Explore the vast range of titles available.

A Poisson process is a commonly used starting point for modeling stochastic variation of ecological count data around a theoretical expectation. However, data typically show more variation than implied by the Poisson distribution. Such overdispersion is often accounted for by using models with different assumptions about how the variance changes with the expectation. The choice of these assumptions can naturally have apparent consequences for statistical inference. We propose a parameterization of the negative binomial distribution, where two overdispersion parameters are introduced to allow for various quadratic mean-–variance relationships, including the ones assumed in the most commonly used approaches. Using bird migration as an example, we present hypothetical scenarios on how overdispersion can arise due to sampling, flocking behavior or aggregation, environmental variability, or combinations of these factors. For all considered scenarios, mean-–variance relationships can be appropriately described by the negative binomial distribution with two overdispersion parameters. To illustrate, we apply the model to empirical migration data with a high level of overdispersion, gaining clearly different model fits with different assumptions about mean-–variance relationships. The proposed framework can be a useful approximation for modeling marginal distributions of independent count data in likelihood-based analyses.

Let Apéry numbers An'=∑k=0n(Cnk)2Cn+kk. We proved a congruence involving An' modulo p4 by using some unknown congruences involving harmonic numbers and Bernoulli numbers, and an identity involving sums of binomial coefficients by Zeilberger’s algorithm.

In the first part of the article, the itinerary of binomials will be briefly illustrated from their first attestations in the classical context up to the forms that they took in the Latin and Romance production of medieval writers like the Stilnovist poets and Dante. In the Divina Commedia, the figure is used with unprecedented versatility, also in relation to the synonymous component. In the second part we will briefly present the different binomial typologies and the corresponding expressive outcomes, with particular reference to the configurations of synonymous links, whose peculiarities frequently open up new semantic perspectives.

Recently, several studies have shown that when \\(q\\equiv3\\pmod{4}\\), the function \\(F_r(x)=x^r+x^{r+\\frac{q-1}{2}}\\) defined over \\(\\mathbb{F}_q\\) is locally-APN and has boomerang uniformity at most~\\(2\\). In this paper, we extend these results by showing that if there is at most one \\(x\\in \\mathbb{F}_q\\) with \\(\\chi(x)=\\chi(x+1)=1\\) satisfying \\((x+1)^r - x^r = b\\) for all \\(b\\in \\mathbb{F}_q^*\\) and \\(\\gcd(r,q-1)\\mid 2\\), then \\(F_r\\) is locally-APN with boomerang uniformity at most \\(2\\). Moreover, we study the differential spectra of \\(F_3\\) and \\(F_{\\frac{2q-1}{3}}\\), and the boomerang spectrum of \\(F_2\\) when \\(p=3\\).

Let q$q$ be a prime power and Fq$\\mathbb {F}_q$ the finite field with q$q$ elements. For a positive integer n$n$, the polynomial Xn−1∈Fq[X]$X^n - 1 \\in \\mathbb {F}_q[X]$ is termed 3‐sparse over Fq$\\mathbb {F}_q$ if all its irreducible factors in Fq[X]$\\mathbb {F}_q[X]$ are either binomials or trinomials. In 2021, Oliveira and Reis characterized all positive integers n$n$ for which Xn−1$X^n - 1$ is 3‐sparse over Fq$\\mathbb {F}_q$ when q=2$q = 2$ and q=4$q = 4$. Recently, the author provided a complete characterization for odd q$q$. This paper extends the investigation to finite fields of characteristic 2, fully determining all n$n$ such that Xn−1$X^n - 1$ is 3‐sparse over Fq$\\mathbb {F}_q$ for even q$q$. This work resolves two open problems posed by Oliveira and Reis for the even‐characteristic case.

Researchers have developed methods to account for imperfect detection of species with either occupancy (presence—absence) or count data using replicated sampling. We show how these approaches can be combined to simultaneously estimate occurrence, abundance, and detection probability by specifying a zero-inflated distribution for abundance. This approach may be particularly appropriate when patterns of occurrence and abundance arise from distinct processes operating at differing spatial or temporal scales. We apply the model to two data sets: (1) previously published data for a species of duck, Anas platyrhynchos, and (2) data for a stream fish species, Etheostoma scotti. We show that in these cases, an incomplete-detection zero-inflated modeling approach yields a superior fit to the data than other models. We propose that zero-inflated abundance models accounting for incomplete detection be considered when replicate count data are available.

Non-Gaussian outcomes are often modeled using members of the so-called exponential family. Notorious members are the Bernoulli model for binary data, leading to logistic regression, and the Poisson model for count data, leading to Poisson regression. Two of the main reasons for extending this family are (1) the occurrence of overdispersion, meaning that the variability in the data is not adequately described by the models, which often exhibit a prescribed mean-variance link, and (2) the accommodation of hierarchical structure in the data, stemming from clustering in the data which, in turn, may result from repeatedly measuring the outcome, for various members of the same family, etc. The first issue is dealt with through a variety of overdispersion models, such as, for example, the beta-binomial model for grouped binary data and the negative-binomial model for counts. Clustering is often accommodated through the inclusion of random subject-specific effects. Though not always, one conventionally assumes such random effects to be normally distributed. While both of these phenomena may occur simultaneously, models combining them are uncommon. This paper proposes a broad class of generalized linear models accommodating overdispersion and clustering through two separate sets of random effects. We place particular emphasis on so-called conjugate random effects at the level of the mean for the first aspect and normal random effects embedded within the linear predictor for the second aspect, even though our family is more general. The binary, count and time-to-event cases are given particular emphasis. Apart from model formulation, we present an overview of estimation methods, and then settle for maximum likelihood estimation with analytic-numerical integration. Implications for the derivation of marginal correlations functions are discussed. The methodology is applied to data from a study in epileptic seizures, a clinical trial in toenail infection named onychomycosis and survival data in children with asthma.

We prove some congruences on sums involving fourth powers of central q -binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]:$$\\sum\\limits_{k = 0}^{((p^r-1)/(2))} {\\displaystyle{{4k + 1} \\over {{256}^k}}} \\left( \\matrix{2k \\cr k} \\right)^4\\equiv p^r\\quad \\left( {\\bmod p^{r + 3}} \\right),$$where p ⩾5 is a prime and r is a positive integer. Our method is similar to but a little different from the WZ method used by Zudilin to prove Ramanujan-type supercongruences.

The arcsine square root transformation has long been standard procedure when analyzing proportional data in ecology, with applications in data sets containing binomial and non-binomial response variables. Here, we argue that the arcsine transform should not be used in either circumstance. For binomial data, logistic regression has greater interpretability and higher power than analyses of transformed data. However, it is important to check the data for additional unexplained variation, i.e., overdispersion, and to account for it via the inclusion of random effects in the model if found. For non-binomial data, the arcsine transform is undesirable on the grounds of interpretability, and because it can produce nonsensical predictions. The logit transformation is proposed as an alternative approach to address these issues. Examples are presented in both cases to illustrate these advantages, comparing various methods of analyzing proportions including untransformed, arcsine- and logit-transformed linear models and logistic regression (with or without random effects). Simulations demonstrate that logistic regression usually provides a gain in power over other methods.