By Hans-Michael Kaltenbach
The textual content supplies a concise advent into primary ideas in statistics. bankruptcy 1: brief exposition of likelihood concept, utilizing general examples. bankruptcy 2: Estimation in idea and perform, utilizing biologically encouraged examples. Maximum-likelihood estimation in coated, together with Fisher details and tool computations. tools for calculating self assurance periods and powerful possible choices to straightforward estimators are given. bankruptcy three: speculation checking out with emphasis on suggestions, really type-I , type-II mistakes, and analyzing try effects. a number of examples are supplied. T-tests are used all through, very important different assessments and robust/nonparametric choices. a number of trying out is mentioned in additional intensity, and mixture of self sustaining assessments is defined. bankruptcy four: Linear regression, with computations completely in response to R. a number of crew comparisons with ANOVA are lined including linear contrasts, back utilizing R for computations.
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For example, consistency guarantees that the estimated value will approach the true value in the limit, but does not give information on how close it is to the true value, given some data with a certain number of samples. For quantifying the quality of a particular estimate, we can compute a confidence interval (CI) around the estimate θˆn , such that this interval covers the true value θ with some high probability 1 − α. The narrower this interval, the closer we are to the true value, with high probability.
Then, by the laws of large numbers, the sample mean approaches the true expectation for increasing b. The same argument still holds if we apply a function h on mean and expectation: 1 b b h(Yi ) → E(h(Y1 )). i=1 For example, we recover the variance estimator by choosing h = (Yi − Y¯ )2 . The key idea on how this helps is the following: let us consider any estimator θˆn and denote by F(x) = P(θˆn ≤ x) its cumulative distribution function. For the beginning, we are interested in calculating Var(θˆn ).
For additionally capturing their different shapes, higher moments are needed (see Sect. 5). Example 8 Let us consider the following model of a random DNA sequence as introduced earlier: we assume independence among the nucleotides and in each position, the probabilities of having a particular nucleotide are p A , pC , pG , pT , respectively. We investigate two sequences of length n by comparing the nucleotides in the same position. Assume that the sequences are completely random and unrelated. At any 2 + p 2 , as both position, the probability of a match is then p := p 2A + pC2 + pG T nucleotides have to be the same.
A Concise Guide to Statistics by Hans-Michael Kaltenbach