7.6. Consent Criteria

Present We will consider one of the questions related to the verification of the likelihood of hypotheses, namely, the question of the consistency of the theoretical and statistical distribution.

Assume that this statistical distribution is aligned with the help of some theoretical curve (fig. 7.6.1). No matter how well the theoretical curve is chosen, some discrepancies are inevitable between it and the statistical distribution. Naturally, the question arises: are these differences explained only by random circumstances associated with a limited number of observations, or are they significant and related to the fact that the curve we have chosen does not evenly align this statistical distribution? To answer this question are the so-called "criteria of consent."

The idea of ​​applying acceptance criteria is as follows.

Based on this statistical material we have to test the hypothesis consisting in that random variable obeys some definite law of distribution. This law can be specified in one form or another: for example, as a distribution function or as a distribution density or in the form of a set of probabilities where - the probability that the magnitude will fall into the limits th discharge.

Fig. 7.6.1

Since of these forms, the distribution function is the most common and defines any other, we will formulate a hypothesis as consisting in that magnitude has a distribution function .

In order to accept or disprove the hypothesis consider some value characterizing the degree of discrepancy between the theoretical and statistical distributions. Magnitude can be selected in various ways; for example, as you can take the sum of the squares of the deviations of theoretical probabilities from relevant frequencies or the sum of the same squares with some coefficients ("weights"), or the maximum deviation of the statistical distribution function from theoretical etc. Let us assume that the quantity selected one way or another. Obviously, this is some random variable. The distribution law of this random variable depends on the distribution law of the random variable. over which experiments were made, and on the number of experiments . If hypothesis true, then the distribution law determined by the law of distribution of magnitude (function ) and number .

Suppose that this distribution law is known to us. As a result of this series of experiments, it was found that the chosen measure of discrepancy took some meaning . The question is whether this can be explained by random reasons or is this discrepancy too great and indicates the presence of a significant difference between the theoretical and statistical distributions and, consequently, the unsuitability of the hypothesis ? To answer this question, suppose the hypothesis is correct, and we calculate in this assumption the probability that the hypothesis is correct, and we calculate in this assumption the probability that due to random reasons associated with an insufficient amount of experimental material, the measure of discrepancy will be no less than the value we observed in the experiment , i.e., we calculate the probability of an event:

.

If this probability is very small, then the hypothesis should reject as little believable; if this probability is significant, it should be recognized that the experimental data do not contradict the hypothesis .

The question arises of how to choose the measure of discrepancy. ? It turns out that with some ways of choosing it, the distribution law has very simple properties with a sufficiently large practically independent of function . It is precisely such measures that discrepancies use in mathematical statistics as criteria for agreement.

Consider one of the most commonly used criteria of consent - the so-called "criterion Pearson.

Suppose that produced independent experiments, in each of which a random variable took a certain meaning. The results of the experiments are summarized in discharges and decorated in the form of statistical series:

It is required to check whether the experimental data are consistent with the hypothesis that the random variable has a given distribution law (given by the distribution function or density ). Let's call this distribution law “theoretical”.

Knowing the distribution law, one can find the theoretical probabilities of a random variable falling into each of the digits:

.

Checking the consistency of the theoretical and statistical distributions, we will proceed from the discrepancies between the theoretical probabilities and observed frequencies . It is natural to choose as a measure of the discrepancy between the theoretical and statistical distributions the sum of squared deviations taken with some weights :

. (7.6.1)

Coefficients (“Weights” of digits) are introduced because, in the general case, deviations related to different digits cannot be considered equal in importance. Indeed, the same in absolute value deviation may be of little significance if the probability itself is small. Therefore, of course "weight" take back proportional to the probabilities of discharges .

The next question is how to choose the coefficient of proportionality.

K. Pearson showed that if we put

(7.6.2)

then for large distribution law It has very simple properties: it practically does not depend on the distribution function and on the number of experiences namely, this law when increasing approaching the so-called “distribution ".

With this choice of coefficients measure of discrepancy is usually denoted :

. (7.6.3)

For ease of calculation (to avoid dealing with fractional values ​​with a large number of zeros), you can enter under the sum sign and given that where - the number of values ​​in th discharge, bring the formula (7.6.3) to the form:

(7.6.4)

Distribution depends on parameter , called the number of "degrees of freedom" distribution. The number of "degrees of freedom" equal to the number of digits minus the number of independent conditions ("connections") imposed on the frequencies . Examples of such conditions can be

,

if we require only that the sum of frequencies be equal to one (this requirement is imposed in all cases);

,

if we select a theoretical distribution with the condition that the theoretical and statistical averages coincide;

,

if we also require a coincidence of theoretical and statistical variances, etc.

For distribution compiled tables (see table. 4 annex). Using these tables, you can for each value and numbers of degrees of freedom find probability the fact that the value distributed by law will surpass this value. In tab. 4 inputs are: probability value and the number of degrees of freedom . The numbers in the table represent the corresponding values. .

Distribution It makes it possible to assess the degree of consistency of the theoretical and statistical distributions. We will proceed from the fact that really distributed by law . Then the probability , determined by the table, there is a probability that, due to purely random reasons, the measure of the discrepancy between the theoretical and statistical distributions (7.6.4) will be no less than that actually observed in this series of experiments . If this probability very small (so small that an event with such a probability can be considered almost impossible), the result of the experiment should be considered contrary to the hypothesis that the law of distribution of magnitude there is . This hypothesis should be discarded as implausible. On the contrary, if the probability relatively large, it is possible to recognize the discrepancies between the theoretical and statistical distributions insignificant and attributed to them due to random reasons. Hypothesis that magnitude distributed by law , can be considered plausible or, at least, not contrary to experimental data.

Thus, the application of the criterion to assessing the consistency of the theoretical and statistical distributions comes down to the following:

1) The measure of discrepancy is determined. according to the formula (7.6.4).

2) The number of degrees of freedom is determined. as the number of digits minus the number of superimposed connections :

.

3) By and using table. 4 determines the probability that the quantity having the distribution with degrees of freedom that exceed this value . If this probability is very small, the hypothesis is rejected as implausible. If this probability is relatively large, the hypothesis can be considered not contradicting the experimental data.

How low should the probability be in order to discard or revise a hypothesis, the question is uncertain; it cannot be solved for mathematical reasons, as well as the question of how small the probability of an event must be in order to consider it practically impossible. In practice, if turns out to be less than 0.1, it is recommended to check the experiment, if possible - to repeat it and in case noticeable discrepancies reappear, trying to find a distribution law that is more suitable for describing statistical data.

It should be noted that using the criterion (or any other consent) it is possible only in some cases to refute the selected hypothesis and discard it as clearly disagree with the experimental data - if the probability is great, this fact alone can by no means be considered proof of the validity of the hypothesis , and only indicates that the hypothesis does not contradict the experimental data.

At first glance it may seem that the greater the probability p, the better the consistency of the theoretical and statistical distributions and the more justified the choice of function as a law of the distribution of a random variable. In fact, it is not. Assume, for example, that, evaluating the agreement of the theoretical and statistical distribution by the criterion , we got . This means that with a probability of 0.99 due to purely random reasons, with a given number of experiments, the discrepancies should be larger than the observed ones. We have received relatively very small discrepancies that are too small to recognize them as plausible. It is more reasonable to recognize that such a close coincidence of the theoretical and statistical distributions is not accidental and can be explained by certain reasons related to the registration and processing of experimental data (in particular, the “cleanup” of experimental data that is very common in practice, when some results are randomly discarded or several vary).

Of course, all these considerations are applicable only in cases where the number of experiments is large enough (of the order of a few hundred) and when it makes sense to apply the criterion itself, based on the limiting distribution of the measure of discrepancy when . Note that when using the criterion not only the total number of experiments should be large enough but the numbers of observations in separate ranks. In practice, it is recommended to have at least 5 to 10 observations in each digit. If the number of observations in individual bits is very small (of the order of 1 - 2), it makes sense to combine some bits.

Example 1. Check consistency of theoretical and statistical distributions for example 1 .

Decision. Using the theoretical normal distribution law with parameters

,

find the probability of falling into the ranks by the formula

,

Where - boundaries th discharge.

Then we make a comparative table of numbers of hits in the bits. and corresponding values .

According to the formula (7.6.4) determine the value of the measure of discrepancy

We determine the number of degrees of freedom as the number of digits minus the number of superimposed bonds. (in this case ):

.

According to the table. 4 applications we find for :

at

at .

Therefore, the desired probability at approximately equal to 0.56.This probability is not small; therefore, the hypothesis that the value is distributed according to the normal law can be considered plausible.

Example 2. Check the consistency of the theoretical and statistical distributions for the conditions of example 2 7.5.

Decision. Meanings we calculate as probabilities of hitting the sections (20; 30). (30; 40), etc. for a random variable distributed according to the law of uniform density on a segment (23.6; 96.6). We make a comparative table of values and :

By the formula (7.6.4) we find :

The number of degrees of freedom:

According to the table. 4 applications we have:

at and .

Consequently, the discrepancy observed by us between the theoretical and statistical distributions could appear for purely random reasons only with probability. .Since this probability is very small, it should be recognized that the experimental data contradict the hypothesis that the value is distributed according to the law of uniform density.

In addition to the criterion , a number of other criteria are used in practice to assess the degree of consistency of the theoretical and statistical distributions. Of these, we briefly discuss the criteria of A.N. Kolmogorov.

As a measure of the discrepancy between the theoretical and statistical distributions, A.N. Kolmogorov considers the maximum modulus of the difference between the statistical distribution function and the corresponding theoretical distribution function:

.

The basis for choosing as a measure of the divergence of the value is the simplicity of its calculation. At the same time, it has a fairly simple distribution law. A. N. Kolmogorov proved that, whatever the distribution function of a continuous random variable , with an unlimited increase in the number of independent observations probability of inequality

tends to the limit

(7.6.5)

The probability values calculated by the formula are given in table 7.6.1.

The scheme of application of the criterion A.N. Kolmogorov is as follows: a statistical distribution function and an estimated theoretical distribution function are constructed , and the maximum modulus of the difference between them is determined (Fig. 7.6.2).

Further, the determined value

and table 7.6.1 is the probability .This is the probability that (if the value is indeed distributed according to the law ) due to purely random reasons, the maximum discrepancy between and will be no less than actually observed. If the probability very small, the hypothesis should be rejected as implausible; at relatively large, it can be considered compatible with the experimental data.

Fig. 7.6.2

Criterion A.N. Kolmogorov its simplicity favorably with the previously described criterion ;therefore, it is very readily applied in practice. However, it should be stipulated that this criterion can be applied only in the case when the hypothetical distribution is completely known in advance from any theoretical considerations, i.e. when not only the type of distribution function is known , but also all parameters included in it. Such a case is relatively rare in practice. Usually, from theoretical considerations, only the general form of the function is known , and the numerical parameters included in it are determined from the given statistical material. When applying the criterion, this circumstance is taken into account by a corresponding decrease in the number of degrees of freedom of distribution. .Criterion A.N. Kolmogorov does not provide for such an agreement. If, however, this criterion is applied in cases where the parameters of the theoretical distribution are selected from statistical data, the criterion gives obviously high values ​​of probability ; therefore, in some cases, we risk accepting as a plausible hypothesis, in reality, which does not agree well with experimental data.