Lecture. Statistical estimates of general population parameters
Quite often, we know the type of feature distribution in advance, but we do not know what parameters this type has.
In other words, it may have some different mathematical expectations, different dispersion, and so on.
The purpose of this section is to evaluate these parameters.
The task is set as follows. Let ushave a certain set of attribute values.
In each experiment, we are going to consider a random variable Хi, whereХiis a random variable, a feature in this experiment, and xi is its value in a particular experiment. That is, we get some value based on the results of the experiment.
We assume that all random variables are independent, that is, the experiments are independent, and all random variables have the same distribution.
That is, we consider the same feature under the same conditions of the experiment.
Then the statistical estimate of the parameter θis some non-random function of random variables x1, x2, and so on xn.
If θ is the mathematical expectation of a random variable, then the probable estimates can be the arithmetic mean of the value (most frequent estimate). It can also be the mode (most likely value), the median (if we sort the values, it is in the middle, that is not mean, but it is in the middle), as well as, half the sum of the maximum and minimum.
The question arises: how do we choose the best one among these estimates? In fact, there is no clear answer to this question. But we are going to consider some criteria by which certain estimates of random variables are compared.
We are also going to consider what features and properties a statistical assessment can have.
Feature 1. Unbiasedness. A statistical estimate is unbiased if its value is equal to its mathematical expectation.
Let us consider the properties of statistical estimates. A statistical estimate is unbiased if the mathematical expectation of the estimate is equal to the value of the property. Statistics tries to consider unbiased estimates, but this is not always possible. In some cases, there are no unbiased estimates, and then asymptotically unbiased estimates are considered.
That is, there is no unbiased estimatefor each specific sample, but in the limit, this estimate is unbiased.
That is, in the limit, the mathematical expectation tends to the value of the property.
Let us take an example. X with a dash is the arithmetic mean of the values of the sample attribute or the sample average. This estimate is unbiased for the mathematical expectation. Let us prove it.
Let the mathematical expectation of X with a dash be the mathematical expectation of this arithmetic mean. By the properties of the mathematical expectation, the constant n can be taken out of bounds, and the mathematical expectation of the sum is equal to the sum of the mathematical expectations. As a result, we get the equation indicated at the bottom of the slide.
Note that all xi are equally distributed.Thus, their mathematical expectation is equal to a, that is, it is equal to the general mathematical expectation.
We get a coefficient of 1/n, n is a quantity, and each value is equal to a. n is reduced, and the mathematical expectation of this estimate is equal to a.
This is the average random unbiased estimate of the mathematical expectation.
Let's check whether the sample dispersion is an unbiased estimate of the general dispersion - squared. What is the general dispersion? The value xi has the mathematical expectation a, as in the previous example, and the dispersion - squared. We are going to use this for calculations. Let us prove it.
The mathematical expectation s2 (we denote the sample dispersion as s2;thus, the sample mean, the square deviation is indicatedby the letter s) is equal to… Replace... Then, according to the properties of the mathematical expectation, 1/n can be taken out of the bracket, and in brackets we write xi minus x with a dash, as xi minus a, minus (x with a dash minus a).
That is, we add and subtract the general average. Next, you need to expand the brackets, transform it, and use the properties of the mathematical expectation.
As a result, you should get that the mathematical expectation s2 is not equal to the general dispersion. It is equal to (n-1) / n -squared. Thus, the sample dispersion is a biased estimate of the general dispersion.
In order to get an unbiased estimate, you need to correct the dispersion, that is, multiply it by the coefficient n/(n-1). The slide shows the sample dispersion and the corrected sample.
The corrected dispersionis denoted by the symbol s02.
Feature 2. An estimate is consistent if its value is likely to converge to the value of the desired feature.
The formula is shown on the slide.
It is inconvenient to prove the consistency of estimates. Sometimes you can use the laws of large numbers. Sometimes we can use the statement shown on the slide to prove the consistency. That is, if the dispersion is limited and tends to 0, then if the estimate is unbiased, it is also consistent.
Note that the estimates of X with a dash s02 are consistent. We won't prove it for s02.
You will do it on your own.
There are quite complex calculations.
Now we are going to prove that the sample average is a consistent estimate of the general average. We know that X with a dash is an unbiased estimate. Therefore, we can use the statement. To do this, let us find the dispersion of the sample average.
By the properties of the dispersion, the coefficient can be squared and taken out, and the dispersion of the sum is equal to the sum of the dispersion. Thus, we get the following expression.
The disperion of the sample mean is - squared by n.
Since - squared is a constant value and n tends to infinity, this ratio tends to 0.
Thus, the dispersion is limited.
Since the estimate is unbiased, according to the statement, it is also consistent.
Feature 3. Efficiency. An unbiased estimate is effective (it is significant that efficiency is considered only for estimates with the same bias; we are going to consider it for unbiased estimates) if its dispersion is minimal among all non-biased estimates studied.
In other words, we consider a class of estimates that have the same bias (in our case, 0) and choose the effective estimate, which has the smallest dispersion among them.
For example, for the mathematical expectation of a random variable, this estimate is the arithmetic mean. However, in practice, inefficient estimates can also be used.
For example, the median efficiency tends to 2/π, it is used in practice due to fewer calculations.
Each rating is considered to have its efficiency. This is the ratio of the dispersion of the effective estimate to the dispersion of the current estimate (this is shown on the slide for the median).
Point estimates are not always enough.
For example, we find that the expectation estimate is 135.5.
Is it true that this mathematical expectation is equal to this value? Of course not.
Another point for us to consider. We can't even say that there is an interval in which this value will necessarily fall with absolute certainty.
We can only say that with some confidence γ, this value is called the confidence probability; the value of the estimate falls into a certain interval.
It's not entirely correct to speak in this way. It is more correct to say that the interval covers the true value. After all, it already exists, it is specified, and the interval is random, because the point estimate is a random variable.
Accordingly, its borders are also random.
γ is also called the reliability of the estimate. Often, in practice, quite large values of γare taken – 0.95, 0.99 or more.
That is, less than 0.95 is usually not considered. Sometimes 0.9 can be taken, but this is a very rare situation.
Accordingly, the interval (θn* -, θn*+) is the confidence interval. The confidence interval is not always symmetric with respect to the point estimate. Sometimes we consider confidence intervals that are symmetric in probability.
When we consider that γ must fall into the interval, and then 1/γ is outside the interval, and 1 - γ must be on both sides.
If we know the distribution of the estimate, then we can find it.
In general, this is required in order to build a confidence interval.
Finding out the distribution of the estimate is the most difficult moment when building interval estimates. It is not always possible to find out the distribution of a feature and, consequently, the distribution of its estimate.
Therefore, we are going to consider the simplest case. Let us assume that the feature has a normal distribution. Then its estimates (parameters) –the mathematical expectation and dispersion – also have a normal distribution for a large sample.
Let'stake the simplest example. Let the feature have a normal distribution.
Its parameters are its mathematical expectation and σ-squared. Since the normal distribution is the most common, we obtain good estimates for its parameters.
The first example is the mathematical expectation. You are to consider the dispersionusing the reference book on your own.
We need to build a confidence interval for the mathematical expectation. If we consider the value (X with the dash – a) * √n/s0, we can see that this value has a Student distribution with (n-1) degree of freedom.
Why Student’s? This distribution is close to normal.
With a large number of tests, it converges to normal.
You can read about its properties in the reference book. There is a table there. Oryou can find additional information.
Mathematical statistics often uses it.
We'll get back to it later.
We have n tests, but due to the fact that our dispersion is normalized, one degree of freedom is bound, so S(n-1).
The confidence interval of the mathematical expectation looks like this (see on the slide), where the values of t are the quantiles of the Student's distribution. The quantiles of this distribution are tabulated, and they can be calculated both in specialized tables and in the Microsoft Excel package, which will be considered at the laboratory class.