Calculation of definite integrals by the fundamental formula of integral calculus, 

$$\int_{a}^{b}{f\left( x \right)dx\,=\,F\left( b \right)\,-\,F\left( a \right)},$$

where f(x), let us say, is a continuous function on [a, b] and F(x) is its primitive function, is made difficult by the fact that the actual determination of F(x) is possible only in rare cases. For this reason, formulas for the approximate calculation of integrals are of great significance. In this chapter, we shall become acquainted with the most important of them. Many formulas for the approximate calculation of definite integrals have the form 

$$\int_{a}^{b}{p\left( x \right)f\left( x \right)dx\,\cong \,\sum\limits_{k\,=\,1}^{n}{A_{k}^{\left( n \right)}f\left( x_{k}^{\left( n \right)} \right)}}$$

(1.1)

and are called mechanical quadrature formulas. The sum on the right hand side of (1.1) is called the quadrature sum. The numbers x k (n) belong to the interval [a, b], and are called the knots of the quadrature formula, and the numbers A k (n) are the coefficients of the quadrature formula. We shall always consider the knots of the quadrature formula to be numbered in increasing order: 

$$x_{1}^{\left( n \right)}\,<\,x_{2}^{\left( n \right)}\,<\,\ldots \,<\,x_{n}^{\left( n \right)}.$$

The interval of integration [a, b] can also be infinite. The integrand is written in the form of the product of the two functions p(x) and f(x). The first of these, p(x), is assumed to be fixed for the given formula (1.1) and is called the weight function. The function f(x) belongs to some sufficiently wide class of functions, for example, continuous and such that the integral on the left hand side of (1.1) exists.

The conditional risk approach to selection and ranking problems

Approximate Calculation of Integrals

A multivariate linear model with latent factor structure

Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.

Linear statistical inference and its applications

(1955). A Multiple Comparison Procedure for Comparing Several Treatments with a Control. Journal of the American Statistical Association: Vol. 50, No. 272, pp. 1096-1121.

https://www.jstor.org/stable/pdfplus/2281208.pdf

background or methods

A Multiple Comparison Procedure for Comparing Several Treatments with a Control

Principles of econometrics

The foundations of a general theory of statistical decision functions, including the classical non-sequential case as well as the sequential case, was discussed by the author in a previous publication [3]. Several assumptions made in [3] appear, however, to be unnecessarily restrictive (see conditions 1-7, pp. 297 in [3]). These assumptions, moreover, are not always fulfilled for statistical problems in their conventional form. In this paper the main results of [3], as well as several new results, are obtained from a considerably weaker set of conditions which are fulfilled for most of the statistical problems treated in the literature. It seemed necessary to abandon most of the methods of proofs used in [3] (particularly those in section 4 of [3]) and to develop the theory from the beginning. To make the present paper self-contained, the basic definitions already given in [3] are briefly restated in section 2.1.

https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730030

Statistical decision functions

In this text about statistical mathematical theory, Harald Cramer joins two major lines of development in the field: while British and American statisticians were developing the science of statistical inference, French and Russian probablists transformed the classical calculus of probability into a rigorous and purely mathematical theory. The result of Cramer's work is an exposition of the mathematical methods of modern statistics that set the standard that others have since sought to follow. The first part of the book is an introduction to the fundamental concept of a distribution and of integration with respect to a distribution. The second part contains the general theory of random variables and probability distributions while the third is devoted to the theory of sampling statistical estimation and tests of significance.

The conditional risk approach to selection and ranking problems

Citations

Approximate Calculation of Integrals

A multivariate linear model with latent factor structure

References

Linear statistical inference and its applications

A Multiple Comparison Procedure for Comparing Several Treatments with a Control

Principles of econometrics

Statistical decision functions

Mathematical methods of statistics

Related Papers (5)

Compromise Ranking Approach with Bootstrap Confidence Intervals for Risk Assessment in Port Management Projects

Value-at-Risk vs. Conditional Value-at-Risk in Risk Management and Optimization

Risk Analysis: Assessing Uncertainties Beyond Expected Values and Probabilities

Foundations of risk analysis : a knowledge and decision-oriented perspective

Risk quantification : management, diagnosis and hedging