Abstract
For the case of sufficiently large sample sizes, many properties of the binomial and multinomial distributions may be well approximated by normal distributions. The theoretical framework for such approximations is convergence in distribution. It is implied by the central limit theorem that when the sample size goes to infinity, appropriately normed binomial and multinomial distributions converge to the normal distribution. If, however, in the case of binomial distributions, also p converges to zero, so that np remains constant, the binomial converges to a Poisson distribution. The most important use of normal approximations is a very useful method to obtain asymptotic variance and covariance formulas for functions of binomial or multinomial variables. This δ-method is widely used in estimation and testing. Asymptotic normality applies to sampling distributions and refers to the deviation of the observed probabilities from their respective expectations. It does not affect the fundamental difference between categorical and normal assumptions with respect to the population distribution.
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Notes
- 1.
Sometimes the distribution function is defined as F n (x) = P(X n < x) but the two definition lead to the same results in this case.
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- 3.
Note that the 0 submatrices in \(\boldsymbol{\varSigma }\) are of different sizes.
- 4.
Because in such cases the population size ranges from a couple of millions to hundreds of millions of people and the sample size is a few thousands, the issue of replacement is irrelevant for practical purposes; see Sect. 2.1.
- 5.
This is a very unrealistic assumption. In real surveys, usually 40–70% of the selected sample persons respond. Some are not found, some reject to participate in the survey, and some choose not to respond to the relevant question. Therefore, considerations that disregard nonresponse, like the ones that follow, are important but do not reflect upon reality sufficiently precisely. Their role in comparing different surveys is similar to that of official fuel consumption figures for cars: one cannot expect to make as many miles per gallon in real life as indicated, but a car with a higher official consumption figure is likely to have a higher real consumption than another one with a lower figure.
- 6.
This will be justified in the next chapter of the book.
References
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Kopylov, I.: Subjective Probability. In Rudas, T. (ed.) Handbook of Probability: Theory and Applications, pp. 35–48. Sage Publications, Thousand Oaks (2008)
Shao, J.: Mathematical Statistics, 2nd ed. Springer, New York (2003)
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Rudas, T. (2018). Normal Approximations. In: Lectures on Categorical Data Analysis. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7693-5_3
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DOI: https://doi.org/10.1007/978-1-4939-7693-5_3
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