You get a bonus - 1 coin for daily activity. Now you have 1 coin

14.4. Exact methods for constructing confidence intervals for random variable parameters

Lecture



Exact methods for constructing confidence intervals for parameters of a random variable distributed according to the normal law

In the previous   14.4.  Exact methods for constructing confidence intervals for random variable parameters We considered roughly approximate methods for constructing confidence intervals for expectation and variance. In this   14.4.  Exact methods for constructing confidence intervals for random variable parameters we will give an idea of ​​the exact methods of solving the same problem. We emphasize that to accurately determine the confidence intervals, it is absolutely necessary to know in advance the form of the distribution law   14.4.  Exact methods for constructing confidence intervals for random variable parameters whereas it is not necessary for the application of approximate methods.

The idea of ​​accurate methods for constructing confidence intervals is as follows. Any confidence interval is found from a condition expressing the probability of certain inequalities that include the estimate we are interested in.   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Distribution law   14.4.  Exact methods for constructing confidence intervals for random variable parameters in general, depends on the unknown parameters themselves   14.4.  Exact methods for constructing confidence intervals for random variable parameters . However, sometimes it is possible to move in inequalities from a random variable   14.4.  Exact methods for constructing confidence intervals for random variable parameters to some other function of the observed values   14.4.  Exact methods for constructing confidence intervals for random variable parameters , the distribution law of which does not depend on unknown parameters, but depends only on the number of experiments   14.4.  Exact methods for constructing confidence intervals for random variable parameters and on the type of distribution law   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Such random variables play a large role in mathematical statistics; they are most studied in the case of a normal distribution of magnitude   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

For example, it is proved that with a normal distribution of   14.4.  Exact methods for constructing confidence intervals for random variable parameters random value

  14.4.  Exact methods for constructing confidence intervals for random variable parameters , (14.4.1)

Where

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ,   14.4.  Exact methods for constructing confidence intervals for random variable parameters ,

obeys the so-called student distribution law with   14.4.  Exact methods for constructing confidence intervals for random variable parameters degrees of freedom; the density of this law is

  14.4.  Exact methods for constructing confidence intervals for random variable parameters (14.4.2)

Where   14.4.  Exact methods for constructing confidence intervals for random variable parameters - known gamma function:

  14.4.  Exact methods for constructing confidence intervals for random variable parameters .

It is also proved that the random variable

  14.4.  Exact methods for constructing confidence intervals for random variable parameters (14.4.3)

has a "distribution   14.4.  Exact methods for constructing confidence intervals for random variable parameters " with   14.4.  Exact methods for constructing confidence intervals for random variable parameters degrees of freedom (see Ch. 7. p. 145), the density of which is expressed by the formula

  14.4.  Exact methods for constructing confidence intervals for random variable parameters (14.4.4)

Without dwelling on the conclusions of the distributions (14.4.2) and (14.4.4), we will show how they can be applied when building confidence intervals for the parameters   14.4.  Exact methods for constructing confidence intervals for random variable parameters and   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

Let produced   14.4.  Exact methods for constructing confidence intervals for random variable parameters independent experiments on a random variable   14.4.  Exact methods for constructing confidence intervals for random variable parameters distributed according to normal law with unknown parameters   14.4.  Exact methods for constructing confidence intervals for random variable parameters and   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Estimates for these parameters

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ,   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

It is required to build confidence intervals for both parameters corresponding to the confidence probability.   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

We first construct the confidence interval for the expectation. Naturally, this interval is symmetric with respect to   14.4.  Exact methods for constructing confidence intervals for random variable parameters ; denote   14.4.  Exact methods for constructing confidence intervals for random variable parameters half the length of the interval. Magnitude   14.4.  Exact methods for constructing confidence intervals for random variable parameters need to choose so that the condition is met

  14.4.  Exact methods for constructing confidence intervals for random variable parameters . (14.4.5)

Let's try to go to the left side of equality (14.4.5) from the random variable   14.4.  Exact methods for constructing confidence intervals for random variable parameters to a random value   14.4.  Exact methods for constructing confidence intervals for random variable parameters distributed by the law of student. To do this, multiply both sides of the inequality   14.4.  Exact methods for constructing confidence intervals for random variable parameters by a positive value   14.4.  Exact methods for constructing confidence intervals for random variable parameters :

  14.4.  Exact methods for constructing confidence intervals for random variable parameters

or, using the designation (14.4.1),

  14.4.  Exact methods for constructing confidence intervals for random variable parameters . (14.4.6)

Find such a number   14.4.  Exact methods for constructing confidence intervals for random variable parameters , what

  14.4.  Exact methods for constructing confidence intervals for random variable parameters . (14.4.7)

Magnitude   14.4.  Exact methods for constructing confidence intervals for random variable parameters there is a condition

  14.4.  Exact methods for constructing confidence intervals for random variable parameters . (14.4.8)

From the formula (14.4.2) it is clear that   14.4.  Exact methods for constructing confidence intervals for random variable parameters - even function; therefore (14.4.8) gives

  14.4.  Exact methods for constructing confidence intervals for random variable parameters . (14.4.9)

Equality (14.4.9) determines the value   14.4.  Exact methods for constructing confidence intervals for random variable parameters depending on the   14.4.  Exact methods for constructing confidence intervals for random variable parameters . If you have at your disposal a table of values ​​of the integral

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ,

This value can be found by reverse interpolation in this table. However, it is more convenient to make a table of values ​​in advance.   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Such a table is given in the annex (see Table 5). This table shows the values   14.4.  Exact methods for constructing confidence intervals for random variable parameters depending on confidence probability   14.4.  Exact methods for constructing confidence intervals for random variable parameters and numbers of degrees of freedom   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Defining   14.4.  Exact methods for constructing confidence intervals for random variable parameters according to table 5 and believing

  14.4.  Exact methods for constructing confidence intervals for random variable parameters (14.4.10)

we will find half the width of the confidence interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters and the interval itself

  14.4.  Exact methods for constructing confidence intervals for random variable parameters . (14.4.11)

Example 1. Produced 5 independent experiments on a random variable   14.4.  Exact methods for constructing confidence intervals for random variable parameters distributed normally with unknown parameters   14.4.  Exact methods for constructing confidence intervals for random variable parameters and   14.4.  Exact methods for constructing confidence intervals for random variable parameters . The results of the experiments are given in table 14.4.1.

Table 14.4.1

  14.4.  Exact methods for constructing confidence intervals for random variable parameters

one

2

3

four

five

  14.4.  Exact methods for constructing confidence intervals for random variable parameters

-2.5

3.4

-2,0

1.0

2.1

Find a rating   14.4.  Exact methods for constructing confidence intervals for random variable parameters for the expectation and build for it a 90% confidence interval (i.e., the interval corresponding to the confidence probability   14.4.  Exact methods for constructing confidence intervals for random variable parameters ).

Decision. We have

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ;   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

According to table 5 of the application for   14.4.  Exact methods for constructing confidence intervals for random variable parameters and   14.4.  Exact methods for constructing confidence intervals for random variable parameters we find

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ,

from where

  14.4.  Exact methods for constructing confidence intervals for random variable parameters .

Confidence interval will be

  14.4.  Exact methods for constructing confidence intervals for random variable parameters .

Example 2. For the conditions of example 1   14.4.  Exact methods for constructing confidence intervals for random variable parameters 14.3, assuming the value   14.4.  Exact methods for constructing confidence intervals for random variable parameters distributed normally, find the exact confidence interval.

Decision. According to table 5 of the application we find when   14.4.  Exact methods for constructing confidence intervals for random variable parameters and   14.4.  Exact methods for constructing confidence intervals for random variable parameters

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ; from here   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

Comparing with the solution of example 1   14.4.  Exact methods for constructing confidence intervals for random variable parameters 14.3 (   14.4.  Exact methods for constructing confidence intervals for random variable parameters ), we are convinced that the discrepancy is very slight. If we keep the accuracy up to the second decimal place, then the confidence intervals found by the exact and approximate methods are the same:

  14.4.  Exact methods for constructing confidence intervals for random variable parameters .

Let us proceed to the construction of a confidence interval for variance.

Consider the unbiased variance estimate.

  14.4.  Exact methods for constructing confidence intervals for random variable parameters

and express the random variable   14.4.  Exact methods for constructing confidence intervals for random variable parameters in terms of   14.4.  Exact methods for constructing confidence intervals for random variable parameters (14.4.3), having a distribution   14.4.  Exact methods for constructing confidence intervals for random variable parameters (14.4.4):

  14.4.  Exact methods for constructing confidence intervals for random variable parameters . (14.4.12)

Knowing the law of distribution of magnitude   14.4.  Exact methods for constructing confidence intervals for random variable parameters , you can find the interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters with which it falls with a given probability   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

Distribution law   14.4.  Exact methods for constructing confidence intervals for random variable parameters magnitudes   14.4.  Exact methods for constructing confidence intervals for random variable parameters has the form shown in fig. 14.4.1.

  14.4.  Exact methods for constructing confidence intervals for random variable parameters

Fig. 14.4.1.

The question arises: how to choose the interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters ? If the distribution law   14.4.  Exact methods for constructing confidence intervals for random variable parameters was symmetrical (like a normal law or student’s t-distribution), it would be natural to take the interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters symmetric with respect to the expectation. In this case, the law   14.4.  Exact methods for constructing confidence intervals for random variable parameters asymmetrical We agree to choose an interval so that the probability of the output value   14.4.  Exact methods for constructing confidence intervals for random variable parameters outside the interval to the right and left (the shaded areas in Fig. 14.4.1) were the same and equal

  14.4.  Exact methods for constructing confidence intervals for random variable parameters .

To build an interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters with this property, we use table 4 of the appendix: it shows the numbers   14.4.  Exact methods for constructing confidence intervals for random variable parameters such that

  14.4.  Exact methods for constructing confidence intervals for random variable parameters

for size   14.4.  Exact methods for constructing confidence intervals for random variable parameters having   14.4.  Exact methods for constructing confidence intervals for random variable parameters distribution with   14.4.  Exact methods for constructing confidence intervals for random variable parameters degrees of freedom. In our case   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Fix   14.4.  Exact methods for constructing confidence intervals for random variable parameters and find in the corresponding row table. 4 two values   14.4.  Exact methods for constructing confidence intervals for random variable parameters ; one that corresponds to probability   14.4.  Exact methods for constructing confidence intervals for random variable parameters , other - probabilities   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Denote these values   14.4.  Exact methods for constructing confidence intervals for random variable parameters and   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters It has   14.4.  Exact methods for constructing confidence intervals for random variable parameters to your left as well   14.4.  Exact methods for constructing confidence intervals for random variable parameters - right end.

Now we find by interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters sought confidence interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters for dispersion with borders   14.4.  Exact methods for constructing confidence intervals for random variable parameters and   14.4.  Exact methods for constructing confidence intervals for random variable parameters that covers the point   14.4.  Exact methods for constructing confidence intervals for random variable parameters with probability   14.4.  Exact methods for constructing confidence intervals for random variable parameters :

  14.4.  Exact methods for constructing confidence intervals for random variable parameters .

We construct such an interval   14.4.  Exact methods for constructing confidence intervals for random variable parameters that covers a point   14.4.  Exact methods for constructing confidence intervals for random variable parameters if and only if the value   14.4.  Exact methods for constructing confidence intervals for random variable parameters falls within the interval  14.4.  Exact methods for constructing confidence intervals for random variable parameters . We show that the interval

  14.4.  Exact methods for constructing confidence intervals for random variable parameters (14.4.13)

satisfies this condition. Indeed, inequalities

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ;   14.4.  Exact methods for constructing confidence intervals for random variable parameters

equal inequalities

  14.4.  Exact methods for constructing confidence intervals for random variable parameters ;   14.4.  Exact methods for constructing confidence intervals for random variable parameters ,

and these inequalities are fulfilled with probability   14.4.  Exact methods for constructing confidence intervals for random variable parameters . Thus, the confidence interval for the dispersion is found and expressed by the formula (14.4.13).

Example 3. Find the confidence interval for the variance under the conditions of Example 2   14.4.  Exact methods for constructing confidence intervals for random variable parameters 14.3 if it is known that the value is   14.4.  Exact methods for constructing confidence intervals for random variable parameters distributed normally.

Decision. We have   14.4.  Exact methods for constructing confidence intervals for random variable parameters ;   14.4.  Exact methods for constructing confidence intervals for random variable parameters ;   14.4.  Exact methods for constructing confidence intervals for random variable parameters . According to table 4 of the application we find when   14.4.  Exact methods for constructing confidence intervals for random variable parameters

for   14.4.  Exact methods for constructing confidence intervals for random variable parameters   14.4.  Exact methods for constructing confidence intervals for random variable parameters ;

for   14.4.  Exact methods for constructing confidence intervals for random variable parameters   14.4.  Exact methods for constructing confidence intervals for random variable parameters .

According to the formula (14.4.13) we find the confidence interval for the variance

  14.4.  Exact methods for constructing confidence intervals for random variable parameters .

The corresponding interval for the standard deviation:   14.4.  Exact methods for constructing confidence intervals for random variable parameters . This interval only slightly exceeds that obtained in example 2   14.4.  Exact methods for constructing confidence intervals for random variable parameters 14.3 approximate spacing method   14.4.  Exact methods for constructing confidence intervals for random variable parameters .


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis