On Generalizations of Tchebychef's Inequality

T HIS PAPER iS purely expository and contains an account of inequalities of the type of Tchebychef's, i.e., inequalities for the value of a distribution function in terms of known facts about the distribution. Such facts may be numerical, e.g., moments or range, or geometrical, e.g., the property of being unimodal or monotonic in some given range. The distributions to which they apply may be classified as singlevariate, distributions of averages, or multivariate distributions. On pages 924-5 there is given a table of the results quoted, showing what data each uses and the type of distribution to which it applies. At the end of the paper are some remarks on possible future developments and on applications.

[1]  ON GENERALISED TCHEBYCHEFF THEOREMS IN THE MATHEMATICAL THEORY OF STATISTICS , 1919 .

[2]  S. Narumi,et al.  ON FURTHER INEQUALITIES WITH POSSIBLE APPLICATION TO PROBLEMS IN THE THEORY OF PROBABILITY , 1923 .

[3]  Birger Meidell.  Une contribution aux lois de fréquences et les erreurs moyennes , 1923 .

[4]  A. Kolmogoroff.  Über die Summen durch den Zufall bestimmter unabhängiger Größen , 1928 .

[5]  A. Kolmogoroff.  Bemerkungen zu meiner Arbeit “Über die Summen zufälliger Größen” , 1930 .

[6]  Clarence DeWitt Smith,et al.  On Generalized Tchebycheff Inequalities in Mathematical Statistics , 2022 .

[7]  C. Craig.  On the Tchebychef Inequality of Bernstein , 1933 .

[8]  Sur l'inégalité fondamentale du calcul des probabilités , 1937 .

[9]  P. Berge.  A Note on a Form of Tohebycheff's Theorem for Two Variables , 1938 .

[10]  A. Wald.  Limits of a distribution function determined by absolute moments and inequalities satisfied by absolute moments , 1939 .

[11]  R. V. Mises.  The Limits of a Distribution Function if Two Expected Values are Given , 1939 .

[12]  C. D. Smith.  On Tchebycheff Approximation for Decreasing Functions , 1939 .

[13]  Henrik L. Selberg.  Über eine Ungleichung der mathematischen Statistik , 1940 .

[14]  A. C. Berry.  The accuracy of the Gaussian approximation to the sum of independent variates , 1941 .

[15]  C. Leser.  INEQUALITIES FOR MULTIVARIATE FREQUENCY DISTRIBUTIONS , 1942 .

[16]  J. Shohat,et al.  The problem of moments , 1943 .

[17]  P. Hsu.  The Approximate Distributions of the Mean and Variance of a Sample of Independent Variables , 1945 .

[18]  A. Offord.  An Inequality for Sums of Independent Random Variables , 1945 .

[19]  C. Winsten.  INEQUALITIES IN TERMS OF MEAN RANGE , 1946 .

[20]  Z. W. Birnbaum,et al.  A Generalization of Tshebyshev's Inequality to Two Dimensions , 1947 .

[21]  L. Guttman.  An Inequality for Kurtosis , 1948 .

[22]  Z. Birnbaum.  On Random Variables with Comparable Peakedness , 1948 .

[23]  Louis Guttman,et al.  A Distribution-Free Confidence Interval for the Mean , 1948 .

[24]  Generalization to $N$ Dimensions of Inequalities of the Tchebycheff Type , 1948 .

[25]  H. Bergström,et al.  On the central limit theorem in the case of not equally distributed random variables , 1949 .

[26]  Une nouvelle généralisation de l'inégalité de Bienaymé (Extrait d'une lettre à M. M. Fréchet) , 1951 .

[27]  I. Richard Savage,et al.  Bibliography of Nonparametric Statistics and Related Topics , 1952 .

[28]  H. Royden,et al.  Bounds on a Distribution Function when its First $n$ Moments are Given , 1953 .

[29]  Salem H. Khamis,et al.  On the Reduced Moment Problem , 1954 .

[30]  M. Zelen,et al.  Bounds on a distribution function that are functions of moments to order four , 1954 .