A Concise Guide to Statistics by Hans-Michael Kaltenbach

By Hans-Michael Kaltenbach

The textual content offers a concise creation into primary techniques in statistics. bankruptcy 1: brief exposition of chance idea, utilizing wide-spread examples. bankruptcy 2: Estimation in conception and perform, utilizing biologically influenced examples. Maximum-likelihood estimation in lined, together with Fisher info and tool computations. tools for calculating self belief durations and powerful possible choices to straightforward estimators are given. bankruptcy three: speculation checking out with emphasis on techniques, fairly type-I , type-II mistakes, and examining try out effects. a number of examples are supplied. T-tests are used all through, very important different assessments and robust/nonparametric choices. a number of checking out is mentioned in additional intensity, and mix of autonomous checks is defined. bankruptcy four: Linear regression, with computations completely according to R. a number of staff comparisons with ANOVA are lined including linear contrasts, back utilizing R for computations.

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Consistency. 2 Constructing Estimators 35 θˆn → θ as n → ∞. While all three estimators ( pˆ n , X¯ , S 2 ) in the binomial and normal examples are consistent, the above estimator θˆn ≡ 0 is obviously not, because the estimate does not get any closer to the true value, no matter how many samples we take. Unbiasedness. Even if an estimator is consistent, it might still be that it systematically over- or underestimates the true value and introduces a bias in the estimate. The bias is given by the difference of expected and true value E(θˆn ) − θ, and an estimator is called unbiased if this difference is zero.

087), which is fairly close to the correct values, taking into account that we do not have many data available. The seemingly ad-hoc estimator in Sect. 2 for the binomial case was derived in this way. 6 Fisher-Information and Cramér-Rao Bound We conclude the chapter by a brief discussion of the idea of Fisher-information, from which we can derive a theoretical lower bound for the variance of an estimator. This bound tells us how precise we can actually estimate a given parameter with a fixed number of samples.

Are they unbiased? Let us start with X¯ ; its unbiasedness is easily established by exploiting the linearity of the expectation: E( X¯ ) = E 1 n n Xi i=1 = 1 n n E(X i ) = i=1 1 nμ = μ. n The calculation for S 2 is slightly more elaborate and we skip some details: E(S 2 ) = 1 n n E (X i − X¯ )2 = i=1 = σ2 − 1 n n E (X i − μ)( X¯ − μ) i=1 1 2 2 nσ 2 σ + 2 = σ2 1 − n n n = σ 2. The MLE for the variance is therefore biased and systematically underestimates the true variance. It is nevertheless consistent, as the bias is proportional to 1/n and decreases rapidly to zero for increasing n.

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