Read e-book online A Concise Guide to Statistics PDF

By Hans-Michael Kaltenbach

ISBN-10: 3642235018

ISBN-13: 9783642235016

The textual content supplies a concise creation into basic strategies in information. bankruptcy 1: brief exposition of likelihood thought, utilizing favourite examples. bankruptcy 2: Estimation in idea and perform, utilizing biologically encouraged examples. Maximum-likelihood estimation in coated, together with Fisher details and tool computations. equipment for calculating self belief periods and strong possible choices to straightforward estimators are given. bankruptcy three: speculation checking out with emphasis on thoughts, rather type-I , type-II error, and studying try effects. numerous examples are supplied. T-tests are used all through, vital different assessments and robust/nonparametric choices. a number of checking out is mentioned in additional intensity, and mixture of self sustaining assessments is defined. bankruptcy four: Linear regression, with computations exclusively in keeping with R. a number of workforce comparisons with ANOVA are coated including linear contrasts, back utilizing R for computations.

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While the normal data fit the line nicely, the exponential data deviate strongly from the expected quantiles of a normal distribution A quantile-quantile plot for a normal sample is given in Fig. 10 (left) together with the theoretical quantiles (solid line). For comparison, sample points from an exponential distribution are plotted together with the normal distribution quantiles in Fig. 10 (right). As we can see, the agreement of the theoretical and the empirical quantiles is quite good for the normal sample, but it is poor for the exponential sample, especially in the tails.

The value of any sample point is completely determined if we know X¯ and the other remaining points. The degrees of freedom in the estimate are therefore n − 1 rather than n, as we already “used” one degree for estimating μ. Indeed, S 2 = σˆ n2 = 1 n−1 n X i − X¯ 2 i=1 is an unbiased estimator for the variance, but not a maximum-likelihood estimator. 36 2 Estimation Properties of ML-Estimators. Conveniently, maximum-likelihood estimators automatically have many desired properties. They are • • • • consistent: they approach the true parameter value with increasing sample size, equivariant: if r is a function, then r (θˆn ) is also the MLE of r (θ ), not necessarily unbiased, so we need to take caution here, ˆ asymptotically normal: √ θn −θ → Norm (0, 1) as n → ∞.

In practice, we usually do not know these probabilities, of course, and sometimes have no control over the possible sample size n. Imagine that we only have a sequence of 20 nucleotides. If it happens not to have any G, we consequently estimate pˆ G = 0. This will have undesired consequences if we use these values for a model to describe the sequence matching probabilities, because the model would assume that G can never occur and will thus incorrectly predict the possible number of matchings. One way of dealing with this problem is to introduce pseudo-counts by pretending that there is a certain number of observations in each category to begin with.

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A Concise Guide to Statistics by Hans-Michael Kaltenbach

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