scholarly journals Best constants for tensor products of Bernstein, Szász and Baskakov operators

2000 ◽  
Vol 62 (2) ◽  
pp. 211-220 ◽  
Author(s):  
Jesús de la Cal ◽  
Ana M. Valle
Keyword(s):  
Tensor Product ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Best Constants ◽  
Moduli Of Continuity ◽  
Lower And Upper Bounds ◽  
Baskakov Operators ◽  
Binomial Distributions ◽  
Families Of Operators ◽  
Crucial Ingredient

We consider tensor product operators and discuss their best constants in preservation inequalities concerning the usual moduli of continuity. In a previous paper, we obtained lower and upper bounds on such constants, under fairly general assumptions on the operators. Here, we concentrate on the l∞-modulus of continuity and three celebrated families of operators. For the tensor product of k identical copies of the Bernstein operator Bn, we show that the best uniform constant coincides with the dimension k when k ≥ 3, while, in case k = 2, it lies in the interval [2, 5/2] but depends upon n. Similar results also hold when Bn is replaced by a univariate Szász or Baskakov operator. The three proofs follow the same pattern, a crucial ingredient being some special properties of the probability distributions involved in the mentioned operators, namely: the binomial, Poisson, and negative binomial distributions.

Recent results on characterization of probability distributions: a unified approach through extensions of Deny&s theorem

1986 ◽  
Vol 18 (03) ◽  
pp. 660-678 ◽  
Author(s):  
C. Radhakrishna Rao ◽  
D. N. Shanbhag
Keyword(s):  
Negative Binomial ◽  
Short Proof ◽  
Probability Distributions ◽  
Direct Link ◽  
Unified Approach ◽  
Binomial Distributions ◽  
Characterization Of Probability Distributions ◽  
Special Cases ◽  
Convolution Equations

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

Recent results on characterization of probability distributions: a unified approach through extensions of Deny&s theorem

1986 ◽  
Vol 18 (3) ◽  
pp. 660-678 ◽  
Author(s):  
C. Radhakrishna Rao ◽  
D. N. Shanbhag
Keyword(s):  
Negative Binomial ◽  
Short Proof ◽  
Probability Distributions ◽  
Direct Link ◽  
Unified Approach ◽  
Binomial Distributions ◽  
Characterization Of Probability Distributions ◽  
Special Cases ◽  
Convolution Equations

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

scholarly journals Bounds for some entropies and special functions

2018 ◽  
Vol 34 (1) ◽  
pp. 09-15
Author(s):  
ADINA BARAR ◽  
◽  
GABRIELA RALUCA MOCANU ◽  
IOAN RASA ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Convex Function ◽  
Negative Binomial ◽  
Special Functions ◽  
Probability Distributions ◽  
Real Parameter ◽  
Binomial Distributions

We consider a family of probability distributions depending on a real parameter and including the binomial, Poisson and negative binomial distributions. The corresponding index of coincidence satisfies a Heun differential equation and is a logarithmically convex function. Combining these facts we get bounds for the index of coincidence, and consequently for Renyi and Tsallis entropies of order 2.

ARRIVAL PATTERNS AND TRAFFIC FLOW CHARACTERISTICS AT SIGNALIZED INTERSECTIONS

2021 ◽  
Vol 15 (1) ◽  
pp. 11-28
Author(s):  
Olanrewaju Oluwafemi Akinfala ◽  
Emmanuel Enyeribe Ege ◽  
Ladi Folorunso Ogunwolu
Keyword(s):  
Time Resolution ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Model Misspecification ◽  
Flow Characteristics ◽  
P Value ◽  
High Time Resolution ◽  
P Values ◽  
Binomial Distributions ◽  
Density Analysis

Traffic arrivals at signal intersection approaches is inherently stochastic. This variability is typically reflected by I-ratio and there is a general consensus that the presence or absence of nearby upstream signal affects Variance to Mean Ratio (I-ratio). However, the effect of time resolution on arrival variability and the interaction effect between upstream signal and time resolution is yet to be examined in detail. This can lead to model misspecification and invariably, erroneous outcomes. This work examines the effect of time resolution and intersection type and their interaction on I-ratio and the resultant probability distributions. Traffic arrivals were measured at high time resolution- 10 seconds interval and then aggregated to lower time resolutions (30-150 seconds) at six intersections. Spectral density analysis showed statistically significant periodicity, specifically at 30 seconds interval with p-values < 0.0001 at all connected intersections while observations at isolated intersections lacked periodicity. Two-way ANOVA using I-ratio as the dependent variable and intersection type and time-resolution as the independent variables was performed. Statistically significant effect with F-value 8.606 at p-value < 0.0001 and R2 value 0.32 were observed. Intersection type, time resolution and the interaction between them were statistically significant, with p-values 0.002, < 0.0001 and 0.000 respectively. The combined effect of these factors led to a wide I-ratio range of 0.37-9.2. Negative Binomial, Poisson, and Binomial distributions represented 76.4, 20.4 and 4.2% of all I-ratios observed. Therefore, in contrast to literature which recommends Poisson, Negative Binomial may be a better suited probability distribution for traffic arrivals.

On characterization of certain probability distributions

1972 ◽  
Vol 71 (2) ◽  
pp. 347-352 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Distribution Function ◽  
Linear Regression ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Random Variable ◽  
The Other ◽  
Linear Statistic ◽  
Binomial Distributions ◽  
Independent Observations

Introduction: Let X1, X2, …, Xn be n (n ≤ 2) independent observations on a random variable X with distribution function F. Also let L = L (X1, X2, …, Xn) be a linear statistic and Q = Q (X1, X2, …, Xn) be a homogeneous quadratic statistic. In this paper, we consider the problem of characterizing a class of probability distributions by the linear regression of the statistic Q on the other statistic L. In section 2, we obtain a characterization of a class of probability distributions, which includes the normal and the Poisson distributions. In section 3, a class of distributions including the gamma, the binomial and the negative binomial distributions is characterized.

scholarly journals INFLUENCE OF STATISTICAL FLUCTUATIONS ON K/π RATIOS IN RELATIVISTIC HEAVY ION COLLISIONS

2001 ◽  
Vol 16 (07) ◽  
pp. 1227-1235 ◽  
Author(s):  
C. B. YANG ◽  
X. CAI
Keyword(s):  
Negative Binomial ◽  
Heavy Ion ◽  
Relativistic Heavy Ion Collisions ◽  
Relative Variance ◽  
Ion Collisions ◽  
Statistical Fluctuations ◽  
Binomial Distributions ◽  
The Mean ◽  
Multiplicity Distributions ◽  
Statistical Background

The influence of pure statistical fluctuations on K/π ratio is investigated in an event-by-event way. Poisson and the modified negative binomial distributions are used as the multiplicity distributions since they both have statistical background. It is shown that the distributions of the ratio in these cases are Gaussian, and the mean and relative variance are given analytically.

Poisson mixtures

1995 ◽  
Vol 1 (2) ◽  
pp. 163-190 ◽  
Author(s):  
Kenneth W. Church ◽  
William A. Gale
Keyword(s):  
Negative Binomial ◽  
Probability Distributions ◽  
Hidden Variables ◽  
Heterogeneous Structure ◽  
Text Compression ◽  
Inverse Document Frequency ◽  
Poisson Mixtures ◽  
Document Frequency ◽  
Wide Range ◽  
Better Than

AbstractShannon (1948) showed that a wide range of practical problems can be reduced to the problem of estimating probability distributions of words and ngrams in text. It has become standard practice in text compression, speech recognition, information retrieval and many other applications of Shannon's theory to introduce a “bag-of-words” assumption. But obviously, word rates vary from genre to genre, author to author, topic to topic, document to document, section to section, and paragraph to paragraph. The proposed Poisson mixture captures much of this heterogeneous structure by allowing the Poisson parameter θ to vary over documents subject to a density function φ. φ is intended to capture dependencies on hidden variables such genre, author, topic, etc. (The Negative Binomial is a well-known special case where φ is a Г distribution.) Poisson mixtures fit the data better than standard Poissons, producing more accurate estimates of the variance over documents (σ2), entropy (H), inverse document frequency (IDF), and adaptation (Pr(x ≥ 2/x ≥ 1)).

scholarly journals Moment based Estimation for the Shape Parameters, Effect it on Some Probability Statistical Distributions Using the Moment Method and Maximum Likelihood Method

2020 ◽  
Vol 5 (6) ◽  
pp. 979-986
Author(s):  
Hassan Tawakol A. Fadol
Keyword(s):  
Maximum Likelihood ◽  
Shape Parameter ◽  
Maximum Likelihood Method ◽  
Probability Distributions ◽  
Estimation Method ◽  
Simulation Method ◽  
Likelihood Method ◽  
Inductive Method ◽  
Binomial Distributions ◽  
Best Estimate

The purpose of this paper was to identify the values of the parameters of the shape of the binomial, bias one and natural distributions. Using the estimation method and maximum likelihood Method, the criterion of differentiation was used to estimate the shape parameter between the probability distributions and to arrive at the best estimate of the parameter of the shape when the sample sizes are small, medium, The problem was to find the best estimate of the characteristics of the society to be estimated so that they are close to the estimated average of the mean error squares and also the effect of the estimation method on estimating the shape parameter of the distributions at the sizes of different samples In the values of the different shape parameter, the descriptive and inductive method was selected in the analysis of the data by generating 1000 random numbers of different sizes using the simulation method through the MATLAB program. A number of results were reached, 10) to estimate the small shape parameter (0.3) for binomial distributions and Poisson and natural and they can use the Poisson distribution because it is the best among the distributions, and to estimate the parameter of figure (0.5), (0.7), (0.9) Because it is better for binomial binomial distributions, when the size of a sample (70) for a teacher estimate The small figure (0.3) of the binomial and boson distributions and natural distributions can be used for normal distribution because it is the best among the distributions.

Sign in / Sign up

Export Citation Format

Share Document