Log‐concavity of multivariate distributions is an important concept in general and has a very special place in the field of Reliability Theory. An attempt has been made in this paper to study ...preservation results for (i) the discrete version of multivariate log‐concavity for multistate series and multistate parallel systems consisting of n$$ n $$ independent components, states of both components and systems being represented by elements in a subset S2={0,1,2}$$ {S}_2=\left\{0,1,2\right\} $$ of SM={0,1,2,…,M},$$ {S}_M=\left\{0,1,2,\dots, M\right\}, $$ and (ii) the continuous version of multivariate log‐concavity under multistate series and multistate parallel systems made up of n$$ n $$ independent components and states of both, systems and components, taking values in the set SM$$ {S}_M $$. These results for discrete and continuous versions of log‐concavity have also been extended to systems that are formed using both multistate series and multistate‐parallel systems. Further, the results in (ii) have been used to obtain important and useful bounds on joint probabilities related to times spent by multistate components, multistate series, multistate parallel systems, and the combinations thereof.
In this paper, we provide a new proof of the preservation of the log concavity by Bernstein operator. It is based on the bivariate characterization of the likelihood ratio order. In addition, we give ...new conditions under which the previous property leads to preservation of ageing properties in coherent systems with independent and identically distributed components.
In recent years, log-concave density estimation via maximum likelihood estimation has emerged as a fascinating alternative to traditional nonparametric smoothing techniques, such as kernel density ...estimation, which require the choice of one or more bandwidths. The purpose of this article is to describe some of the properties of the class of log-concave densities on ℝ𝑑 which make it so attractive from a statistical perspective, and to outline the latest methodological, theoretical and computational advances in the area.
Given a sequence
α
=
(
a
k
)
k
≥
0
of nonnegative numbers, define a new sequence
L
(
α
)
=
(
b
k
)
k
≥
0
by
b
k
=
a
k
2
-
a
k
-
1
a
k
+
1
. The sequence
α
is called
r
-
log-concave
if
L
i
(
α
)
=
L
(
...L
i
-
1
(
α
)
)
is a nonnegative sequence for all
1
≤
i
≤
r
. In this paper, we study the
r
-log-concavity and its
q
-analogue for
r
=
2
,
3
using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann
ξ
-function. We give some criteria for
r
-
q
-log-concavity for
r
=
2
,
3
. As applications, we get 3-
q
-log-concavity of
q
-binomial coefficients and different
q
-Stirling numbers of two kinds, which extends results for
q
-log-concavity. We also present some results for
r
-
q
-log-concavity from the linear transformations. Finally, we pose an interesting question.
We study nonparametric maximum likelihood estimation for two classes of multivariate distributions that imply strong forms of positive dependence; namely log‐supermodular (MTP2) distributions and ...log‐L♮‐concave (LLC) distributions. In both cases we also assume log‐concavity in order to ensure boundedness of the likelihood function. Given n independent and identically distributed random vectors from one of our distributions, the maximum likelihood estimator (MLE) exists a.s. and is unique a.e. with probability one when n≥3. This holds independently of the ambient dimension d. We conjecture that the MLE is always the exponential of a tent function. We prove this result for samples in {0,1}d or in ℝ2 under MTP2, and for samples in ℚd under LLC. Finally, we provide a conditional gradient algorithm for computing the maximum likelihood estimate.
We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ...ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a single cycle in the first permutation and gluing two cycles in the second permutation, while all the other cycles are left untouched.
In the analysis of real integer‐valued time series data, we often encounter negative values and negative correlations. For integer‐valued autoregressive time series, there are many parametric models ...to choose from, but some of them are relatively complex. With little information about the background of real data, we hope that a simple and effective semiparametric model can be used to obtain more information that usually cannot be provided by parametric models, such as the confidence interval of the innovation distribution. But the only existing semiparametric model based on thinning operators can only deal with non‐negative data with positive correlation coefficients. In addition, it has two drawbacks: first, an initial distribution of the innovation is required, but different initial values may lead to different results; second, the confidence interval of the innovation distribution is not available, which is essential in low‐valued data. To overcome these drawbacks, we propose a rounded semiparametric autoregressive model with a log‐concave innovation, which can deal with ℤ‐valued time series with autoregressive coefficients of arbitrary sign. The consistencies of the estimators for the parametric and nonparametric parts of the model are also discussed. We illustrate the superior performance of the proposed model based on three real datasets.
Résumé
Dans l'analyse de données réelles provenant de séries chronologiques entières, les valeurs et les corrélations négatives sont fréquentes. Pour les séries chronologiques entières autorégressives, plusieurs modèles paramétriques sont disponibles, mais certains s'avèrent plutôt complexes. Avec peu d'information sur la provenance de données réelles, on pourrait espérer qu'un modèle semi‐paramétrique simple et efficace puisse servir à obtenir l'information que les modèles paramétriques ne peuvent pas exprimer, notamment un intervalle de confiance pour la distribution des innovations. Le seul modèle semi‐paramétrique qui existe utilise des opérateurs d'éclaircissement et il convient seulement à des données non négatives et à des coefficients de corrélation positifs. De plus, il comporte deux inconvénients : il nécessite une distribution initiale pour les innovations, et il ne permet pas d'obtenir d'intervalles de confiance pour la distribution des innovations, alors que ces informations sont essentielles pour des données de faibles valeurs. Afin de surmonter ces inconvénients, les auteurs proposent un modèle autorégressif semi‐paramétrique arrondi avec des innovations log‐concaves permettant de gérer des séries chronologiques prenant leurs valeurs dans ℤ avec des coefficients autorégressifs de signes arbitraires. Les auteurs discutent de la convergence des estimateurs pour les parties paramétrique et semi‐paramétrique du modèle. Ils illustrent la performance supérieure du modèle proposé avec trois jeux de données réelles.
We study sharp frame bounds of Gabor frames for integer redundancy with the standard Gaussian window and prove that the square lattice optimizes both the lower and the upper frame bound among all ...rectangular lattices. This proves a conjecture of Floch, Alard & Berrou (as reformulated by Strohmer & Beaver). The proof is based on refined log-convexity/concavity estimates for the Jacobi theta functions θ3 and θ4.
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