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Iosif Pinelis

Publications and source records attributed to Iosif Pinelis.

At least 37 records · Page 2Linked to original sources
arXiv2021

Monotonicity preservation properties of kernel regression estimators

Three common classes of kernel regression estimators are considered: the Nadaraya--Watson (NW) estimator, the Priestley--Chao (PC) estimator, and the Gasser--Müller (GM) estimator. It is shown that (i) the GM estimator has a certain monotonicity preservation property for any kernel $K$, (ii) the NW estimator has this property if and only the kernel $K$ is log concave, and (iii) the PC estimator does not have this property for any kernel $K$. Other related properties of these regression estimators are discussed.

math.ST↗
arXiv2021

Exact converses to a reverse AM--GM inequality, with applications to sums of independent random variables and (super)martingales

For every given real value of the ratio $μ:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $\mathsf{P}(X/G_X\ge v)$ and $\mathsf{P}(X/G_X\le v)$ are obtained, in terms of $μ$ and $v$. In particular, these bounds imply that $X/G_X\to1$ in probability as $A_X/G_X\downarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=\ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a reverse Jensen inequality for the strictly concave quadratic function $f(x) \equiv -x^2$. As applications of the mentioned new results, improvements of the Markov, Bernstein--Chernoff, sub-Gaussian, and Bennett--Hoeffding probability inequalities are given.

math.PR↗
arXiv2020

Monotonicity properties of the gamma family of distributions

For real $a>0$, let $X_a$ denote a random variable with the gamma distribution with parameters $a$ and $1$. Then $\mathsf P(X_a-a>c)$ is increasing in $a$ for each real $c\ge0$; non-increasing in $a$ for each real $c\le-1/3$; and non-monotonic in $a$ for each $c\in(-1/3,0)$. This extends and/or refines certain previously established results.

math.PR↗
arXiv2020

Exact lower and upper bounds on the incomplete gamma function

Lower and upper bounds $B_a(x)$ on the incomplete gamma function $Γ(a,x)$ are given for all real $a$ and all real $x>0$. These bounds $B_a(x)$ are exact in the sense that $B_a(x)\underset{x\downarrow0}\simΓ(a,x)$ and $B_a(x)\underset{x\to\infty}\simΓ(a,x)$. Moreover, the relative errors of these bounds are rather small for other values of $x$, away from $0$ and $\infty$.

math.CA↗
arXiv2019

An asymptotically optimal transform of Pearson's correlation statistic

It is shown that for any correlation-parametrized model of dependence and any given significance level $α\in(0,1)$, there is an asymptotically optimal transform of Pearson's correlation statistic $R$, for which the generally leading error term for the normal approximation vanishes for all values $ρ\in(-1,1)$ of the correlation coefficient. This general result is then applied to the bivariate normal (BVN) model of dependence and to what is referred to in this paper as the SquareV model. In the BVN model, Pearson's $R$ turns out to be asymptotically optimal for a rather unusual significance level $α\approx0.240$, whereas Fisher's transform $R_F$ of $R$ is asymptotically optimal for the limit significance level $α=0$. In the SquareV model, Pearson's $R$ is asymptotically optimal for a still rather high significance level $α\approx0.159$, whereas Fisher's transform $R_F$ of $R$ is not asymptotically optimal for any $α\in[0,1]$. Moreover, it is shown that in both the BVN model and the SquareV model, the transform optimal for a given value of $α$ is in fact asymptotically better than $R$ and $R_F$ in wide ranges of values of the significance level, including $α$ itself. Extensive computer simulations for the BVN and SquareV models of dependence are presented, which suggest that, for sample sizes $n\ge100$ and significance levels $α\in\{0.01,0.05\}$, the mentioned asymptotically optimal transform of $R$ generally outperforms both Pearson's $R$ and Fisher's transform $R_F$ of $R$, the latter appearing generally much inferior to both $R$ and the asymptotically optimal transform of $R$ in the SquareV model.

math.ST↗
arXiv2019

Generalized semimodularity: order statistics

A notion of generalized $n$-semimodularity is introduced, which extends that of (sub/super)mod\-ularity in four ways at once. The main result of this paper, stating that every generalized $(n\colon\!2)$-semimodular function on the $n$th Cartesian power of a distributive lattice is generalized $n$-semimodular, may be considered a multi/infinite-dimensional analogue of the well-known Muirhead lemma in the theory of Schur majorization. This result is also similar to a discretized version of the well-known theorem due to Lorentz, which latter was given only for additive-type functions. Illustrations of our main result are presented for counts of combinations of faces of a polytope; one-sided potentials; multiadditive forms, including multilinear ones -- in particular, permanents of rectangular matrices and elementary symmetric functions; and association inequalities for order statistics. Based on an extension of the FKG inequality due to Rinott \& Saks and Aharoni \& Keich, applications to correlation inequalities for order statistics are given as well.

math.PR↗
arXiv2019

A simple and more general approach to Stokes' theorem

Oftentimes, Stokes' theorem is derived by using, more or less explicitly, the invariance of the curl of the vector field with respect to translations and rotations. However, this invariance -- which is oftentimes described as the curl being a "physical" vector -- does not seem quite easy to verify, especially for undergraduate students. An even bigger problem with Stokes' theorem is to rigorously define such notions as ``the boundary curve remains to the left of the surface''. Here an apparently simpler and more general approach is suggested.

math.HO↗
arXiv2018

A simpler $\mathbb R^3$ realization of the Möbius strip

A very simple $\mathbb R^3$ realization of the Möbius strip, significantly simpler than the common one, is given. For any, however large width/length ratio of the strip, it is shown that this realization, in contrast with the common one, is the union of a vertical segment and the graph of a simple rational function on a simple polynomially defined subset of $\mathbb R^2$.

math.DG↗
arXiv2018

The exp-normal distribution is infinitely divisible

Let $Z$ be a standard normal random variable (r.v.). It is shown that the distribution of the r.v. $\ln|Z|$ is infinitely divisible; equivalently, the standard normal distribution considered as the distribution on the multiplicative group over $\mathbb{R}\setminus\{0\}$ is infinitely divisible.

math.PR↗
arXiv2018

Exact upper and lower bounds on the misclassification probability

Exact lower and upper bounds on the best possible misclassification probability for a finite number of classes are obtained in terms of the total variation norms of the differences between the sub-distributions over the classes. These bounds are compared with the exact bounds in terms of the conditional entropy obtained by Feder and Merhav.

math.ST↗