is some probability that the system will "break" the model...
Thank you for your opinion!
Here are my counterarguments.
Stationarity is a characteristic of a time series. Figure 2 is a variation series. The article does not talk about time series! Although I agree that time is a useful characteristic.....
As far as I understand, ergodicity means a certain stability of the system under study....
So, I would like to note an important point. If the system, let's talk about a financial time series, is not stationary, we can still use econometrics to find a stable model (e.g. GARCH) describing the behaviour of the model. And in this I see the constancy of the system - behaviour according to the model.... but with the condition that there is some probability that the system will "break" the model.....
Some time, a few years ago, I published an article here on the site in which I substantiated one idea that is completely unacceptable to most people. Namely.
There are a lot of indicators. Everyone thinks that if an indicator is drawn, it is the same - after all, we see this very thing. At the same time, it does not occur to most people that what we see in reality may not exist! The reason is banal. If we take the regression corresponding to the indicator, it can easily turn out that some of its coefficients have such wide confidence intervals that it is impossible to speak about the value of such a coefficient at all, and if we throw out such a defective coefficient, the indicator pattern will be completely different. When they say: there is truth, there is falsehood, and there is statistics, they mean this sad and very unaccustomed circumstance - nothing can be trusted, including confidence intervals.
That is why I left parametric models and got involved in machine learning based models. There are no problems with stationarity there, but the problems with overtraining are in full glory.
And I liked the article.
Yes, San Sanych's and Reshetov's remarks are legitimate - if the compared system (or system) changes its parameters, the test results will be useless.
But the very demonstration of methods application is pleasing. It is rare for Forex!
I would say something else, as a person who applies similar methods exactly for quote prices. It is possible to check in advance whether the environment is homogeneous (on two independent large samples) and then trust the results of hypothesis testing with a certain degree of calmness. This can also be done thanks to the same tests.

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New article Trader's Statistical Cookbook: Hypotheses has been published:
This article considers hypothesis - one of the basic ideas of mathematical statistics. Various hypotheses are examined and verified through examples using methods of mathematical statistics. The actual data is generalized using nonparametric methods. The Statistica package and the ported ALGLIB MQL5 numerical analysis library are used for processing data.
Any trader willing to create their own trading system is going to become an analyst sooner or later. They are permanently trying to find market trends and testing trading ideas. Testing an idea can be based on different approaches - from a usual search of best parameter values in the optimization mode of the Strategy Tester to scientific (sometimes pseudo scientific) market research.
In this article I suggest considering statistical hypothesis - an instrument of statistical analysis for research and inference verification. Let us test various hypotheses with the Statistica package and ported numerical analysis library ALGLIB MQL5 using examples.
2. Testing Hypotheses. Theory
The hypothesis to be tested is called a null hypothesis (Н0). A competing hypothesis (Н1) is its alternative. It is on the flip side of the Н0's coin, i.e. it logically refuses the null hypothesis.
Imagine, that there is a population of data on Stop Losses of some trading system. We are going to state two hypotheses making a basis for testing.
Н0 – average Stop Loss value equal to 30 points;
Н1 – average Stop Loss value not equal to 30 points.
Variants of acceptance and rejection of hypotheses:
The last two variants are connected with errors.
Now the value of significance level is to be specified. It is the probability that the alternative hypothesis will be accepted whereas the true hypothesis is the null one (third variant). This probability is preferable to be minimized.
In our case such error will occur if we assume that Stop Loss at the average is not equal to 30 points even though that it actually is.
Usually the significance level (α) is equal to 0.05. That means that the test statistic value of the null hypothesis can populate the critical region in no more that 5 cases out of 100.
In our case the test statistic value will be evaluated on a classical chart (Fig.1).
Fig.1. Test statistic value distribution by normal probability law
Author: Dennis Kirichenko