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2022 | OriginalPaper | Buchkapitel

5. Estimation

verfasst von : Maurits Kaptein, Edwin van den Heuvel

Erschienen in: Statistics for Data Scientists

Verlag: Springer International Publishing

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Abstract

The field of inferential statistics tries to use the information from a sample to make statements or decisions about the population of interest. It takes into account the uncertainty that the information is coming from sampling and does not perfectly represent the population, since another sample would give different outcomes. An important aspect of inferential statistics is estimation of the population parameters of interest. We have discussed the step from descriptions of a sample to those of a population already in Chap. 2; however, now that we have the theory of random variables at our disposal we can do much more than we did before. This is what we explore in this chapter. This chapter can be split up into two parts: in Sects. 5.2–5.4 we consider the distribution functions of sample statistics or estimators given assumptions regarding the distribution of the variables of interest in the population. Sample statistics themselves are random variables, and hence we can study their distribution functions, expectations, and higher moments. We first study the distributions of sample statistics in general, assuming that the variable of interest has some distribution in the population but without further specifying the shape of this distribution function. Next, we study the distributions of sample statistics when we assume the variable of interest to be either normally or log normally distributed in the population. We devote more attention to so-called normal populations because of their prominence in statistical theory. The second part of this chapter is Sect. 5.5, where we change our focus to estimation: in the subsections we discuss two different methods to obtain estimates \(\hat{\boldsymbol{\theta }}\) of the parameters of a population distribution \(F_{\boldsymbol{\theta }}(x)\) given sample data. The methods we discuss are the method of moments and the maximum likelihood method. In these sections, to provide a concrete example, we study the log normal distribution function, as this is one of the distribution functions for which the estimates originating from the two estimation methods differ.

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Vorheriges Kapitel Random Variables and Distributions

Nächstes Kapitel Multiple Random Variables

Higher moments of the population density f are difficult to determine or to estimate from a sample with small sample sizes. Thus, whenever data are sparse, higher moments will be considered less relevant, but in cases with big data, we anticipate that higher moments may become more important since the large sample size would make it possible to estimate these population characteristics better.

We used the notation \(\mu \left( f\right) \), \(\sigma \left( f\right) \), \(\gamma _{1}(f)\), \(\gamma _{2}(f)\), and \(x_{p}\left( f\right) \) to indicate that these population characteristics are dependent on the density function f. In many texts however, the explicit dependence on \(\left( f\right) \) is omitted for convenience, as we already did in Chap. 4.

Note that the random variables \(X_{1}\), \(X_{2},\ldots ,X_{n}\) would indicate the order of sampling. In the order of sampling there is no guarantee that they also represent the order of size. Furthermore, it should be noted that ordering the random variables (like we do for the minimum and maximum) is only unique when the random variables are continuous, since each realization will produce different values. In practice though, rounding may violate uniqueness.

Note that we are assuming simple random sampling here; when assuming other types of sampling procedures \(F_{T_{n}}\) might change. Thus \(F_{T_{n}}\) depends also on the sampling plan.

As we mentioned in Chap. 2, for normal population densities f the second moment of the sample variance is \(2\sigma ^{4}\left( f\right) /\left( n-1\right) \), which is just a function of the sample variance. Indeed, under the assumption of normality, the excess kurtosis is equal to zero.

It is interesting to think about what would happen if this was not true; how would we then go about stating something regarding a population based on a sample?

There exist several formulations of the central limit theorem. We chose the classical theorem which is the Lindeberg-Levy formulation.

The central limit theorem holds true for sums of random variables, as we just indicated, but there are other examples that demonstrate that the large sample distribution of \(T_{n}\) can be normal. For instance, it is shown that the large sample distribution of \(\sqrt{n}\left( X_{(\left\lceil np\right\rceil )}-x_{p}\right) \) converges to a normal distribution \(N\left( 0,p(1-p)/(f(x_{p}))^{2}\right) \), with f the population density, \(x_{p}\) the pth quantile, and \(X_{(k)}\) the kth-order statistic. Thus the sample distribution function of the sample statistic \(T_{n}\) can sometimes be approximated by a normal distribution function, even if it is not always the sum of independent random variables.

Appropriate confidence intervals means that the confidence interval contains the parameter of interest with the correct level of confidence. If we construct 95% confidence intervals, we would like the probability that the parameter is inside the confidence interval to be equal to 95%. If we would use asymptotic confidence intervals, and the normal approximation is not close yet due to relatively small sample sizes, the confidence level could deviate from 95%.

Note that the MLE and MME are not always the same. This depends on the particular density.

The theory of maximum likelihood estimation was developed by Sir Ronald Fisher.

B. Patrick, Probability and Measure (A Wiley-Interscience Publication, Wiley, Hoboken, 1995)

Titel: Estimation
verfasst von: Maurits Kaptein
Edwin van den Heuvel
Verlag: Springer International Publishing
Buch: Statistics for Data Scientists
Print ISBN: 978-3-030-10530-3

Electronic ISBN: 978-3-030-10531-0

Copyright-Jahr: 2022
DOI: https://doi.org/10.1007/978-3-030-10531-0_5

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