scholarly journals Quantile Jensen’s inequalities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mu Zhao ◽  
Xinai Yang ◽  
Qi He ◽  
Zunrong Zhou ◽  
Xiangyu Ge
Keyword(s):  
Confidence Interval ◽  
Jensen’S Inequality ◽  
Random Variable ◽  
Pivotal Quantity ◽  
Jensen's Inequality ◽  
Rigorous Description ◽  
The Relationship ◽  
Special Case

AbstractQuantiles of random variable are crucial quantities that give more delicate information about distribution than mean and median and so on. We establish Jensen’s inequality for q-quantile ($q\geq 0.5$ q ≥ 0.5 ) of a random variable, which includes as a special case Merkle (Stat. Probab. Lett. 71(3):277–281, 2005) where Jensen’s inequality about median (i.e. $q= 0.5$ q = 0.5 ) was given. We also refine this inequality in the case where $q<0.5$ q < 0.5 . An application to the confidence interval of parameters in pivotal quantity is also considered by virtue of the rigorous description on the relationship between quantiles and intervals that have required probability.

On Some Integral Inequalities Related to Hardy's

1977 ◽  
Vol 20 (3) ◽  
pp. 307-312 ◽  
Author(s):  
Christopher Olutunde Imoru
Keyword(s):  
Integral Inequality ◽  
Convex Functions ◽  
Jensen’S Inequality ◽  
Integral Inequalities ◽  
Hardy’S Inequality ◽  
Jensen's Inequality ◽  
Hardy's Inequality ◽  
Special Case

AbstractWe obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.

On measures of non-degeneracy

1992 ◽  
Vol 29 (3) ◽  
pp. 733-739
Author(s):  
K. B. Athreya
Keyword(s):  
Convex Function ◽  
Branching Processes ◽  
Jensen’S Inequality ◽  
Random Variable ◽  
Jensen's Inequality ◽  
Unique Role

If φ is a convex function and X a random variable then (by Jensen's inequality) ψ φ (X) = Eφ (X) – φ (EX) is non-negative and 0 iff either φ is linear in the range of X or X is degenerate. So if φ is not linear then ψ φ (X) is a measure of non-degeneracy of the random variable X. For φ (x) = x2, ψ φ (X) is simply the variance V(X) which is additive in the sense that V(X + Y) = V(X) + V(Y) if X and Y are uncorrelated. In this note it is shown that if φ ″(·) is monotone non-increasing then ψ φ is sub-additive for all (X, Y) such that EX ≧ 0, P(Y ≧ 0) = 1 and E(X | Y) = EX w.p.l, and is additive essentially only if φ is quadratic. Thus, it confirms the unique role of variance as a measure of non-degeneracy. An application to branching processes is also given.

scholarly journals Novel Fractional Dynamic Hardy–Hilbert-Type Inequalities on Time Scales with Applications

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2964
Author(s):  
Ahmed A. El-Deeb ◽  
Jan Awrejcewicz
Keyword(s):  
Present Article ◽  
Time Scales ◽  
Jensen’S Inequality ◽  
Legendre Transform ◽  
Jensen's Inequality ◽  
Discrete Inequalities ◽  
Special Case ◽  
Hilbert Type

The main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert type on time scales. We present and prove very important generalized results with the help of Fenchel–Legendre transform, submultiplicative functions. We prove the (γ,a)-nabla conformable Hölder’s and Jensen’s inequality on time scales. We prove several inequalities due to Hardy–Hilbert inequalities on time scales. Furthermore, we introduce the continuous inequalities and discrete inequalities as special case.

scholarly journals Some Dynamic Inequalities of Hilbert’s Type

2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
A. M. Ahmed ◽  
Ghada AlNemer ◽  
M. Zakarya ◽  
H. M. Rezk
Keyword(s):  
Time Scales ◽  
Hölder’S Inequality ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Hölder's Inequality ◽  
Discrete Inequalities ◽  
Special Case ◽  
Hilbert Type

This paper is concerned with deriving some new dynamic Hilbert-type inequalities on time scales. The basic idea in proving the results is using some algebraic inequalities, Hölder’s inequality and Jensen’s inequality, on time scales. As a special case of our results, we will obtain some integrals and their corresponding discrete inequalities of Hilbert’s type.

On measures of non-degeneracy

1992 ◽  
Vol 29 (03) ◽  
pp. 733-739
Author(s):  
K. B. Athreya
Keyword(s):  
Convex Function ◽  
Branching Processes ◽  
Jensen’S Inequality ◽  
Random Variable ◽  
Jensen's Inequality ◽  
Unique Role

If φ is a convex function and X a random variable then (by Jensen's inequality) ψ φ (X) = Eφ (X) – φ (EX) is non-negative and 0 iff either φ is linear in the range of X or X is degenerate. So if φ is not linear then ψ φ (X) is a measure of non-degeneracy of the random variable X. For φ (x) = x 2, ψ φ (X) is simply the variance V(X) which is additive in the sense that V(X + Y) = V(X) + V(Y) if X and Y are uncorrelated. In this note it is shown that if φ ″(·) is monotone non-increasing then ψ φ is sub-additive for all (X, Y) such that EX ≧ 0, P(Y ≧ 0) = 1 and E(X | Y) = EX w.p.l, and is additive essentially only if φ is quadratic. Thus, it confirms the unique role of variance as a measure of non-degeneracy. An application to branching processes is also given.

scholarly journals A new form of Jensen's inequality and its application to statistical experiments

1995 ◽  
Vol 36 (4) ◽  
pp. 389-398
Author(s):  
D. J. F. Nonnenmacher ◽  
R. Zagst
Keyword(s):  
Jensen’S Inequality ◽  
Random Variable ◽  
Probability Measures ◽  
Jensen's Inequality ◽  
New Production ◽  
Statistical Experiments ◽  
Expected Information ◽  
Monotonicity Result ◽  
Production Technologies ◽  
Blackwell Sufficiency

AbstractJensen's inequality for the expectation of a convex function of a random variable is proved for a wide class of convex functions defined on a space of probability measures. The result is applied to statistical experiments using the concept of Blackwell-sufficiency. In particular, we show a monotonicity result for the expected information of Poisson-experiments. As an application to economics we consider the introduction of new production technologies.

scholarly journals Convexity, Jensen's inequality and benefits of noisy mechanical ventilation

2005 ◽  
Vol 2 (4) ◽  
pp. 393-396 ◽  
Author(s):  
John F Brewster ◽  
M. Ruth Graham ◽  
W. Alan C Mutch
Keyword(s):  
Airway Pressure ◽  
Life Support ◽  
Jensen’S Inequality ◽  
Local Convexity ◽  
Mechanical Ventilators ◽  
Jensen's Inequality ◽  
Probabilistic Argument ◽  
Mean Pressure ◽  
The Relationship ◽  
Support Devices

Mechanical ventilators breathe for you when you cannot or when your lungs are too sick to do their job. Most ventilators monotonously deliver the same-sized breaths, like clockwork; however, healthy people do not breathe this way. This has led to the development of a biologically variable ventilator—one that incorporates noise. There are indications that such a noisy ventilator may be beneficial for patients with very sick lungs. In this paper we use a probabilistic argument, based on Jensen's inequality, to identify the circumstances in which the addition of noise may be beneficial and, equally important, the circumstances in which it may not be beneficial. Using the local convexity of the relationship between airway pressure and tidal volume in the lung, we show that the addition of noise at low volume or low pressure results in higher mean volume (at the same mean pressure) or lower mean pressure (at the same mean volume). The consequence is enhanced gas exchange or less stress on the lungs, both clinically desirable. The argument has implications for other life support devices, such as cardiopulmonary bypass pumps. This paper illustrates the benefits of research that takes place at the interface between mathematics and medicine.

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