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**Contents:**

Introduction to Bayesian Econometrics. Edward Greenberg. Robert P. Applied Statistical Inference. Leonhard Held. Statistical Methods for Fuzzy Data.

Reinhard Viertl. Probability Theory. Nikolai Dokuchaev. The Fascination of Probability, Statistics and their Applications. Mark Podolskij. The Improbability Principle. David J.

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Vladimir Spokoiny, Thorsten Dickhaus. Basics of Modern Mathematical. Statistics. ∗. – Textbook –. April 26, Springer. Berlin Heidelberg. This textbook provides a unified and self-contained presentation of the main approaches to and ideas of mathematical statistics. It collects the basic.

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Probability, Statistics, and Stochastic Processes. Peter Olofsson. Understanding Demographic Transitions. Claude Diebolt. Bayesian Essentials with R. Christian P. The question of a rational choice of the significance level under given concrete conditions for example, in the development of rules for statistical quality control in mass production is very essential.

In this connection the desire to apply only rules with a very high close to 1 significance level faces the situation that for a restricted number of observations such rules only allow inferences with poor precision it may not be possible to establish the inequality of probabilities even given a noticeable inequality of the frequencies, etc.

The above-mentioned methods of parameter estimation and hypotheses testing are based on the assumption that the number of observations required to attain a given precision in the conclusions is determined in advance before carrying out the sampling. However, frequently an a priori determination of the number of observations is inconvenient, since by not fixing the number of trials in advance, but by determining it during the experiment, it is possible to decrease the expected number of trials.

This situation was first observed in the example of choosing between one of two hypotheses in a sequence of independent trials.

The corresponding procedure first proposed in connection with problems of statistical sampling is as follows: at each step decide, by the results of the observations already carried out, whether to a conduct the next trial, or b stop the trials and accept the first hypothesis, or c stop the trials and accept the second hypothesis.

With an appropriate choice of the quantitative characteristics such a procedure can secure with the same precision in the calculations a reduction in the average number of observations to almost half that of the fixed size sampling procedure see Sequential analysis. The development of the methods of sequential analysis led, on the one hand, to the study of controlled stochastic processes cf. Controlled stochastic process and, on the other, to the appearance of statistical decision theory.

This theory arises because the results of sequentially carrying out observations serve as a basis for the adoption of certain decisions intermediate — to continue the trial, and final — when the trials are stopped. In problems on parameter estimation the final decisions are numbers the values of the estimators , in problems on hypotheses testing they are the accepted hypothesis.

The aim of the theory is to give rules for the acceptance of decisions which minimise the mean loss or risk the risk depends on the probability distributions of the results of the observations, on the final decision, on the expense of conducting the trials, etc. Questions on the expedient distribution of effort in carrying out a statistical analysis of phenomena are considered in the theory of design of experiments, which plays a major part in modern mathematical statistics.

Side by side with the development and elaboration of the general ideas of mathematical statistics there have evolved various specialized branches such as dispersion analysis ; covariance analysis ; multi-dimensional statistical analysis ; the statistical analysis of stochastic processes ; and factor analysis. New considerations in regression analysis have appeared see also Stochastic approximation.

A major part in problems of mathematical statistics is played by the Bayesian approach to statistical problems. The first elements of mathematical statistics can already be found in the writings of the originators of probability theory — J. Bernoulli, P. Laplace and S. In Russia the methods of mathematical statistics in the application to demography and actuarial work were developed by V. Bunyakovskii Of key importance for all subsequent development of mathematical statistics was the work of the classical Russian school of probability theory in the second half of the 19th century and beginning of the 20th century P.

Chebyshev, A. Markov, A. Lyapunov, and S. Many questions of statistical estimation theory were essentially devised on the basis of the theory of errors and the method of least squares C. Gauss and Markov. The work of A. Galton and K.

Pearson has great significance, but in terms of utilizing the achievements of probability theory they lagged behind that of the Russian school. Pearson widely expanded the work on the formation of tables of functions necessary for applying the methods of mathematical statistics. Slutskii, N.

Smirnov and L. In the creation of small sample theory, the general theory of statistical estimation and hypotheses testing free of assumptions on the presence of a priori distributions , and sequential analysis, the role of the Anglo-American school Student, the pseudonym of W. Gosset, R. Fisher, Pearson, and J. Neyman , whose activity began in the 's, was very significant. Romanovskii, A.

Kolmogorov and Slutskii, to whom belongs important work on the statistics of dependent stationary series, Smirnov, who laid the foundations of the theory of non-parametric methods in statistics , and Yu.

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