Faucets

The concept of statistical evaluation. Definition of statistical evaluation

Let it be required to study the quantitative sign of the general population. Assume that, from theoretical considerations, it was possible to establish which distribution has a feature. The problem arises of estimating the parameters that determine this distribution. For example, if it is known in advance that the trait under study is distributed in the general population according to the normal law, then it is necessary to estimate the mathematical expectation and standard deviation, since these two parameters completely determine the normal distribution. If there are reasons to believe that the feature has a Poisson distribution, then it is necessary to estimate the parameter by which this distribution is determined. Usually there are only sample data obtained as a result of observations: , , ... , . Through these data and express the estimated parameter. Considering , , ... , as values of independent random variables , , ... , , we can say that to find a statistical estimate of the unknown parameter of the theoretical distribution means to find a function of the observed random variables, which gives an approximate value of the estimated parameter.

So, statistical evaluation unknown parameter of the theoretical distribution is called a function of the observed random variables. The statistical evaluation of an unknown parameter of the general population by one number is called point. The following point estimates are considered below: biased and unbiased, effective and consistent.

In order for statistical estimates to give good approximations of the estimated parameters, they must satisfy certain requirements. Let's specify these requirements. Let there be a statistical estimate of the unknown parameter of the theoretical distribution. Assume that an estimate is found based on the volume sample. Let's repeat the experiment, i.e., we will extract from the general population another sample of the same size and, using its data, we will find an estimate, etc. We will get numbers , , ... , , which will be different from each other. Thus, the estimate can be considered as a random variable, and the numbers , , ... , - as its possible values.

If the estimate gives an approximate value with excess, then the number found from the sample data ( ) will be greater than the true value. Consequently, the mathematical expectation (mean value) of the random variable will be greater than , i.e. . If gives an approximate value with a disadvantage, then .

So the use statistical evaluation, the mathematical expectation of which is not equal to the estimated parameter, would lead to systematic errors. Therefore, it is necessary to require that the mathematical expectation of the estimate be equal to the estimated parameter. Compliance eliminates systematic errors.

unbiased called a statistical estimate, the mathematical expectation of which is equal to the estimated parameter, i.e. .

Displaced called a statistical estimate, the mathematical expectation of which is not equal to the estimated parameter.

However, it is a mistake to assume that an unbiased estimate always gives a good approximation of the estimated parameter. Indeed, the possible values may be highly scattered around their mean value, i.e., the dispersion of the value may be significant. In this case, the estimate found from the data of one sample, for example, may turn out to be very far from its average value , and hence from the estimated parameter itself. Taking as an approximate value, we would make a big mistake. If you require that the dispersion of the value was small, then the possibility of making a large error will be excluded. Therefore, efficiency requirements are imposed on statistical evaluation.

efficient called the statistical estimate that (for a given sample size ) has the smallest possible variance. When considering samples of a large volume, statistical estimates are subject to the requirement of consistency.

Wealthy is called a statistical estimate, which tends in probability to the estimated parameter. For example, if the variance of the unbiased estimator at tends to zero, then such an estimator also turns out to be consistent.

Let us consider the question of which sample characteristics best estimate the general mean and variance in terms of unbiasedness, efficiency, and consistency.

Let a discrete general population be studied with respect to a quantitative attribute. General secondary is called the arithmetic mean of the values of the feature of the general population. It can be calculated using formulas or , where are the values of the sign of the general population of volume , are the corresponding frequencies, and .

Let from the general population, as a result of independent observations on a quantitative trait, a sample of the volume with the values of the trait . Sample mean is called the arithmetic mean of the sample. It can be calculated using formulas or , where are the values of the attribute in the sample set of volume , are the corresponding frequencies, and .

If the general mean is unknown and it is required to estimate it from the sample data, then the sample mean, which is an unbiased and consistent estimate, is taken as an estimate of the general mean. It follows that if several samples of a sufficiently large volume from the same general population are used to find sample means, then they will be approximately equal to each other. This is the property stability of sample means.

In order to characterize the dispersion of the values of a quantitative attribute of the general population around its average value, a summary characteristic is introduced - the general variance. General variance called the arithmetic mean of the squared deviations of the values of the sign of the general population from their mean value, which is calculated by the formulas: , or .

In order to characterize the scatter of the observed values of a sample quantitative attribute around its mean value , a summary characteristic is introduced - the sample variance. Sample variance called the arithmetic mean of the squared deviations of the observed values of the feature from their mean value, which is calculated by the formulas: , or .

In addition to dispersion, to characterize the dispersion of the values of a feature of the general (sample) population around its average value, they use a summary characteristic - the standard deviation. General standard deviation called the square root of the general variance: . Sample standard deviation called the square root of the sample variance:

Let a sample of volume be extracted from the general population as a result of independent observations on a quantitative trait. It is required to estimate the unknown general variance from the sample data. If we take the sample variance as an estimate of the general variance, then this estimate will lead to systematic errors, giving an underestimated value of the general variance. This is explained by the fact that the sample variance is a biased estimate; in other words, the mean of the sample variance is not equal to the estimated general variance, but is equal to .

It is easy to correct the sample variance so that its mean is equal to the general variance. To do this, it is enough to multiply by a fraction. As a result, we obtain the corrected variance, which is usually denoted by . The corrected variance will be an unbiased estimate of the general variance: .

2. Interval estimates.

Along with point estimation, the statistical theory of parameter estimation deals with questions of interval estimation. The problem of interval estimation can be formulated as follows: based on the sample data, construct a numerical interval, relative to which, with a pre-selected probability, we can say that the estimated parameter is located inside this interval. Interval estimation is especially necessary for a small number of observations, when the point estimate is largely random and, therefore, not very reliable.

Confidence interval for a parameter, such an interval is called, with respect to which it is possible, with a pre-selected probability close to one, to assert that it contains an unknown value of the parameter , i.e. . The smaller the number for the selected probability, the more accurate the estimate of the unknown parameter. And vice versa, if this number is large, then the estimate made using this interval is of little use for practice. Since the ends of the confidence interval depend on the elements of the sample, the values of and can change from sample to sample. The probability is usually called the confidence probability (reliability). Usually, the reliability of the estimate is set in advance, and a number close to one is taken as the value. The choice of confidence probability is not mathematical problem, but is determined by the specific problem being solved. Most often, reliability is set to ; ; .

Let us give without derivation the confidence interval for the general mean with a known value of the standard deviation, provided that the random variable (quantitative attribute) is normally distributed:

where is a predetermined number close to one, and the values of the function are given in Appendix 2.

The meaning of this relation is as follows: it can be stated with reliability that the confidence interval ( ) covers the unknown parameter , the accuracy of the estimate is . The number is determined from the equality , or . According to the table (Appendix 2), an argument is found that corresponds to the value of the Laplace function equal to .

Example 1. The random variable has a normal distribution with a known standard deviation. Find the confidence intervals for estimating the unknown general mean from the sample means if the sample size and the reliability of the estimate are given.

Solution. Let's find . From the ratio we get that . According to the table (Appendix 2) we find. Find the accuracy of the estimate . The confidence intervals will be: . For example, if , then the confidence interval has the following confidence limits: ; . Thus, the values of the unknown parameter , consistent with the sample data, satisfy the inequality .

Confidence interval for the general mean of the normal distribution of the trait at unknown value the standard deviation is given by the expression .

It follows from this that it can be stated with reliability that the confidence interval covers the unknown parameter.

There are ready-made tables (Appendix 4), using which, for given and find the probability , and vice versa, for given and can be found.

Example 2. The quantitative sign of the general population is normally distributed. Based on the volume sample, the sample mean and the corrected standard deviation were found. Estimate the unknown population mean using a confidence interval with reliability .

Solution. Let's find . Using the table (Appendix 4) for and we find:. Let's find the confidence limits:

So, with reliability, the unknown parameter is enclosed in a confidence interval.

3. The concept of a statistical hypothesis. General Statement of the Problem of Hypothesis Testing.

Statistical hypothesis testing is closely related to the theory of parameter estimation. In natural science, technology, and economics, often in order to clarify one or another random fact, they resort to stating hypotheses that can be tested statistically, that is, based on the results of observations in a random sample. Under statistical hypotheses such hypotheses are meant that refer either to the type or to individual parameters of the distribution of a random variable. So, for example, the statistical hypothesis is that the distribution of labor productivity of workers performing the same work in the same conditions has a normal distribution law. Statistical will also be the hypothesis that the average dimensions of parts produced on the same type of parallel machines do not differ from each other.

The statistical hypothesis is called simple if it uniquely determines the distribution of the random variable , otherwise the hypothesis is called complex. For example, a simple hypothesis is the assumption that a random variable is distributed according to the normal law with mathematical expectation, equal to zero, and dispersion, equal to one. If an assumption is made that a random variable has a normal distribution with a variance equal to one, and the mathematical expectation is a number from the segment , then this is a complex hypothesis. Another example of a complex hypothesis is the assumption that a continuous random variable takes a value from the interval with probability, in this case the distribution of the random variable can be any of the class of continuous distributions.

Often the distribution of a quantity is known, and it is necessary to test the assumptions about the value of the parameters of this distribution using a sample of observations. Such hypotheses are called parametric.

The hypothesis to be tested is called null hypothesis and is denoted. Along with the hypothesis, one of the alternative (competing) hypotheses is considered. For example, if the hypothesis is being tested that the parameter is equal to some given value , i.e. : , then one of the following hypotheses can be considered as an alternative hypothesis: : ; : ; : ; : , where is the set value, . The choice of an alternative hypothesis is determined by the specific formulation of the problem.

The rule by which a decision is made to accept or reject a hypothesis is called criterion. Since the decision is made on the basis of a sample of observations of the random variable , it is necessary to choose an appropriate statistic, called in this case the test statistic . When testing a simple parametric hypothesis : the same statistic is chosen as the criterion statistic as for the parameter estimate .

Statistical hypothesis testing is based on the principle that low-probability events are considered impossible, and events with a high probability are considered certain. This principle can be implemented in the following way. Before the analysis of the sample, some small probability is fixed, called significance level. Let be a set of values of statistics , and be a subset such that, under the condition that the hypothesis is true, the probability that the criterion statistic falls into is equal to , i.e. .

Denote by the sample value of the statistic calculated from the sample of observations. The criterion is formulated as follows: reject the hypothesis if ; accept the hypothesis if . A test based on the use of a predetermined level of significance is called significance criterion. The set of all values of the criterion statistic for which a decision is made to reject the hypothesis is called critical area; the area is called acceptance area hypotheses.

The significance level determines the size of the critical region. The position of the critical region on the set of values of the statistic depends on the formulation of the alternative hypothesis. For example, if the hypothesis is tested : , and the alternative hypothesis is formulated as : (), then the critical region is located on the right (left) “tail” of the distribution of statistics , i.e., it has the form of inequality: (), where and are those values of statistics that are accepted with probabilities, respectively, and provided that the hypothesis is true. In this case, the criterion is called unilateral, respectively, right-handed and left-handed. If the alternative hypothesis is formulated as : , then the critical region is located on both “tails” of the distribution , i.e., it is determined by the set of inequalities and ; in this case the criterion is called bilateral.

On fig. 30 shows the location of the critical region for various alternative hypotheses. Here is the distribution density of the criterion statistics provided that the hypothesis is true, is the area of acceptance of the hypothesis, .

Thus, testing a parametric statistical hypothesis using a significance test can be divided into the following steps:

1) formulate a testable () and alternative () hypotheses;

2) assign a significance level ; as inconsistent with the results of observations; if , then accept the hypothesis , i.e., assume that the hypothesis does not contradict the results of observations.

Usually, when performing items 4 - 7, statistics are used, the quantiles of which are tabulated: statistics with a normal distribution, Student's statistics, Fisher's statistics.

Example 3. According to the passport data of an automobile engine, fuel consumption per 100 km mileage is 10 l. As a result of the redesign of the engine, fuel consumption is expected to decrease. Tests are being carried out to verify 25 randomly selected vehicles with an upgraded engine, and the sample average of fuel consumption per 100 km mileage according to the test results was 9.3 l. Assume that the sample of fuel consumption is obtained from a normally distributed population with mean and variance. Provided that the critical region hypothesis for the original statistic is true, i.e. equal to the significance level. Find the probabilities of errors of the first and second kind for a criterion with such a critical region. has a normal distribution with mean equal to and variance equal to . We find the probability of an error of the second kind by the formula (11.2):

Therefore, in accordance with the accepted criterion, 13.6% of vehicles with fuel consumption 9 l on the 100 km mileage are classified as vehicles with fuel consumption 10 l.

4. Theoretical and empirical frequencies. Consent Criteria.

Empirical Frequencies- frequencies obtained as a result of experience (observation). Theoretical frequencies calculated by formulas. For a normal distribution, they can be found as follows:

, (11.3)

for self-preparation for a practical lesson in mathematics

Topic: Statistical distribution of the sample, discrete and interval variation series. Point and interval estimates of distribution parameters. Measurement errors and their estimates.

Relevance of the topic: familiarization with the basic concepts and methods of mathematical statistics as a means of solving problems of a physical, chemical, biological and other nature, encountered both in the process of studying profile disciplines and in further professional activities

The purpose of the lesson: to learn how to build statistical series for discrete and continuous random variables and calculate point estimates of general parameters, calculate errors in direct and indirect measurements.

Topic study plan

1. The main tasks of mathematical statistics.

2. General and sample populations.

3. Discrete variation series and its graphic representation.

4. Interval variation series and its graphic representation. Types of statistical estimates.

5. Requirements for statistical evaluations.

6. The concepts of general and sample averages.

7. The concepts of general, sample and corrected variances.

8. The concepts of general, sample and corrected standard deviation.

Main literature:

1. Morozov, Yu.V. Fundamentals of higher mathematics and statistics: textbook. for medical students and a pharmacist. universities and faculty / Yu.V. Morozov.-

M.: Medicine, 2004.-232 p.

2. Fundamentals of higher mathematics and mathematical statistics: textbook. for medical students and a pharmacist. universities / I.V. Pavlushkov, L.V. Rozovsky, A.E. Kapultsevich and other-2nd ed., Rev.-M.: GOETAR-

Media, 2006.-423 p.

Additional literature:

∙ Guidelines for practical training in higher mathematics [Electronic resource]: study method. allowance for universities / ed.-comp. : T.A.Novichkova; GOU VPO "Kursk. state. medical. un-t", department. physics, informatics and mathematics.-Kursk: KSMU, 2009.

∙ Gmurman V.E. Theory and mathematical statistics. M. "Higher School", ed. 5, 2004.

Questions for self-control:

1) Definition of a statistical series.

2) Definition of the general population.

3) Determination of the sample population.

4) The representativeness of the sample.

5) Types of samples.

6) What is a variant?

7) Ranking definition.

8) Definition of frequency, relative frequency, accumulated frequency.

9) Algorithm for constructing an interval variation series.

10) Definition of polygon, cumulates (discrete variation series).

11) Determination of the histogram, cumulates (interval variation series) determination of the statistical evaluation.

12) what are the requirements for statistical evaluations.

13) What statistical estimate is called biased, unbiased?

14) formulas for calculating the general and sample mean for grouped and ungrouped data.

15) formulas for calculating the general and sample variance for grouped and ungrouped data.

16) What is the sample mean for the general mean?

17) What is the sample variance for the general estimate?

18) Formula for calculating the corrected standard deviation.

19) What are the direct measurements?

20) What is meant by the true absolute error of X?

21) What is taken as the true value of X?

22) What serves as a point estimate of the true value of X?

23) What serves as an estimate of the variance of X?

25) How to find the boundaries of the confidence interval for the true value of X ?

26) What measurements are called indirect?

27) If y = f(x1, x2, ..., xn), then what formula is used to calculate the root mean square error of the mean value y?

28) What formula is used to find the absolute error y: at ?

29) How to find the relative error y:ε y?

Self-study assignments:

1. As a result of separate tests of the activity of tetracycline, the following values \u200b\u200bare obtained (in units of action per 1 mg): 925, 940, 760, 905, 995, 965, 940, 925, 940, 905. make up a distribution series. Build a landfill, cumulate.

2. Construct a histogram of relative frequencies according to the sample distribution: 11, 15, 16, 18, 15.5, 19, 20.1, 20.9, 23, 24.5, 23, 21, 23.9, 24.6, 25.5, 26, 29, 28.6, 30.1, 32.

3. Find the corrected standard deviation for the given sample distribution

Guidelines for action:

1. Learn the basic concepts on the topic

2. Answer questions for self-control

3. Work out examples of solving problems on the topic

4. Complete tasks for self-control

5. Solve control tasks on the topic

After studying this topic, the student should know: the concept of a variational series, its types and their graphic representation,

concepts of statistical evaluation, their types, requirements for estimates, concepts of general and sample mean, general and sample variances. be able to: build statistical series for discrete and continuous random variables and calculate point estimates of general parameters, calculate errors in direct and indirect measurements.

Brief theory

Mathematical statistics is a branch of applied mathematics devoted to the methods of collecting, grouping and analyzing statistical information obtained as a result of observations or experiments.

From here follow the tasks of mathematical statistics:

ways of selecting statistical data.

ways of grouping statistical data.

data analysis methods:

∙ estimation of known distribution parameters;

∙ estimation of the unknown distribution function;

∙ assessment of the dependence of one random variable on others;

∙ testing of statistical hypotheses.

methods for determining the number of observations (experiment planning).

making decisions.

IN mathematical statistics the study of a random variable is related

from performing a series of independent experiments in which it takes on certain values.

Population- a set of objects that are homogeneous with respect to some qualitative or quantitative attribute.

For example, if there is a series of tablets of a medicinal substance, then the standardity of the tablet can serve as a qualitative sign, and the controlled weight of the tablet can serve as a quantitative sign.

Population- a set of all objects that can be attributed to it.

Theoretically, this might be. an infinitely large or approaching infinity set.

N-r, all patients with rheumatism on the globe- general population. In reality, this is within specific limits (city, region).

The number of objects in the general population is called its volume and denoted by N.

Sample population- a set of objects randomly selected from the general population.

The number of objects in the sample is called its size and denoted by n.

In order for the properties of the sample to reflect the properties of the population well enough, the sample must be representative (representative).

This requirement ensures the randomness of the selection of elements in the sample, i.e. the equiprobability of any object being included in the sample.

Depending on the technique for selecting objects from the general population, the samples are divided into:



Repeated					Non-repeating
(the selected object is returned					(the selected object is not returned
to the general population)					to the general population)

In practice, non-repetitive sampling is used.

With large volumes N of the general population and a small relative size n / N of the sample, the differences in the formulas describing both samples in terms of their selection technique are small.

Discrete distribution range

The observed values of a feature are called variants. Ranking - the arrangement of the option in ascending order, or

descending.

variation series called a ranged number of options and their corresponding frequencies.

The statistical distribution of the sample call the list of options and their corresponding frequencies or relative frequencies.

Let a sample of size n be taken from the general population. The quantitative value of the studied trait x1 appeared m1 times, x2 - m 2

times, …, x k – m k times.

Moreover, ∑ m i = n

i=1

The numbers mi are called frequencies, and their ratios to the sample size n are called relative frequencies pi =mi /n. Moreover, Σpi =1.

For the case when a quantitative attribute is discrete, its values and their corresponding frequencies or relative frequencies are presented in the form of a table.



pi =mi/n
pi*=	m1 /n	(m1+m2 )/n
mi*/n	m1 /n	(m1+m2 )/n

When studying variational series, along with the concept of frequency, the accumulated frequency (mi * ) is used. The accumulated frequency shows how many variants were observed with a feature value less than x.

The ratio of the knee frequency mi * to the total number of observations n is called the relative frequency pi * = mi * /n.

A graphic representation of a discrete statistical series is a polygon of frequencies (relative).

The polygon serves to display a discrete variational series and is a broken line in which the ends of the straight line segments have coordinates (xi , mi ) or (xi , pi ) in the case of a polygon of relative frequencies.

Interval statistical series.

In the case of a large number of variants (n>50) and a continuous distribution of the feature, the statistical distribution of the feature can be specified as a sequence of intervals and their corresponding frequencies.

More often, an equal-interval series is used.

You need to choose the right width of the class interval. The number of intervals should depend on the scope of the sample and its size.

Algorithm for constructing a histogram.

1. Given a sample X = (x 1 , x 2 , ..., x n ) ; n is its volume

Sample range D = x max – x min

2. Number of classes

K \u003d 1 + 3.32 × lg n (Sturgess formula for n< 100 )

K \u003d 5 × lg n (Brooks formula for n> 100)

3. The value of the class interval D x \u003d D / K

4. Boundaries and Midpoints of Partial Intervals

x1l \u003d xmin - D x / 2

x1pr \u003d x2l \u003d xmin + D x / 2

x 1 = x min

x 2 \u003d x 1 + D x

5. Frequency of hitting the interval:

variational series and is a stepped figure of rectangles with bases equal to intervals of feature values xi =xi+1 -xi , i=1,2,…,k and heights equal to frequencies (relative frequencies) mi (pi ) of intervals.

If you connect the midpoints of the upper bases of the rectangles with straight line segments, you can get a polygon of the same distribution.

Empirical distribution function To get an idea of the distribution of a random

quantities X, for which the distribution law is unknown, build an empirical distribution function.

The empirical distribution function (sample distribution function) is the function F* (x), which determines for each value x the relative frequency of the event X

, where m* is the number of observations in which the value of the feature X was observed less than x.

The population distribution function is called the theoretical function.

The difference between empirical and theoretical functions is that the theoretical function determines the probability of the event X<х, а эмпирическая – относительную частоту данного события.

The concept of statistical evaluation.

It is required to study the quantitative sign of the general population. Suppose we know the law of distribution of the general population. This law is determined by several parameters. Sample data are used to estimate the unknown parameters of the general population.

Statistical evaluation unknown distribution parameter of the general population is called a function of the observed random variables.

Denote:

θ is an unknown parameter; θ* – statistical estimation of the unknown parameter; θ* = f (x 1, x 2, ..., x n)

The statistical estimate θ* is random variable, therefore, it has a variance and a standard deviation, as well as a representativeness error (deviation of the sample indicator from the general one).

There are two types of statistical estimates: point and interval.

An estimate by a single number that depends on sample data is called a point estimate.

An estimate by two numbers that are the ends of an interval is called an interval estimate.

Requirements for point statistical estimates.

The quality of an estimate is determined not by one specific sample, but by

to the entire conceivable set of specific samples, i.e. throughout the multitude

point estimates θ i * of the unknown parameter θ .

In order for statistical evaluations to give a good

approximation of the estimated parameters, they must satisfy

the following requirements:

unbiased (absence of systematic errors in

any sample size М(θ *) = θ );

efficiency (among all possible estimates, effective

the estimate has the smallest variance min D(θ *) ).

solvency

(pursuit

probabilities

estimated parameter as n → ∞ , i.e. θ * ¾¾ ¾ ® θ );

n→∞

General

Point Estimation

Properties

parameter

point estimate

M(X) = xr =

non-displaceable

x in = ∑ x i

= ∑ m i x i selective

Effective

∑ x i

i = 1

Wealthy

N i = 1

Asymptotically

− x

unbiased, i.e.

M(Dv) ¹ σ g 2, but

ni = 1

D(X) = σ r =

sample variance

) = σ

− x i )

n→∞

N i = 1

S2 =

D corrected

n - 1

non-displaceable

dispersion

δ in =

displaceable

(standard)

σ g =

σ g 2

corrected

rms

unbiased

deviation

is a random variable, then it has a variance -

sample mean variance:

× n × S 2 =

) = D(

∑ xi ) =

D(∑xi) =

∑ D(xi) =

∑ (xi −

n(n − 1) i =1

Accuracy, reliability of estimates

interval estimation called the estimate, which is determined by two numbers - the ends of the interval.

Interval estimates allow you to establish the accuracy and reliability of a point estimate.

Let q * be a point estimate of the unknown parameter q , which is a random variable.

The smaller ½q - q * ½ , the more precisely q * determines the parameter q .

If δ > 0 and ½q - q * ½< δ , то чем меньше δ , тем точнее оценка. Число

δ is called assessment accuracy.

IN force of chance q * we can only talk about the probability of the inequality ½q - q * ½< e .

Reliability (confidence probability) of estimate q * called probability g , with which the inequality½q - q * ½< δ .

Usually g = 0.95; 0.99; 0.999…P(|Θ-Θ*|< δ)=γ

It is sometimes said that the confidence level g characterizes the degree of our confidence that the confidence interval will cover the parameter q .

P (q * - e< q < q * + e} = g означает, что вероятность того, что интервал (q * - e ; q * + e ) заключает в себе неизвестный параметр q , равна g :

The probability that the unknown parameter does not fall into the interval ½q - q * ½< e , равна 1 - g = a (уровень значимости).

The level of significance (risk) is the probability that the module of deviation of the empirical characteristic from the theoretical one will exceed the marginal error P(|Θ-Θ*|< ∆)=γ , предельная ошибка – максимально допустимая |Θ-Θ*|< ∆

Student's distribution

Let X ~ N(µ,σ), and the distribution parameters are unknown.

Consider the distribution of the quantity T = x in − μ .

The distribution of T with f=n-1 degrees of freedom is called the t-distribution or Student's distribution.

The probability density function φ(t) depends on the number of degrees of freedom and does not depend on the dispersion of random variables.

With an increase in the number of degrees of freedom, the distribution of this quantity approaches normal

The interval estimate of the mathematical expectation with an unknown variance is the interval

(x - tγ (f ) × Sx ; x + tγ (f ) × Sx )

An interval estimate of the mathematical expectation with a known

variance is the interval

(x - uα × Sx ; x + uα × Sx )

Ф (u α ) = 1− α - Laplace function.

Examples of problem solving

1) Represent in the form of a statistical discrete series, construct a polygon of frequencies, relative frequencies, a cumulative curve (cumulative frequency curve): 6.7; 6.8; 7; 6.5; 7.3; 7; 7.2; 6.9; 7.1; 6.8; 7.1; 6.8; 7.1; 7.2; 6.8; 6.9;

7; 6,7; 6,6; 6,3; 7,5; 6,9.

Solution. mi – frequency, p – relative frequency, pi * - accumulated relative frequency




pi *
					Frequency polygon

Statistical estimates of the parameters of the general population. Statistical hypotheses

LECTURE 16

Usually, in the distribution, the researcher has only sample data, for example, the values of a quantitative trait obtained as a result of observations (hereinafter, the observations are assumed to be independent). Through these data and express the estimated parameter.

Considering as values of independent random variables , we can say that to find a statistical estimate of an unknown parameter of a theoretical distribution means to find a function of the observed random variables, which gives an approximate value of the estimated parameter. For example, as will be shown below, to estimate the mathematical expectation of a normal distribution, the function (arithmetic mean of the observed values of a feature) is used:

So, statistical evaluation unknown parameter of the theoretical distribution is called a function of the observed random variables. The statistical estimate of an unknown parameter of the general population, written as a single number, is called point. Consider the following point estimates: biased and unbiased, effective and consistent.

In order for statistical estimates to give “good” approximations of the estimated parameters, they must satisfy certain requirements. Let's specify these requirements.

Let there be a statistical estimate of the unknown parameter of the theoretical distribution. Assume that when sampling the volume, an estimate is found. Let's repeat the experiment, that is, we will extract another sample of the same size from the general population and, using its data, we will find an estimate, etc. Repeating the experiment many times, we get the numbers , which, generally speaking, will differ from each other. Thus, the estimate can be considered as a random variable, and the numbers as possible values.

It is clear that if the estimate gives an approximate value with an excess, then each number found from the data of the samples will be greater than the true value of . Therefore, in this case, the mathematical (mean value) of the random variable will be greater than , that is, . Obviously, if it gives an approximate value with a disadvantage, then .

Therefore, the use of a statistical estimate, the mathematical expectation of which is not equal to the estimated parameter, leads to systematic (one sign) errors. For this reason, it is natural to require that the mathematical expectation of the estimate be equal to the estimated parameter. Although compliance with this requirement will not, in general, eliminate errors (some values are greater than and others less than ), errors of different signs will occur equally often. However, compliance with the requirement guarantees the impossibility of obtaining systematic errors, that is, eliminates systematic errors.

unbiased called a statistical estimate (error), the mathematical expectation of which is equal to the estimated parameter for any sample size, that is, .

Displaced called a statistical estimate, the mathematical expectation of which is not equal to the estimated parameter for any sample size, that is.

However, it would be erroneous to assume that an unbiased estimate always gives a good approximation of the estimated parameter. Indeed, the possible values may be highly scattered around their mean, i.e. the variance may be significant. In this case, the estimate found from the data of one sample, for example, may turn out to be very remote from the average value , and hence from the estimated parameter itself. Thus, taking as an approximate value, we will make a big mistake. If, however, the variance is required to be small, then the possibility of making a large error will be excluded. For this reason, the requirement of efficiency is imposed on the statistical evaluation.

efficient called a statistical estimate, which (for a given sample size ) has the smallest possible variance.

Wealthy is called a statistical estimate, which tends in probability to the estimated parameter, that is, the equality is true:

For example, if the variance of the unbiased estimator at tends to zero, then such an estimator also turns out to be consistent.

Consider the question of which sample characteristics best estimate the general mean and variance in terms of unbiasedness, efficiency, and consistency.

Let a discrete general population be studied with respect to some quantitative attribute .

General secondary is called the arithmetic mean of the values of the feature of the general population. It is calculated by the formula:

§ - if all values of the sign of the general population of volume are different;

§ – if the values of the sign of the general population have frequencies, respectively, and . That is, the general average is the weighted average of the trait values with weights equal to the corresponding frequencies.

Comment: let the population of the volume contain objects with different values of the attribute . Imagine that one object is randomly selected from this collection. The probability that an object with a feature value, for example , will be retrieved is obviously equal to . Any other object can be extracted with the same probability. Thus, the value of a feature can be considered as a random variable, the possible values of which have the same probabilities equal to . It is not difficult, in this case, to find the mathematical expectation:

So, if we consider the examined feature of the general population as a random variable, then the mathematical expectation of the feature is equal to the general average of this feature: . We obtained this conclusion, assuming that all objects of the general population have different values of the feature. The same result will be obtained if we assume that the general population contains several objects with the same attribute value.

Generalizing the result obtained to the general population with a continuous distribution of the attribute , we define the general average as the mathematical expectation of the attribute: .

Let a sample of volume be extracted to study the general population with respect to a quantitative attribute.

Sample mean called the arithmetic mean of the values of the feature of the sample population. It is calculated by the formula:

§ - if all values of the sign of the sample population of volume are different;

§ – if the values of the feature of the sampling set have, respectively, frequencies , and . That is, the sample mean is the weighted average of the trait values with weights equal to the corresponding frequencies.

Comment: the sample mean found from the data of one sample is obviously a certain number. If we extract other samples of the same size from the same general population, then the sample mean will change from sample to sample. Thus, the sample mean can be considered as a random variable, and therefore, we can talk about the distributions (theoretical and empirical) of the sample mean and the numerical characteristics of this distribution, in particular, the mean and variance of the sample distribution.

Further, if the general mean is unknown and it is required to estimate it from the sample data, then the sample mean is taken as an estimate of the general mean, which is an unbiased and consistent estimate (we propose to prove this statement on our own). It follows from the foregoing that if several samples of a sufficiently large volume from the same general population are used to find sample means, then they will be approximately equal to each other. This is the property stability of sample means.

Note that if the variances of two populations are the same, then the proximity of the sample means to the general ones does not depend on the ratio of the sample size to the size of the general population. It depends on the sample size: the larger the sample size, the less the sample mean differs from the general one. For example, if 1% of objects are selected from one set, and 4% of objects are selected from another set, and the volume of the first sample turned out to be larger than the second, then the first sample mean will differ less from the corresponding general mean than the second.

After studying this chapter, the student will know, that the sample can be considered as an empirical analogue of the general population, that with the help of sample data it is possible to judge the properties of the general population and evaluate its characteristics, the basic laws of the distribution of statistical estimates, be able to make point and interval estimates of the parameters of the general population by the method of moments and maximum likelihood, own methods for determining the accuracy and reliability of the estimates obtained.

Types of statistical estimates

We know about the parameters of the general population that they objectively exist, but it is impossible to determine them directly due to the fact that the general population is either infinite or excessively large. Therefore, the question can only be about the evaluation of these characteristics.

Previously, it was found that for a sample drawn from the general population, subject to the conditions of representativeness, it is possible to determine characteristics that are analogous to the characteristics of the general population.

cjp Definition 8.1. Approximate values of the distribution parameters found from the sample are called parameter estimates.

Let us designate the estimated parameter of a random variable (general population) as 0, and its estimate obtained using the sample as 0.

The score 0 is a random value, because any sample is random. Estimates obtained for different samples will differ from each other. Therefore, we will consider 0 a function that depends on the sample: 0 = 0(X c).

SHR Definition 8.2. The statistical evaluation is called wealthy if it tends in probability to the estimated parameter:

This equality means that the event 0=0 becomes reliable with an unlimited increase in the sample size.

An example is the relative frequency of some event BUT, which is a consistent estimate of the probability of this event in accordance with Poisson's theorem (see formula (6.1), part 1).

Definition 8.3. A statistical estimate is said to be efficient if it has the smallest variance for the same sample sizes.

Consider the estimate M x mathematical expectation M x random variable x. As such an estimate, we choose x. Let's find the mathematical expectation of a random variable x.

Let us first make an important assertion: given that all random variables x, drawn from the same population x, so they have the same distribution as x, can be written:

Now let's find M(X in):

Thus, the sample mean is a statistical estimate of the mathematical expectation of a random variable. This estimate is consistent since, in accordance with the corollary of Chebyshev's theorem, it converges in probability to the expectation (6.3).

We found that in the case under consideration, the mathematical expectation of the estimate (random variable) chosen by us is equal to the estimated parameter itself. Estimates with this property occupy a special place in mathematical statistics; they are called unbiased.

Definition 8.4. A statistical estimate © is called unbiased if its mathematical expectation is equal to the estimated parameter

If this requirement is not met, then the estimate is called biased.

Thus, the sample mean is an unbiased estimate of the mathematical expectation.

Let us analyze the bias of the sample variance D, if it is chosen as an estimate of the general variance D x . To do this, we check the feasibility of condition (8.2) for?) :

We transform each of the two terms obtained:

Here we have used the equality M(X.) = M(X 2), valid for the same reason as (8.1).

Let's consider the second term. Using the sum square formula P terms we get

taking into account equality (8.1) again, and also the fact that X. and X are independent random variables, we write

and finally we get:

Let us substitute the obtained results into (8.3)

After transformation we get

Thus, it can be concluded that the sample variance is displaced estimation of the general variance.

Taking into account the result obtained, we set the task of constructing an estimate of the general variance that would satisfy the unbiasedness condition (8.2). To do this, consider a random variable

It is easy to see that condition (8.2) is satisfied for this quantity:

Note that the difference between the sample variance and corrected sample variance becomes insignificant at large sample sizes.

When choosing estimates for the characteristics of random variables, it is important to know their accuracy. In some cases, high accuracy is required, and sometimes it is sufficient to have a rough estimate. For example, when planning a flight with a transfer, it is important for us to know as accurately as possible the planned time of arrival at the place of flight connection. In another situation, for example, being at home and waiting for the courier with the goods ordered by us, the high accuracy of the time of his arrival is not important for us. In both cases, the random variable is the time of arrival, and the characteristic of the random variable of interest to us is the average travel time.

Ratings are of two types. In the first case, the task is to get a specific numerical value of the parameter. In another case, an interval is determined in which the parameter of interest to us falls with a given probability.

) problems of mathematical statistics.

Let us assume that there is a parametric family of probability distributions (for simplicity, we will consider the distribution of random variables and the case of one parameter). Here, is a numeric parameter whose value is unknown. It is required to estimate it by the available sample of values generated by this distribution.

There are two main types of assessments: point estimates And confidence intervals.

Point Estimation

Point estimation is a type of statistical estimation in which the value of an unknown parameter is approximated by a single number. That is, you must specify the function of the sample (statistics)

whose value will be considered as an approximation to the unknown true value .

Common methods for constructing point estimates of parameters include: maximum likelihood method, method of moments, quantile method.

Below are some properties that point estimates may or may not have.

solvency

One of the most obvious requirements for a point estimate is that one can expect a reasonably good approximation to the true value of the parameter for sufficiently large values of the sample size . This means that the estimate must converge to the true value at . This evaluation property is called solvency. Since we are talking about random variables for which there are different types of convergence, this property can also be precisely formulated in different ways:

When just using the term solvency, then we usually mean weak consistency, i.e., convergence in probability.

The consistency condition is practically obligatory for all estimates used in practice. Inconsistent estimates are rarely used.

Unbiasedness and asymptotic unbiasedness

The parameter estimate is called unbiased, if its mathematical expectation is equal to the true value of the estimated parameter:

The weaker condition is asymptotic unbiasedness, which means that the mathematical expectation of the estimate converges to the true value of the parameter with an increase in the sample size:

Unbiasedness is a recommended property of estimators. However, its importance should not be overestimated. Most often, unbiased parameter estimates exist, and then one tries to consider only them. However, there may be some statistical problems in which unbiased estimates do not exist. The most famous example is the following: consider a Poisson distribution with a parameter and set the problem of estimating the parameter . It can be proved that there is no unbiased estimator for this problem.

Grade Comparison and Efficiency

To compare different estimates of the same parameter with each other, the following method is used: choose some risk function, which measures the deviation of the estimate from the true value of the parameter, and the best one is considered to be the one for which this function takes a smaller value.

Most often, the mathematical expectation of the squared deviation of the estimate from the true value is considered as a risk function

For unbiased estimators, this is simply the variance.

There is a lower bound on this risk function called Cramer-Rao inequality.

(Unbiased) estimators for which this lower bound is met (i.e. having the smallest possible variance) are called efficient. However, the existence of an effective estimate is a rather strong requirement for the problem, which is by no means always the case.

The weaker condition is asymptotic efficiency, which means that the ratio of the variance of the unbiased estimate to the lower Cramer-Rao bound tends to unity at .

Note that under sufficiently broad assumptions about the distribution under study, the maximum likelihood method gives an asymptotically efficient estimate of the parameter, and if there is an effective estimate, then it gives an efficient estimate.

Sufficient statistics

The statistic is called sufficient for the parameter if the conditional distribution of the sample provided that , does not depend on the parameter for all .

The importance of the concept of sufficient statistics is due to the following approval. If is a sufficient statistic and is an unbiased estimate of the parameter , then the conditional expectation is also an unbiased estimate of the parameter , and its variance is less than or equal to the variance of the original estimate .

Recall that the conditional expectation is a random variable that is a function of . Thus, in the class of unbiased estimators, it suffices to consider only those that are functions of a sufficient statistic (provided that such a statistic exists for the given problem).

The (unbiased) effective parameter estimate is always a sufficient statistic.

We can say that a sufficient statistic contains all the information about the estimated parameter that is contained in the sample.