Effective assessment. Probability and statistics are basic facts. What is the subject of econometrics

So far we have talked about the optimality of estimates in the sense of the minimum of the quadratic criterion. It turns out that if the Gauss-Markov conditions are met, they are also optimal in the sense of minimum variance.

An estimate is called effective if it has minimal variance compared to other estimates of a given class.

Thus, least squares estimators are efficient, i.e. the best in terms of minimal dyspepsia, in the class of all linear unbiased parameter estimates.

Let us consider the probability density functions of both a single observation and the sample average .

The value x is considered distributed. The distributions and are symmetrical with respect to the theoretical mean. The difference is that the distribution is narrower and higher. The value is closer to than the value of a single observation, since its random component is the average of the purely random components in the sample and they seem to “cancel out” each other when calculating the average.

Subtract from (1) (2):

That is, the estimate of the theoretical variance depends on (and only on) the number of random components of observations x in the sample. Since these components change from sample to sample, the value of the estimate also changes from sample to sample.

Undisplaced.

Since the estimates are random variables, their values ​​can only by chance coincide exactly with the characteristics of the population. There will usually be some error present, which may be large or small, positive or negative, depending on the purely random components of the x values ​​in the sample.

It is desirable that the estimate be accurate on average over a sufficiently long period. That is, the mathematical expectation of the estimate = the corresponding characteristic of the general population. Such the estimate is called unbiased . If this is not the case, then the estimate is called displaced and the difference between its M.O. and the corresponding theoretical characteristic of the general population is called displacement.

The resulting estimate is not the only possible unbiased estimate. Consider a sample of only two observations and . Any weighted average of observations would be an unbiased estimate if the sum of the weights is equal to 1. Let us prove this. Let's consider the generalized form of assessment:

That ,

An unbiased statistical estimator whose variance is the same as the lower bound in the Cramer–Rao inequality.

Definition

Grade $\backslash widehat(\backslash theta\_1)\; \backslash in\; \backslash Kappa$ parameter $\backslash theta$ called effective classroom assessment $\backslash Kappa$ , if for any other estimate $\backslash widehat(\backslash theta\_2)\; \backslash in\; \backslash Kappa$ inequality holds $M\_(\backslash theta)(\backslash widehat(\backslash theta\_1)-\backslash theta)^2\backslash leqslant\; M\_(\backslash theta)(\backslash widehat(\backslash theta\_2)-\backslash theta)^2$ for anyone $\backslash theta$.

Unbiased estimators play a special role in mathematical statistics. If the unbiased estimate $\backslash widehat(\backslash theta\_1)$ is an effective estimator in the class of unbiased ones, then such statistics are usually called simply effective.

Uniqueness

Effective assessment $\backslash widehat(\backslash theta)$ in class $\backslash Kappa\_b\; =\; \backslash (\; E(\backslash widehat(\backslash theta))\; =\; c(\backslash theta)\backslash )$, Where $c(\backslash theta)$- some function exists and is unique up to values ​​on the set $A$, the probability of getting into which is zero ( $P(x\backslash in\; A)=0$).

Asymptotic efficiency

Some estimates may not be the most effective in small samples, but may have advantages in large samples. Typically, consistent estimates are considered, the variance of which tends to zero as the sample size increases. Therefore, such estimates can be compared by the speed of convergence, that is, in fact, by the dispersion (covariance matrix) of the random variable (vector) $\backslash sqrt(n)\backslash hat(\backslash theta)$. In particular, the asymptotically normal estimate

$\backslash sqrt(n)(\backslash hat(\backslash theta)-\backslash theta)\backslash xrightarrow\; d\; N(0,V)$

is asymptotically efficient if the asymptotic covariance matrix V is minimal in a given class of estimates.

see also

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Excerpt describing Effective assessment

“I’m very glad to meet you here, Count,” he told him loudly and without being embarrassed by the presence of strangers, with particular decisiveness and solemnity. “On the eve of the day on which God knows which of us is destined to survive, I am glad to have the opportunity to tell you that I regret the misunderstandings that existed between us, and I would like you not to have anything against me.” Please forgive me.
Pierre, smiling, looked at Dolokhov, not knowing what to say to him. Dolokhov, with tears welling up in his eyes, hugged and kissed Pierre.
Boris said something to his general, and Count Bennigsen turned to Pierre and offered to go with him along the line.
“This will be interesting for you,” he said.
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Half an hour later, Kutuzov left for Tatarinova, and Bennigsen and his retinue, including Pierre, went along the line.

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Pierre did not pay much attention to this redoubt. He did not know that this place would be more memorable for him than all the places in the Borodino field. Then they drove through the ravine to Semenovsky, in which the soldiers were taking away the last logs of the huts and barns. Then, downhill and uphill, they drove forward through broken rye, knocked out like hail, along a road newly laid by artillery along the ridges of arable land to the flushes [a type of fortification. (Note by L.N. Tolstoy.) ], also still being dug at that time.

One of the main requirements when constructing estimates is to obtain estimates with minimum variance or minimum dispersion (if they exist). In this regard, in mathematical statistics the concept of effective estimates was introduced,

In relation to biased estimates of a signal parameter, the estimate is called effective if the average value of the squared deviation of the estimate from the true value of the estimated parameter I does not exceed the average value of the squared deviation of any other estimate y, i.e., the inequality is satisfied

For an unbiased estimator, the dispersion of the estimator is the same as its variance; therefore, the effective unbiased estimator is defined as the estimator with the minimum variance.

S. Rao and Cramer independently obtained expressions for the lower bounds of the conditional variances and dispersion of estimates, which are the dispersion and dispersion of effective estimates, provided that they exist for the given parameters.

Let us present the derivation of this expression, assuming that the necessary assumptions are valid.

We present the estimate of parameter y in abbreviated form where X is a multidimensional sample from the implementation over a time interval

Let's average the expression

for all possible values ​​of a multidimensional sample X, which is described by a conditional probability density. Taking into account the known relation for the derivative of the natural logarithm after averaging, we obtain

Due to the normalization property of the probability density, the last term in (1.3.3) is equal to zero. The integral of the first term represents the average value of the estimate

Taking the latter into account, the averaged value can be written in the form

The left side of this expression is the average of the product of two random variables with finite values ​​of the first two moments. Under these conditions, the Bunyakovsky-Schwartz inequality, known from mathematical statistics, is valid for random variables

which turns into equality if the random variables are related by a deterministic dependence. Taking (1.3.6) into account, from expression (1.3.5) we can obtain

For unbiased and constant-biased estimators, the estimator variance satisfies the Rao-Kramer inequality

It should be noted that in all relationships, averaging is performed over a multidimensional sample of observed data X (with continuous processing - over all possible implementations of a

derivatives are taken at the point of the true value of the estimated parameter.

The equal sign in expressions (1.3.7) and (1-3.8) is achieved only for effective estimates.

In relation to expression (1.3.7), we consider the conditions under which the inequality turns into equality, i.e., the parameter estimate is an effective biased estimate. According to (1.3.6), for this it is necessary that the cross-correlation coefficient between be equal to one, i.e., so that these random functions are related by a deterministic linear relationship.

Indeed, let us represent the derivative of the logarithm of the likelihood function in the form

where is a function that does not depend on the estimate of y and the sample of observed data, but may depend on the estimated parameter. When substituting (1.3.5) and (1.3.9) into inequality (1.3.7), it turns into equality. However, representation of the derivative of the logarithm of the likelihood function in the form (1.3.9) is possible if the sufficiency condition (1.2.9) is satisfied to estimate y, from which it follows that

and therefore, if the derivative of the logarithm of the likelihood ratio depends linearly on the sufficient estimate, then the proportionality coefficient does not depend on the sample

Thus, for the existence of a biased effective estimate, two conditions must be met: the estimate must be sufficient (1.2.9) and relation (1.3.9) must be satisfied. Similar restrictions are imposed on the existence of effective unbiased estimates, under which in expression (1.3.8) the inequality sign turns into equality.

The expression obtained above for the lower bound of the dispersion of the biased estimate is also valid for the lower bound of the dispersion of the biased estimate, since i.e.

The last inequality turns into equality if, in addition to the condition of sufficiency of the estimate, the following relation is true:

where has the same meaning as in expression (1.3.9).

Formula (1.3.10) is derived similarly to (1.3.7), if in the original expression (1.3.2) instead of considering

From the nature of conditions (1.2.9) and (1.3.9) it is clear that effective estimates exist only in very specific cases. It should also be noted that an effective estimate necessarily belongs to the class of sufficient estimates, while a sufficient estimate is not necessarily effective.

Analysis of the expression for the variance of the effective mixed estimator 1.3.7) shows that there may be biased estimators that provide less estimator variance than unbiased ones. To do this, it is necessary that the derivative of the offset have a negative value and be close to unity in absolute value at the point of the true value of the parameter.

Since in most cases the average square of the resulting estimation error (dispersion) is of interest, it makes sense to talk about the average square of the estimation error, which for any estimate is bounded from below:

In this case, for effective estimates there is an equal sign.

It is easy to show that relations (1.3.10) and (1.3.12) coincide if conditions (1.3.11) and (1.3.9) are satisfied, respectively. Indeed, substituting the values ​​expressed through functions into the numerator and denominator (1.3.10) we obtain (1.3.12).

Using the properties of effective estimates discussed above, we will clarify their definition. We will call an estimate y effective if either conditions (1.2.9) and (1.3.11) are satisfied for it, or if it has a dispersion for a given bias

or scattering

or with zero bias this estimate has variance

Note that the characteristics of the effective estimate (1.3.13) - (1.3.15) can also be calculated for those parameters for which there is no effective estimate. In this case, values ​​(1.3.13) - (1.3.15) determine the lower limit (unattainable) for the corresponding assessment characteristics.

To compare real estimates with effective ones in mathematical statistics, the concept of relative efficiency of estimates has been introduced, representing the ratio of the mean square deviation of the effective estimate relative to the true value of the parameter to the mean square deviation of the real estimate relative to the true value of the parameter:

Here y is the real estimate, the effectiveness of which is equal to the effective estimate.

From the definition of the variance of the effective estimate (1.3.1) it is clear that the relative efficiency of the estimate varies within

In addition to the concept of effective estimates, there is the concept of asymptotically effective estimates. It is assumed that for a sufficiently long observation time or an unlimited increase in the signal-to-noise ratio limit value the relative efficiency of the real assessment is equal to one. This means that with an asymptotically efficient estimate, the variance of the estimate for a given bias is determined by expression (1.3.13), and in the absence of a bias, by expression (1.3.15).

In order for statistical estimates to provide a good approximation of the estimated parameters, they must be unbiased, efficient and consistent.

Unbiased is called a statistical parameter estimate , the mathematical expectation of which is equal to the estimated parameter for any sample size.

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

Effective called statistical estimation
parameter , which for a given sample size has the smallest dispersion.

Wealthy called statistical estimation
parameter , which at
tends in probability to the estimated parameter.

i.e. for any

.

For samples of different sizes, different values ​​of the arithmetic mean and statistical dispersion are obtained. Therefore, the arithmetic mean and statistical variance are random variables for which there is a mathematical expectation and variance.

Let's calculate the mathematical expectation of the arithmetic mean and variance. Let us denote by mathematical expectation of a random variable

Here the following are considered as random variables: – S.V., the values ​​of which are equal to the first values ​​obtained for various volume samples from the general population,
–S.V., the values ​​of which are equal to the second values ​​obtained for various volume samples from the general population, ...,
– S.V., whose values ​​are equal -th values ​​obtained for various volume samples from the general population. All these random variables are distributed according to the same law and have the same mathematical expectation.

From formula (1) it follows that the arithmetic mean is an unbiased estimate of the mathematical expectation, since the mathematical expectation of the arithmetic mean is equal to the mathematical expectation of the random variable. This assessment is also valid. The effectiveness of this estimate depends on the type of distribution of the random variable
.
If, for example,

Let us now find a statistical estimate of the dispersion.

The expression for statistical variance can be transformed as follows

(2)

Let us now find the mathematical expectation of the statistical dispersion

. (3)

Considering that
(4)

we obtain from (3) -

From formula (6) it is clear that the mathematical expectation of the statistical dispersion differs by a factor from the dispersion, i.e. is a biased estimate of the population variance. This is due to the fact that instead of the true value
, which is unknown, the statistical mean is used in estimating the variance .

Therefore, we introduce the corrected statistical variance

(7)

Then the mathematical expectation of the corrected statistical variance is equal to

those. the corrected statistical variance is an unbiased estimate of the population variance. The resulting estimate is also consistent.

- score from the minimum variance for a given sample size. An analysis that has a similar property for an unlimitedly increasing sample size is called asymptotically efficient. The property of efficiency must be taken into account in geology depending on the circumstances under which the estimate is obtained. In some cases, lithology uses an ineffective estimate (median, quartiles) due to the fact that their calculation is simpler than the corresponding O. e. The coefficient of variation, which is widely used in reserve calculations, is also an ineffective estimate. In the latter case, its use is sometimes not justified.

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