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We present a self improvement to the Cauchy–Schwarz inequality, which in the probability case yields [ E ( X Y ) ] 2 ≤ E ( X 2 ) E ( Y 2 ) − ( | E ( X ) | Var ( Y ) − | E ( Y ) | Var ( X ) ) 2 . It is to be noted that the additional term to the inequality only involves the marginal first two moments for X and Y , and not any joint property. We also provide the discrete improvement to the inequality.

Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement beyond the unstructured Rosenthal inequality of quadratic forms when dimension of the vectors increases without bound. Applications to test the block diagonal structures and detect the sparsity in the high-dimensional covariance matrix are presented., Comment: 12 pages, 0 figure

A maximal moment inequality for partial sums of the α -mixing random variable sequence is established. The inequality uses some moment summations as upper bound. To show the applications of the inequality, we discuss the convergence for α -mixing sequences, which improves some known results.

A useful inequality involving correlations between three random variables with mean zero and finite second moments is presented. The inequality is applied to show that the entries on the main diagonal of the spectral density of a periodically correlated Markov process, as derived in Nematollahi and Soltani [2000. Discrete time periodically correlated Markov processes. Probab. Math. Statist. 20 (1) 127–140], are nonnegative. The inequality, when two variables with the same second moments are involved, is compared to the Cauchy–Schwartz inequality.

A well-known bound for the absolute value of the mean a function g ( X ) of a random variable X , | E ( g ( X ) ) | , is E ( g ( X ) ) 2 . Here, we use a sharper bound of Cauchy's inequality, proved by Hovenier (J. Math. Appl. 186 (1994) 156–160), to give upper bounds for the absolute of mean value, | E ( g ( X ) ) | . Some examples for particular distributions are also provided.

Under mild conditions we obtain the usual Berry–Esseen inequality for linear processes in Hilbert spaces.

We define an asymptotic mean version of Tukey's sensitivity curve. For this we show that some of the more important features of Hampel's influence function hold for L- and M-estimators also. Therefore, we have an alternative approach to the influence function using only expectations and limits.

The sharp inequality for squared skewness minus kurtosis is derived for the class of unimodal distributions.

An interesting result from the point of view of upper variance bounds is the inequality of Chernoff [Chernoff, H., 1981. A note on an inequality involving the normal distribution. Annals of Probability 9, 533–535]. Namely, that for every absolutely continuous function g with derivative g ′ such that Var { | g ( ξ ) | } ∞ , and for standard normal r.v. ξ , Var ( | g ( ξ ) | ) ≤ E { ( g ′ ( ξ ) ) 2 } . Both the usefulness and simplicity of this inequality have generated a great deal of extensions, as well as alternative proofs. Particularly, Olkin and Shepp [Olkin, I., Shepp, L., 2005. A matrix variance inequality. Journal of Statistical Planning and Inference 130, 351–358] obtained an inequality for the covariance matrix of k functions. However, all the previous papers have focused on univariate function and univariate random variable. We provide here a covariance matrix inequality for multivariate function of multivariate normal distribution.

Let Y k , n denote the n th k -record value (upper) of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F . We derive bounds for the expected values of Y k , n based on greatest convex minorants (Moriguti's method). We also present numerical comparisons between bounds obtained by Cauchy-Schwarz inequality and Moriguti's method.

A Bernstein-type inequality for non-degenerated U-statistics is presented. As the Bernstein inequality for sums of independent identically distributed random variables, in the limit, its tail has the same order as the tail of the limit. We also consider the case of U-processes indexed by a uniformly bounded VC subgraph class of functions. This is applied to obtain exponential inequalities for the local oscillations of the empirical distribution function of U-statistical structure.

In this note, we present an alternate way of obtaining the sharp bound for expected values of order statistics derived by Moriguti (1953) based on the principle of greatest convex minorants. The formula presented here using an application of binomial—negative binomial relationship, in addition to being simpler to use for numerical computations, also yields explicit bounds for a few extreme order statistics.

A simple method is presented for improving Hartley-David-Gumbel's universal bound for the mean of extreme order statistics.

Asymmetric sharp bounds are obtained for the expected values of order statistics using Moriguti's idea of the greatest convex minorant. It rectifies some results in the literature.

A refinement of the Cauchy-Schwarz inequality is suitably exploited to yield upper and lower bounds for the variance of a function of a continuous random variable.

The upper bounds for the variance of a function g of a random variable X obtained in Cacoullos (1982) (for short CP) are improved in the case μ = E ( X ) ≠ 0. A main feature of these bounds is that they involve the second moment of the derivative or the difference of g . A multivariate extension for functions of independent random variables is also given.