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Bounds on variance for symmetric unimodal distributions

Hye Won Chung, Brian M. Sadler, Alfred O. Hero
2015 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)  
Combining this bound with the well-known entropy power lower bound on variance, we prove that for the general class of symmetric unimodal distributions the variance can be bounded below and above by the  ...  We show a direct relationship between the variance and the differential entropy for the general class of symmetric unimodal distributions by providing an upper bound on variance in terms of entropy power  ...  UPPER BOUND ON THE VARIANCE OF SYMMETRIC UNIMODAL DISTRIBUTIONS WITH ENTROPY In this section, we exhibit two classes of symmetric unimodal densities whose variance can be bounded above by its entropy power  ... 
doi:10.1109/allerton.2015.7447149 dblp:conf/allerton/ChungSH15 fatcat:cbf5d66m25fm7hxwcygswvwiui

Bounds on Variance for Unimodal Distributions

Hye Won Chung, Brian M. Sadler, Alfred O. Hero
2017 IEEE Transactions on Information Theory  
We show a direct relationship between the variance and the differential entropy for subclasses of symmetric and asymmetric unimodal distributions by providing an upper bound on variance in terms of entropy  ...  Combining this bound with the well-known entropy power lower bound on variance, we prove that the variance of the appropriate subclasses of unimodal distributions can be bounded below and above by the  ...  UPPER BOUND ON THE VARIANCE OF SYMMETRIC UNIMODAL MIXTURE DENSITIES In Section III-A, we consider generalized Gaussian distributions and show that for this class of distributions variance and entropy power  ... 
doi:10.1109/tit.2017.2749310 fatcat:iodd5uhw7rc4xgg336x5hzhfvu

ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz

Eric Clarkson, J. L. Denny, Larry Shepp
2009 The Annals of Applied Probability  
For independent X and Y in the inequality P(X≤ Y+μ), we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite  ...  The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz.  ...  If X has a symmetric unimodal distribution supported on where the F i are distribution functions having boxcar densities and 0 < β < 1, and the variance of (3) and of (4) equals 1.  ... 
doi:10.1214/08-aap536 pmid:20191100 pmcid:PMC2828638 fatcat:jtramjcegrddlagqzcnnpnsowy

Possibility transformation of the sum of two symmetric unimodal independent/comonotone random variables

Gilles Mauris
2013 Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology  
The paper extends author's previous works on a probability/possibility transformation based on a maximum specificity principle to the case of the sum of two identical unimodal symmetric random variables  ...  It often leads to the worst case in terms of specificity of the corresponding possibility distribution, but it may arise that the independent case is worse than the comonotone case, e.g. for symmetric  ...  Indeed, the uniform distribution is the least specific one among the symmetric unimodal distributions with bounded support [11] .  ... 
doi:10.2991/eusflat.2013.93 dblp:conf/eusflat/Mauris13 fatcat:kkmdgpj5cngbvfqvtjvqqoorra

Covariance bounds for multivariate unimodal distributions and a characterization of uniformity

L Mattner
1991 Journal of Multivariate Analysis  
Thanks go to the referee for several thoughtful remarks, in particular for suggesting a treatment of central convex unimodal distributions.  ...  ACKNOWLEDGMENTS An earlier version of this paper was based on a part of the author's dissertation written at the Fachbereich Mathematik der Universitlt Hannover.  ...  Key words and phrases: characterization of multivariate uniformity, covariance bounds, generalized variance, mean variance, multivariate unimodality.  ... 
doi:10.1016/0047-259x(91)90040-9 fatcat:hgo5hfoxargnvilzverletdlfq

Page 931 of Mathematical Reviews Vol. 40, Issue 4 [page]

1970 Mathematical Reviews  
It is shown in this paper that the requirement of unimodality restricts the variance to (b—a)?/9, and that this is a least upper bound. Note that this bound exceeds the variance (b—a)?  ...  Merely knowing that a distribution is restricted to (a, 6] serves to bound its variance by }(b—a)?  ... 

The variance upper bound for a mixed random variable

Martín Egozcue, Luis Fuentes García
2017 Communications in Statistics - Theory and Methods  
In this note, we derive upper bounds on the variance of a mixed random variable. Our results are an extension of previous results for unimodal and symmetric random variables.  ...  The novelty of our findings is that this mixed random variable does not necessary need to be symmetric and is multimodal. We also characterize the cases when these bounds are optimal.  ...  Corollary 1 Let X i be unimodal random variables with the same variance σ 2 and symmetric on [a i , b i ] for i = 1, 2, ..., n. Let Z be a random variable mixture of X 1 , . . . , X n .  ... 
doi:10.1080/03610926.2017.1390136 fatcat:ckhaisucefalhjsdxtr3k3pnfu

Optimal Asymmetric Binary Quantization for Estimation Under Symmetrically Distributed Noise [article]

Rodrigo Cabral Farias, Eric Moisan, Jean-Marc Brossier
2013 arXiv   pre-print
We study the behavior of the Cramer-Rao bound as a function of the quantizer threshold for different symmetric unimodal noise distributions.  ...  Estimation of a location parameter based on noisy and binary quantized measurements is considered in this letter.  ...  ACKNOWLEDGMENT The authors would like to thank Steeve Zozor for his helpful comments.  ... 
arXiv:1310.6938v1 fatcat:uowizdsg6jhgpekqxgu3k6zjgy

Page 5196 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
Statist. 4 (1976), no. 3, 607-613; MR 54 #3799] whether symmetric, star and linear unimodality necessarily implies monotone unimodality.  ...  Seaman, John (1-WACO); Odell, Pat (1-TXD) 86k:60030 On Goldstein’s variance bound. Adv. in Appl. Probab. 17 (1985), no. 3, 679-681. Author summary: “M. Goldstein [J. Appl.  ... 

A note on adaptation in garch models

Gloria González-Rivera
1997 Econometric Reviews  
Among unimodal densities, I have searched for instances in which, even though (Cl) and (C2) NOTE ON ADAPTATION IN GARCH MODELS financial literature are: TABLE I I Symmetric Unimodal Densities variance  ...  If the error probability density function is fully known, the maximum likelihood estimator will have an asymptotic variance which achieves a lower bound, the Cramer-Rao bound, for all regular estimators  ... 
doi:10.1080/07474939708800372 fatcat:67mpnz3qorae5g3dcvlhu4lthq

Page 7349 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews  
even in the case of unimodal discrete distributions; (ii) the assumption of unimodality introduces, in general, smaller bounds on the variance of the distribution.  ...  with applications to variance upper bounds.  ... 

Service-oriented performance of inventory models with partial information on unimodal demand lead-time distributions

Gerrit K. Janssens, Lotte Verdonck, Katrien Ramaekers
2020 Reserche operationelle  
The knowledge of the demand distribution during lead-time serves to determine the safety inventory level. Many times the distribution is not fully known, except maybe for its range, mean or variance.  ...  This research develops a technique to derive the safety stock for unimodal demand distributions of which the mode either is known or can be estimated.  ...  The convolution of two symmetric unimodal distributions on is symmetric and unimodal.  ... 
doi:10.1051/ro/2020026 fatcat:h4mnhcnxm5fchgoic3wfmt5kxu

Optimal-Partitioning Inequalities in Classification and Multi-Hypotheses Testing

Theodore P. Hill, Y. L. Tong
1989 Annals of Statistics  
Best possible bounds are derived for the minimax risk for location parameter families, based on the tail concentrations and Levy concentrations of the distributions.  ...  Best possible bounds are derived for the minimax risk for location parameter families, based on the tail concentrations and Levy concentrations of the distributions.  ...  The authors are grateful to John Elton for a suggestion concerning the proof of Theorem 3.2 and to the Associate Editor and the referee for a number of suggestions.  ... 
doi:10.1214/aos/1176347272 fatcat:fkx246nesncunmugqzngic7y6q

Page 4675 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
One of the most interesting problems in the theory of unimodal probability distributions is to determine conditions under which the convolution * of a unimodal distribution yu with itself is also unimodal  ...  unimodal distributions.  ... 

Stochastic Model Predictive Control for Linear Systems Using Probabilistic Reachable Sets

Lukas Hewing, Melanie N. Zeilinger
2018 2018 IEEE Conference on Decision and Control (CDC)  
under a unimodality assumption on the disturbance distribution.  ...  A control scheme reverting to a backup solution from a previous time step in case of infeasibility is proposed, for which an asymptotic average performance bound is derived.  ...  Under a unimodality assumption on the disturbance distribution and for symmetric PRS, the method is shown to also guarantee chance constraint satisfaction in a stronger sense, termed closed-loop satisfaction  ... 
doi:10.1109/cdc.2018.8619554 dblp:conf/cdc/HewingZ18 fatcat:b6agji6znvghjkkms6vdr7anpi
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