Efficiency is an asymptotic criterion. Asymptotic directions

Definition. The direction determined by a non-zero vector is called asymptotic direction relative to the second order line, if any a straight line of this direction (that is, parallel to the vector) either has at most one common point with the line, or is contained in this line.

? How many common points can a second-order line and a straight line of asymptotic direction have with respect to this line?

In the general theory of second order lines it is proven that if

Then the non-zero vector ( specifies the asymptotic direction relative to the line

(general criterion for asymptotic direction).

For second order lines

if , then there are no asymptotic directions,

if then there are two asymptotic directions,

if then there is only one asymptotic direction.

The following lemma turns out to be useful ( criterion for the asymptotic direction of a line of parabolic type).

Lemma . Let be a line of parabolic type.

The non-zero vector has an asymptotic direction

relatively . (5)

(Problem: Prove the lemma.)

Definition. The straight line of the asymptotic direction is called asymptote line of the second order, if this line either does not intersect with or is contained in it.

Theorem . If it has an asymptotic direction relative to , then the asymptote parallel to the vector is determined by the equation

Let's fill out the table.

TASKS.

1. Find the vectors of asymptotic directions for the following second order lines:

4 - hyperbolic type two asymptotic directions.

Let's use the asymptotic direction criterion:

Has an asymptotic direction relative to this line 4.

If =0, then =0, that is, zero. Then Divide by We get quadratic equation: , where t = . We solve this quadratic equation and find two solutions: t = 4 and t = 1. Then the asymptotic directions of the line .

(Two methods can be considered, since the line is of a parabolic type.)

2. Find out whether the coordinate axes have asymptotic directions relative to the second-order lines:

3. Write the general equation of the second order line for which

a) the x-axis has an asymptotic direction;

b) Both coordinate axes have asymptotic directions;

c) the coordinate axes have asymptotic directions and O is the center of the line.

4. Write the equations of the asymptotes for the lines:

a) ng w:val="EN-US"/>y=0"> ;

5. Prove that if a second-order line has two non-parallel asymptotes, then their intersection point is the center of this line.

Note: Since there are two non-parallel asymptotes, there are two asymptotic directions, then , and, therefore, the line is central.

Write the equations of the asymptotes in general view and a system for finding the center. Everything is obvious.

6.(No. 920) Write the equation of a hyperbola passing through point A(0, -5) and having asymptotes x – 1 = 0 and 2x – y + 1 = 0.

Note. Use the statement from the previous problem.

Homework . , No. 915 (c, e, f), No. 916 (c, d, e), No. 920 (if you didn’t have time);

Cribs;

Silaev, Timoshenko. Practical tasks in geometry,

1st semester. P.67, questions 1-8, p.70, questions 1-3 (oral).

DIAMETERS OF SECOND ORDER LINES.

CONNECTED DIAMETERS.

An affine coordinate system is given.

Definition. Diameter a second-order line conjugate to a vector of non-asymptotic direction with respect to , is the set of midpoints of all chords of the line parallel to the vector .

During the lecture it was proven that diameter is a straight line and its equation was obtained

Recommendations: Show (on an ellipse) how it is constructed (we set a non-asymptotic direction; draw [two] straight lines of this direction intersecting the line; find the midpoints of the chords to be cut off; draw a straight line through the midpoints - this is the diameter).

Discuss:

1. Why in determining the diameter is a vector of a non-asymptotic direction taken. If they cannot answer, then ask them to construct the diameter, for example, for a parabola.

2. Does any second-order line have at least one diameter? Why?

3. During the lecture it was proven that diameter is a straight line. The midpoint of which chord is point M in the figure?

4. Look at the parentheses in equation (7). What do they remind you of?

Conclusion: 1) each center belongs to each diameter;

2) if there is a line of centers, then there is a single diameter.

5. What direction do the diameters of a parabolic line have? (Asymptotic)

Proof (probably in lecture).

Let the diameter d, given by equation (7`), be conjugate to a vector of non-asymptotic direction. Then its direction vector

(-(), ). Let us show that this vector has an asymptotic direction. Let us use the criterion of the asymptotic direction vector for a line of parabolic type (see (5)). Let’s substitute and make sure (don’t forget that .

6. How many diameters does a parabola have? Their relative position? How many diameters do the remaining parabolic lines have? Why?

7. How to construct the total diameter of some pairs of second-order lines (see questions 30, 31 below).

8. We fill out the table and be sure to make drawings.

1. . Write an equation for the set of midpoints of all chords parallel to the vector

2. Write the equation for the diameter d passing through the point K(1,-2) for the line.

Solution steps:

1st method.

1. Determine the type (to know how the diameters of this line behave).

In this case, the line is central, then all diameters pass through center C.

2. We compose the equation of a straight line passing through two points K and C. This is the desired diameter.

2nd method.

1. We write the equation for diameter d in the form (7`).

2. Substituting the coordinates of point K into this equation, we find the relationship between the coordinates of the vector conjugate to the diameter d.

3. We set this vector, taking into account the found dependence, and compose an equation for diameter d.

In this problem, it is easier to calculate using the second method.

3. . Write an equation for the diameter parallel to the x-axis.

4. Find the midpoint of the chord cut off by the line

on the straight line x + 3y – 12 =0.

Directions to the solution: Of course, you can find the points of intersection of the straight line and line data, and then the middle of the resulting segment. The desire to do this disappears if we take, for example, a straight line with the equation x +3y – 2009 =0.

IN modern conditions Interest in data analysis is constantly and intensively growing in completely different fields, such as biology, linguistics, economics, and, of course, IT. The basis of this analysis is statistical methods, and every self-respecting data mining specialist needs to understand them.

Unfortunately, truly good literature, the kind that can provide both mathematically rigorous proofs and clear intuitive explanations, is not very common. And these lectures, in my opinion, are unusually good for mathematicians who understand probability theory precisely for this reason. They are taught to masters at the German Christian-Albrecht University in the Mathematics and Financial Mathematics programs. And for those who are interested in how this subject is taught abroad, I translated these lectures. It took me several months to translate, I diluted the lectures with illustrations, exercises and footnotes on some theorems. I note that I am not a professional translator, but simply an altruist and amateur in this field, so I will accept any criticism if it is constructive.

In short, this is what the lectures are about:

Conditional mathematical expectation

This chapter does not directly relate to statistics, however, it is ideal for starting to study it. Conditional expectation is the best choice for predicting a random outcome based on information already available. And this is also a random variable. Here we consider its various properties, such as linearity, monotonicity, monotonic convergence and others.

Point Estimation Basics

How to estimate the distribution parameter? What criterion should I choose for this? What methods should I use? This chapter helps answer all these questions. Here we introduce the concepts of unbiased estimator and uniformly unbiased minimum variance estimator. Explains where the chi-square and t-distributions come from and why they are important in estimating the parameters of a normal distribution. Explains what the Rao-Kramer inequality and Fisher information are. The concept of an exponential family is also introduced, which greatly facilitates obtaining a good estimate.

Bayesian and minimax parameter estimation

A different philosophical approach to evaluation is described here. In this case, the parameter is considered unknown because it is a realization of a certain random variable with a known (a priori) distribution. By observing the result of the experiment, we calculate the so-called posterior distribution of the parameter. Based on this, we can obtain a Bayesian estimator, where the criterion is the minimum loss on average, or a minimax estimator, which minimizes the maximum possible loss.

Sufficiency and completeness

This chapter has serious practical significance. A sufficient statistic is a function of the sample such that it is sufficient to store only the result of this function in order to estimate the parameter. There are many such functions, and among them are the so-called minimum sufficient statistics. For example, to estimate the median of a normal distribution, it is enough to store only one number - the arithmetic mean over the entire sample. Does this also work for other distributions, such as the Cauchy distribution? How do sufficient statistics help in choosing estimates? Here you can find answers to these questions.

Asymptotic properties of estimates

Perhaps the most important and necessary property of an assessment is its consistency, that is, the tendency towards a true parameter as the sample size increases. This chapter describes what properties the estimates we know, obtained by the statistical methods described in previous chapters, have. The concepts of asymptotic unbiasedness, asymptotic efficiency and Kullback-Leibler distance are introduced.

Testing Basics

In addition to the question of how to estimate a parameter unknown to us, we must somehow check whether it satisfies the required properties. For example, an experiment is being conducted to test a new drug. How do you know if the likelihood of recovery is higher with it than with using old medications? This chapter explains how such tests are constructed. You will learn what the uniformly most powerful test is, the Neyman-Pearson test, the significance level, the confidence interval, and where the well-known Gaussian test and t-test come from.

Asymptotic properties of criteria

Like grades, criteria must satisfy certain asymptotic properties. Sometimes situations may arise when it is impossible to construct the required criterion, however, using the well-known central limit theorem, we construct a criterion that asymptotically tends to the necessary one. Here you will learn what the asymptotic significance level is, the likelihood ratio method, and how the Bartlett test and the chi-square test of independence are constructed.

Linear model

This chapter can be seen as a complement, namely the application of statistics in the case of linear regression. You will understand what grades are good and under what conditions. You will learn where the least squares method came from, how to construct tests, and why the F-distribution is needed.

Exact Tests provides two additional methods for calculating significance levels for the statistics available through the Crosstabs and Nonparametric Tests procedures. These methods, the exact and Monte Carlo methods, provide a means for obtaining accurate results when your data fail to meet any of the underlying assumptions necessary for reliable results using the standard asymptotic method. Available only if you have purchased the Exact Tests Options.

Example. Asymptotic results obtained from small datasets or sparse or unbalanced tables can be misleading. Exact tests enable you to obtain an accurate significance level without relying on assumptions that might not be met by your data. For example, results of an entrance exam for 20 fire fighters in a small township show that all five white applicants received a pass result, whereas the results for Black, Asian and Hispanic applicants are mixed. A Pearson chi-square testing the null hypothesis that results are independent of race produces an asymptotic significance level of 0.07. This result leads to the conclusion that exam results are independent of the race of the examinee. However, because the data contains only 20 cases and the cells have expected frequencies of less than 5, this result is not trustworthy. The exact significance of the Pearson chi-square is 0.04, which leads to the opposite conclusion. Based on the exact significance, you would conclude that exam results and race of the examinee are related. This demonstrates the importance of obtaining exact results when the assumptions of the asymptotic method cannot be met. The exact significance is always reliable, regardless of the size, distribution, sparseness, or balance of the data.

Statistics. Asymptotic significance. Monte Carlo approximation with confidence level, or exact significance.

• Asymptotic. The significance level based on the asymptotic distribution of a test statistic. Typically, a value of less than 0.05 is considered significant. The asymptotic significance is based on the assumption that the data set is large. If the data set is small or poorly distributed, this may not be a good indication of significance.
• Monte Carlo Estimate. An unbiased estimate of the exact significance level, calculated by repeatedly sampling from a reference set of tables with the same dimensions and row and column margins as the observed table. The Monte Carlo method allows you to estimate exact significance without relying on the assumptions required for the asymptotic method. This method is most useful when the data set is too large to compute exact significance, but the data do not meet the assumptions of the asymptotic method.
• Exact. The probability of the observed outcome or an outcome more extreme is calculated exactly. , a significance level less than 0.05 is considered significant, indicating that typically there is some relationship between the row and column variables.

EFFICIENCY ASYMPTOTIC CRITERIA

A concept that allows, in the case of large samples, to quantify two different statistics. criteria used to check false and the same statistics. hypotheses. The need to measure the effectiveness of criteria arose in the 30-40s, when simple in terms of calculations, but ineffective

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

See what "EFFICIENT ASYMPTOTIC CRITERION" is in other dictionaries:

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