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Download An Essay on the Principle of Population (Oxford World's by Thomas Malthus PDF

By Thomas Malthus

ISBN-10: 0191605948

ISBN-13: 9780191605949

Malthus's Essay appears to be like on the perennial tendency of people to outstrip their assets: replica continually exceeds meals creation. at the present time Malthus is still a byword for drawback approximately man's demographic and ecological clients.

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Extra info for An Essay on the Principle of Population (Oxford World's Classics)

Sample text

The central mathematical result is given in the following wellknown theorem, which we state without proof. 2. Let θˆ1 , . . , θˆk be independent and identically distributed normal (θ, σ 2 ) random variables. Then (i) the statistic z = (θ¯ˆ − θ )/ σ 2 /k is distributed as a normal (0, 1) random variable; and (ii) the statistic ¯ˆ t = (θ¯ˆ − θ)/ v(θ) is distributed as Student’s t with k − 1 degrees of freedom. If the variance of θ¯ˆ is essentially known without error or if k is very large, then a (1 − α) 100% confidence interval for θ is (θ¯ˆ − z α/2 v(θ¯ˆ),θ¯ˆ + z α/2 v(θ¯ˆ)), where z α/2 is the upper α/2 percentage point of the N (0, 1) distribution.

2. Although this variation of the method is not frequently employed in practice, there is exact theory regarding the properties of the estimators whenever the samples are independent. In practical applications, the samples are often dependent in some sense, and exact statistical theory for the estimators is generally not available. 3. 21 P1: OTE/SPH P2: OTE SVNY318-Wolter December 13, 2006 22 19:54 2. 2. The Case of Independent Random Groups Mutual independence of the various samples (or, more properly, of estimators derived from the samples) arises when one sample is replaced into the population before selecting the next sample.

1 displays the bias in v(θ¯ˆ) as an estimator of Var{θ}. lations and small sampling fractions, however, 2 kα kβ>α Cov{θˆα , θˆβ }/{k(k − 1)} P1: OTE/SPH P2: OTE SVNY318-Wolter December 13, 2006 19:54 36 2. The Method of Random Groups will tend to be relatively small and negative. The quantity 1 k(k − 1) k (μα − μ) ¯ 2 α=1 ¯ˆ will tend to be relatively small when μα = ˙ μ(α ¯ = 1, . . , k). Thus, the bias of v(θ) will be unimportant in many large-scale surveys and tend to be slightly positive. When the estimator θˆ is linear, various special results and estimators are available.

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An Essay on the Principle of Population (Oxford World's Classics) by Thomas Malthus


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