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Finite mixture representations of zero-&- N-inflated distributions for count-compositional data Andr´ e F. B. Menezesa,∗, Andrew C. Parnellb, Keefe Murphya aHamilton Institute and Department of Mathematics and Statistics, Maynooth University bSchool of Mathematics and Statistics, Insight Centre for Data Analytics Unive... | https://arxiv.org/abs/2501.13879v1 |
associated derivations and inference schemes for both distributions are deferred to the Appendices. We also provide further results in the Supplementary Material. For now, we begin by describing some related proposals. 1.1. Related work Many extensions of the multinomial distribution have been proposed in the literatur... | https://arxiv.org/abs/2501.13879v1 |
the following joint PMF for Yandϕ: Pr[Y=y, ϕ;λ] =N!ϕN−1 Γ(N)dY j=1" λyj je−λjϕ yj!# , (3) where the marginal PMF of yfollows that of Equation (1). Note that the augmented likelihood factors into independent terms for each λj. The same likelihood, up to a multiplicative constant, can be obtained via the Poisson-multinom... | https://arxiv.org/abs/2501.13879v1 |
(7) +dX j=1η(j) N" 10 X k:k̸=jyk!# (8) +X K∈KηK 10 X k∈Kyk!N {yj}j /∈KY j /∈K θK jyj (9) +η0dY j=110(yj) for y∈Ω0 d,N, (10) where θ= (θ1, . . . , θ d), with θj≥0andPd j=1θj= 1,θK j=θj/(1−P ℓ∈Kθℓ), and ζ= (ζ1, . . . , ζ d), with ζj∈[0,1]. The mixture weights ηare functions of ζ(see Definition 2.2). Proof. See A... | https://arxiv.org/abs/2501.13879v1 |
Proof. Follows from the fact that the mixture weights defined in Definition 2.2 are products of Bernoulli probabilities. From the two stochastic representations above, we can easily generate values from ZANIM. Obviously, the representation in Proposition 2.3 is more efficient. We note that this represen- tation has sim... | https://arxiv.org/abs/2501.13879v1 |
αK 0has empty entries in αfor all j∈ K. Proof. Follows by identifying each mixture component in the PMF of ZANIDM. Remark 2.2. Although ZANIM and ZANIDM have 2dcomponents, each distribution has only 2dparameters, since the component weights are functions of ζand the θandαparameters fully determine the non-degenerate co... | https://arxiv.org/abs/2501.13879v1 |
components each contribute a degenerate mass at 0 with probability η(k) N, for k̸=j. Secondly, the purely de- generate components in Equations (10) and (15) also contribute a degenerate mass at 0 with probability η0. Therefore, the marginal distribution of Yjis degenerate at 0 with probability η0+P k̸=jη(k) N, which si... | https://arxiv.org/abs/2501.13879v1 |
of the probability mass is concentrated at lower values of k, there is also slight but nonetheless visible N-inflation atk=N= 30, since the corresponding mixture weight, η(1) N= (1−ζ1)ζ2ζ3= 0.00675, is non- zero. Interestingly, the ZANIDM marginal is more right-skewed and overdispersed than that of 8 ZANIM, which also ... | https://arxiv.org/abs/2501.13879v1 |
h . Proof. Follows from the moment properties of finite mixture distributions. 9 Proposition 3.4. LetY∼ZANIDM d[N,α,ζ], then the mean and variance of the j-th entry of the random vector Y, i.e., the random variable Yj, are given by E[Yj] =N η(j) N+ηdαj αs+X Sj∈SjηSjαj αj+αSj s Var[Yj] =η(j) NN2+ηdN α j[N(1 +αj) +α(... | https://arxiv.org/abs/2501.13879v1 |
particularly interesting in light of the assertion of Koslovsky [1] that ZANIDM is limited to modelling negative correlations and that further extensions would be required to accommodate positive dependence among counts. We can now see that this is false; integrating out the latent structure already yields a finite mix... | https://arxiv.org/abs/2501.13879v1 |
, n, where Ni=dX j=1yij. Given the latent variables λij, zij, and ϕi, along with the observed vector yi, the augmented likelihood of the i-th observation factors into dindependent terms, as per ZANIM, as follows p(yi,λi,zi, ϕi|α,ζ) =p(yi|λi)p(ϕi|yi,λi)dY j=1[p(λij|zij, αj)p(zij|ζj)] =ϕNi−1 i Γ(Ni)N y1, . . . , y ddY ... | https://arxiv.org/abs/2501.13879v1 |
π(βj)∝1 Γ(eβj)tjexp eβjX i:λij>0log (λij) +βjs−2 j(mj−0.5βj) . (25) To sample from π(βj), we consider two well-known general schemes; a MH algorithm where the proposals follow a Gaussian random walk, as used by Koslovsky [1], and slice sampling with the stepping-out and shrinkage procedures as proposed by Neal [16]... | https://arxiv.org/abs/2501.13879v1 |
the posterior mean, and the overall coverage probability of the 95% credible interval. Letting ϑ= (ϑ1, . . . , ϑ d) denote the true values of the parameter vector of interest, we compute these metrics as follows: Bias(ϑ) =1 RdRX r=1dX j=1E[ϑj|y(r)] ϑj−1 and CP 95%(ϑ) =1 RdRX r=1dX j=11 ϑj∈CI95%[ϑj|y(r)] , where E[ϑ... | https://arxiv.org/abs/2501.13879v1 |
ζj∼Beta[1 ,1]. For ZANIDM, we run our MCMC scheme for 110,000 iterations, with the first 10 ,000 discarded as burn-in and a thinning interval of 100 applied. This setup helps to ensure reliable posterior inference and reduce autocorrelation in theαchains, in particular. The prior for ζjis set as per the ZANIM model and... | https://arxiv.org/abs/2501.13879v1 |
p(yi|ϑ(m)) is the model likelihood evaluated at the observation yi. The higher the ELPD, the better the model. Table 4 gives the ELPD results for different models and both DGPs. We also include the multinomial and DM distributions for comparison purposes, for which we use Stan via the Rpackage cmdstanr [18] in each cas... | https://arxiv.org/abs/2501.13879v1 |
the corresponding marginal distributions. We showed that the distributions can accommodate overdispersion and positive correlations, which can be attributed to their mixture structure and zero-inflation properties. We subsequently developed Bayesian inference frameworks for both distributions. Specifi- cally, for ZANID... | https://arxiv.org/abs/2501.13879v1 |
the four types of mixture component in ZANIM. •Standard multinomial: Z p(y, ϕ) dϕ=ZN!ϕN−1 Γ(N)dY j=1" (1−ζj)λyj je−λjϕ yj!# dϕ =N y1. . . y d"dY j=1(1−ζj)λyj j#Z1 Γ(N)ϕN−1e−ϕPd j=1λjdϕ 19 =N y1. . . y ddY j=1(1−ζj) λjPd k=1λk!yj =N y1. . . y ddY j=1(1−ζj)θyj j. •δ0dcomponent: Z p(y, ϕ) dϕ="dY j=1ζj10(yj)# N!Z1 Γ(... | https://arxiv.org/abs/2501.13879v1 |
λ j∈ {0,R+}. Note that the augmented likelihood for the ZANIDM distribution can be written as L(y,λ;α,ζ) =N y1, . . . , y ddY j=1 λjPd k=1λk!yj ζ10(λj) j (1−ζj)1−10(λj) λαj−1 je−λj Γ(αj)!1−10(λj) =cdY j=1h ζ10(λj) j (1−ζj)1−10(λj)idY j=1 λjPd k=1λk!yj λαj−1 je−λj Γ(αj)!1−10(λj) =cdY j=1h ζ10(λj) j (1−ζj)1−... | https://arxiv.org/abs/2501.13879v1 |
j e−λj Γ(αj)!#1−10(λj) dλ1. . .dλd =cY k∈KζkY j /∈K(1−ζj)ZY j /∈K"1P ℓ/∈Kλℓyjλyj+αj−1 j e−λj Γ(αj)# dλ(K) =Γ(αK s)Γ(N+ 1) Γ(N+αK s)Y k∈KζhY j /∈K(1−ζj)Γ(yj+αj) Γ(αj)Γ(yj+ 1), where the multivariate integral is over the set λ(K)={λj:λj/∈ K} . Collecting the terms while accounting for the restriction that yj= 0 when λj... | https://arxiv.org/abs/2501.13879v1 |
augmented likelihood in Equation (6), from which ZANIM was derived, can be recovered from this expression. For this derivation, we adopt the notation c= N y1···yd and drop the subscript i, for simplicity. Letz= (z1, . . . , z d), where zj= 0 corresponds to a structural zero count and zj= 1 represents a count obtained ... | https://arxiv.org/abs/2501.13879v1 |
a given category jwhen yj>0, we have that p(zj|yj>0, ϕ)∝zj(1−ζj)yλj je−ϕλj. It is evident that p(zj= 1|yj>0, ϕ) = 1, hence ( zj|yj>0, ϕ) is a degenerate distribution at 1 when yj>0. On the other hand, when yj= 0, we have that p(zj|yj= 0, ϕ)∝(1−zj)ζj+zj(1−ζj)e−ϕλj. Since zj∈ {0,1}, we obtain p(zj|yj= 0, ϕ) =(1−zj)ζj+zj(... | https://arxiv.org/abs/2501.13879v1 |
data augmentation strategies proposed by Hamura et al. [15], who present a general scheme for cases where the parameter of interest appears as the argument of a gamma function, as occurs with the αparameter in the Dirichlet, Dirichlet-multinomial, and indeed ZANIDM distributions. Recall that under the prior αj∼Gamma[ c... | https://arxiv.org/abs/2501.13879v1 |
generating functions via mixture properties We can find the moment generating function (MGF) for both distributions using the moment properties of mixtures with g(Y) =et·Y. By identifying the component-specific distributions based on our novel stochastic representations of ZANIM and ZANIDM in terms of finite mix- tures... | https://arxiv.org/abs/2501.13879v1 |
MH-RW, slice sampling, and ZIDM perform similarly in terms of parameter recovery, although only DA-PTN has credible intervals which contain the true values ofζin each case. As per Section 5.2, inference for αis poor, under all approaches, in this scenario with ZANIM as the data-generating process. In Table S.2, where Z... | https://arxiv.org/abs/2501.13879v1 |
is fitted to data generated from ZANIDM are approximately 1 /3. 31 Table S.3: Posterior summaries of ZANIM and ZANIDM under two balanced data-generating processes. DGP Model Parameter Mean 95% LCI 95% UCI ESS ratio ZANIM: θ= (0.05,0.70,0.25), ζ= (0.05,0.15,0.10), andN= 30 trials.ZANIDMα1 9.590 6.773 14.080 0.276 α2 9.6... | https://arxiv.org/abs/2501.13879v1 |
A new class of tests for convex-ordered families based on expected order statistics Tommaso Lando1,∗, Mohammed Es-Salih Benjrada1 Abstract Consider a pair of cumulative distribution functions FandG, where Fis unknown and G is a known reference distribution. Given a sample from F, we propose tests to detect the convexit... | https://arxiv.org/abs/2501.14075v1 |
improve the nonparametric estimate of F. This approach corresponds to the scope of shape-constrained inference; see, for instance, the books by Robertson et al. (1988) and Groeneboom and Jongbloed (2014). For these reasons, nonparametric tests for convex-ordered families are particularly inter- esting and have been stu... | https://arxiv.org/abs/2501.14075v1 |
in Section 5. In the IHR and DHR cases, we compare our tests with the well-known test of Proschan and Pyke (1967), which has been shown to satisfy the same theoretical properties of our class of tests (Bickel and Doksum, 1969). Moreover, we apply our tests to other important families of distributions, for which the app... | https://arxiv.org/abs/2501.14075v1 |
increasing, or decreasing, respectively. The conditions F∈ Fcx GandF∈ Fcv Ghave an effect on the expectation of Xj:m, denoted withEXj:m=µj:m(F) =µj:m. Indeed, for F∈ Fcx G, Jensen’s inequality implies that EXj:m≤F−1◦G(E(G−1◦F(Xj:m))) = F−1◦G(E(G−1(Bj:m))), therefore, P(X≤µj:m)≤G(E(G−1(Bj:m)).Similar results hold when F... | https://arxiv.org/abs/2501.14075v1 |
an observed sample x1, ..., x n. Our testing approach leverages Proposition 1, therefore we need to estimate the expected order statistics µj:mbased on a sample of size n, where nandmare generally different. Given that Xj:mhas CDF FBj:m◦F, we can express the functional µj:m(F) as the integral µj:m(F) =R RxdF Bj:m◦F(x).... | https://arxiv.org/abs/2501.14075v1 |
D+andD−can be analysed symmetrically, using the relation min {X1, . . . , X n}=−max{−X1, . . . ,−Xn},therefore it is sufficient to focus on the behaviour of the right tail. A necessary and sufficient condition for F∈ D(Φα) is that lim t→∞1−F(tx) 1−F(t)=x−α (de Haan and Ferreira, 2006, Theorem 1.2.1). The parameter αdet... | https://arxiv.org/abs/2501.14075v1 |
test HG 1−with TG− m,p(Fn) =||(vG m−ˆVm)−||p. The expressions of TG+ m,pandTG− m,pdefine new families of test statistics, parameterised by the number of order statistics involved, m, and by the order of the Lpnorm, p. The effect of such parameters on the tests’ performance will be addressed by simulations, while in thi... | https://arxiv.org/abs/2501.14075v1 |
,the probability of rejecting the null hypothesis tends to 1 when the alternative is true. This result will be generalised later to the infinite mean case, leveraging Theorem 1. Proposition 5. Assume that FandGhave finite expectations. 1.Under HG 1+,P(TG+ m,p(Fn)≥c+ α,n)→1. 2.Under HG 1−,P(TG− m,p(Fn)≥c− α,n)→1. If we ... | https://arxiv.org/abs/2501.14075v1 |
comparison is restricted to G=E, as the P&P approach is designed exclusively for testing IHR/DHR alternatives. We will mainly simulate from the following models: Weibull distribution with shape parameter aand scale parameter b, represented as W(a, b); log-logistic distribution with shape parameter aand scale parameter ... | https://arxiv.org/abs/2501.14075v1 |
as “unexpected stochastic dominance”, was recently proved by Chen et al. (2024). We conduct simulations using the log-logistic distribution L(a,1), which is IOR for a≥1 and DOR for a≤1. For the IOR test, we compute TG+ m,ℓ,p with ℓ=m−2, in order to guarantee consistency. The plots in Figure 2 summarise the output of th... | https://arxiv.org/abs/2501.14075v1 |
lead to rejecting exponentiality in favour of both IHR and DHR alternatives. Since these alternatives are contradictory, one can infer that the distribution being tested is IHR in some interval, but DHR in some other interval, as is the case for St(1.1). Similar results, not reported here, 16 hold simulating from the S... | https://arxiv.org/abs/2501.14075v1 |
it. As we will discuss, this approach allows testing multiple hypotheses to uncover the key property of the distribution of interest, ultimately aiding in the selection of the appropriate model to fit the data. We apply our method to a dataset in Bryson (1974), which reports annual flows of the Weldon River at Mill Gro... | https://arxiv.org/abs/2501.14075v1 |
j=1EX∗ j:m=E 1 mmX j=1X∗ j:m =EX∗. Now, the conclusion follows from noticing that EX∗ j:m=µj:m(Fn) andEX∗=Pn i=1xi/n. 2. By definition of order statistics, X∗ j:m≤X∗ j+1:m, then, taking expectations, µj:m(Fn)≤ µj+1:m(Fn). 3. It’s easy to check that Bj:m+1≤stBj:m, hence, for every realisation Fn,FBj:m+1◦Fn(x)≥ FBj:m◦Fn... | https://arxiv.org/abs/2501.14075v1 |
+|F−1(Un:n)|(1−FBj:m(1−r)) can be made arbitrarily small, with probability 1, by the choice of r. Hence, it is enough to investigate the behaviour for n→ ∞ andr→0. Letting r= 1/n,we need to show that F−1(Un:n)(1−FBj:m(n−1 n))−F−1(U1:n)FBj:m(1 n)→a.s.0, (1) asn→ ∞ . The terms FBj:m(1 n) and (1 −FBj:m(n−1 n)) converge to... | https://arxiv.org/abs/2501.14075v1 |
≤eHnZ xdF Bj:m◦Hn(x) . This inequality holds for every possible realization HnofHn. Therefore, Theorem 1.A.1 of Shaked and Shantikumar (2007) yields that eFn(µj:m(Fn))≤steHn(µj:m(Hn)). Proof of Theorem 2. We prove just part 1. Given random variables UjandZj,j= 1, . . . , m , such that Uj≤stZjfor every j= 1, . . . , m... | https://arxiv.org/abs/2501.14075v1 |
H., 1998. The reversed hazard rate function. Probability in the Engineering and informational Sciences 12, 69–90. Bryson, M.C., 1974. Heavy-tailed distributions: properties and tests. Technometrics 16, 61–68. Carolan, C.A., 2002. The least concave majorant of the empirical distribution function. Canadian Journal of Sta... | https://arxiv.org/abs/2501.14075v1 |
0.3512 0.6204 0.8914 - 20 0.193 0.2912 0.4736 0.751 ∞ 1 0.325 0.4804 0.7174 0.9124 - 5 0.2352 0.3465 0.5164 0.7596 - 10 0.1956 0.2702 0.3544 0.5426 - 20 0.1598 0.2006 0.2494 0.3406 Table 1: Rejection rates for the IHR case, where F∼W(1.5,1) (a) P&P test n= 25 n= 50 n= 100 n= 200 n= 500 0.493 (0.009) 0.8848 (0) 0.9792 (... | https://arxiv.org/abs/2501.14075v1 |
STATISTICAL VERIFICATION OF LINEAR CLASSIFIERS ZHIYANOV A.P.1, SHKLYAEV A.V.2, GALATENKO A.V.2, GALATENKO V.V.2AND TONEVITKSY A.G.1,3,4 1Faculty of Biology and Biotechnology, HSE University, Moscow, 101000, Russia 2Faculty of Mechanics and Mathematics, Lomonosov MSU, Moscow, 119991, Russia 3Shemyakin-Ovchinnikov Instit... | https://arxiv.org/abs/2501.14430v1 |
small testing sets can be sufficient to confirm statistical significance of classification “non-randomness”. Formally, let Y= (Yi, i≤k),k∈N, and Z= (Zi, i≤l),l∈N, be two independent samples in Rd,d∈N. We assume that YandZconsist of independent identically distributed (i.i.d.) random vectors (r.v.) distributed according... | https://arxiv.org/abs/2501.14430v1 |
sizes k >0andl=n−k >0, where k≤l. For a fixed S, these partitions are equiprobable and have probability 1/ n k . Definition 1. The subsets YandZare called near-linearly seaparable with m errorsif there exists a half-plane H+such that |H+∩Y|=k−mY,|H−∩Z|= l−mZandmax( mY, mZ) =m, where H−=R2\H+. IfmY+mZ=m, we call the su... | https://arxiv.org/abs/2501.14430v1 |
For in- stanse, if outliers are removed using a procedure based on X(i.e., without consider- ing labels YandZ), the modified sample X′consists of exchangeable but not i.i.d. random vectors. The first test is based on near-linear seaparability of YandZ. It rejects H0if the number of errors mis small enough. Corollary 1 ... | https://arxiv.org/abs/2501.14430v1 |
probability estimation of linear separability was computed, with each experiment repeated 10 times for each n. A B Figure 1. A:Theupperbound(4)of P(A0)inthenormalcaseistight. B: The upper bound from Lemma 1 approximates P(A≤m)reasonably well. As shown in Figure 1 (A), the upper bound (4) was accurate for normally dis- ... | https://arxiv.org/abs/2501.14430v1 |
exceeding 0.05), amounting to 13 gene pairs. In comparison, Validation and Training datasets resulted in 180 and 267 pairs, respectively, while for Filtration 1 dataset, almost all classifiers (559 out of 570) were estimated to be “random”. The exact values of the upper bounds on FDRs and p-values for all classifiers a... | https://arxiv.org/abs/2501.14430v1 |
involves testing them on an independent dataset. However, due to the inherent heterogeneity of biomedical data, classifiers achieving relatively low classification accuracy are still considered acceptable in practice. We developed and characterized statistical tests answering the question whether a given classifier is ... | https://arxiv.org/abs/2501.14430v1 |
that mY andmZboth equal m, andA≤m=Am. Note that the line ℓcorresponds to an error-free partition p′, dividing the points into two classes, i.e., those above ℓand those below. Thus, we build a mapping from nearly separable partitions to partitions without errors. It remains to note that for any error-free partition p′, ... | https://arxiv.org/abs/2501.14430v1 |
plicative constant in Theorem 1 for specific kandn. Proof of Theorem 2. Recallthatallpartitionsof Sofsizes kandlareequiprobable. LetDφ mbe the event that φis a near-separating directed angle with merrors or less. Using Fubini’s theorem, we obtain Eµ≤m=EZ2π 0I{Dφ m}dφ=Z2π 0P(Dφ m)dφ, whereI{Dφ m}is the indicator functio... | https://arxiv.org/abs/2501.14430v1 |
2 possible hyperplanes ℓ+andℓ−in the definition of F(ℓ). Therefore, the resulting upper bound should be 2 times greater, i.e., G(t+ ∆t)−G(t)≤2d2κd−2κd−2 κd−1d−1 ∆t. Using the proof of [15, Theorem 1] with the described modification and (12), we obtain EN(S)≤E2(k−1, n)≤24κ2 0 κ1(n−1) n 2 n−2 k−1 n−2 k−1=8n π, wher... | https://arxiv.org/abs/2501.14430v1 |
Oncology , vol. 27, no. 8, pp. 1160–1167, 2009, issn: 1527-7755. [9] V. V. Galatenko et al., “Highly informative marker sets consisting of genes withlowindividualdegreeofdifferentialexpression,” Scientific Reports ,vol.5, no. 1, pp. 1–8, 2015, issn: 2045-2322. [10] G. Unger and B. Chor, “Linear separability of gene exp... | https://arxiv.org/abs/2501.14430v1 |
vol. 5, no. 7, pp. 621–628, 2008, issn: 1548-7105. [26] Y. Benjamini and Y. Hochberg, “Controlling the false discovery rate: A prac- tical and powerful approach to multiple testing,” Journal of the Royal Sta- tistical Society Series B: Statistical Methodology , vol. 57, no. 1, pp. 289–300, 1995, issn: 1467-9868. [27] F... | https://arxiv.org/abs/2501.14430v1 |
Models Parametric Analysis via Adaptive Kernel L earning Vladimir Norkin [0000 -0003 -3255 -0405 ] and Alois Pichler [0000 -0001-8876 -2429 ] V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine, Kyiv, 03178 Ukraine & National Technical University of Ukraine “Igor Sikorsky Kyiv Polytech... | https://arxiv.org/abs/2501.14485v1 |
on certain parameters. Howe v- er, the problem is that there can be a lot of parameters (e.g., a matrix), and they cohe r- ently affect the solution nonlinearly. An important application of kernel method is visualization of spatial data [1, 2] , where the selection of kernel width is essential for smooth representation... | https://arxiv.org/abs/2501.14485v1 |
arises for such an adaptive support vector machine. Second, t he paper proposes to consider a regression problem not in RKHS [25, 26] but in a kernel subspace of 2()nL with a standard inner product in this space and with a standard 2L -norm in the regularization term . Unlike [17], where the kernel matrix is o btained ... | https://arxiv.org/abs/2501.14485v1 |
methods. EXAMPLE (a toy parametric problem) . Consider an input -output type model in the form of a quadratic equation 2y x ax b , where x is the input of the model, and y is the output of the model. Suppose we are looking for an input x , at which the output is zero, i.e. 0y . For this, we need to solve the equa... | https://arxiv.org/abs/2501.14485v1 |
since ()zx is just some average value of quantities 1m iiy . There is a room for further modification of the NW-regression by adjusting parameter to dada. For example, i f K is an one-dimensional function, i ixx xxKK and ( , )iix x x x , then (3) becomes just an arithmetic average... | https://arxiv.org/abs/2501.14485v1 |
, ) , ; ( , ) ; , minmii abimmm mm i i i i m a b j m j j a b i ii iijmx z a b a b . This is a high -dimensional optimization problem with r espect to variables 1( , )mii abi . It is non -convex and ( possibly ) has many extremes. Various optimiz a- tion methods can be used to sol... | https://arxiv.org/abs/2501.14485v1 |
the ide ntity matrix , 1KI is the inverse of matrix KI , T 1,...,m y y y . The opt imal value for problem (6) is 1 T minmR y K I y [17, 29] . Remark 1. In [29] the kernel estimate of the regression function had the form 1 1( ) ( , )m iiif m k x and the erro r minimization problem had the for... | https://arxiv.org/abs/2501.14485v1 |
x x k x x k x x dx . Let function 2 ( ) ( )nfL and ( ) 0fx for xX , X is a bounded Borell measurable subset in n . Define the averaged function ( ) ( ) ( ) ( ) ( )nnf x f y k y x dy f x z z dz . 9 If ()f is continuous in a vicinity of x , then 0 lim ( ) ( ) f x f x , i.e. fun c- tion... | https://arxiv.org/abs/2501.14485v1 |
10 where , ,1ml ijijKk is a matrix (to be more specified later) with entries ˆˆ ( , ), ( , ) ( , ) ( , )ni j i j ij i j i jk k x k x k x x k x x dx . The norm of ,,()xfx in this subspace is: 21212T ,, ,1( ) ( , ), ( , ) ijm x i j i j ij Lf k x ... | https://arxiv.org/abs/2501.14485v1 |
widths 1,...,m ), 11 2 2 1( ) ( ) (2 ) ,, 2 1 1()(2 ) ...n l i l illmxxi x n i i inf x e (with anisotropic variable kernel widths 11 1,...,n n j m jj j . The correspo nding kernel regression problem for a sample , , 1,...,iix y i m takes on the form ... | https://arxiv.org/abs/2501.14485v1 |
(13) To solve this problem one has t o 12 1) Calculate the matrix ,1( , ), ( , ) ijm ijijK k x k x as a function of T 1,...,m ; 2) Solve the corresponding minimization problem (13) with respect to T 12, ,...,m m and T 12, ,...,m . Let us calculate matrix K (9) in ... | https://arxiv.org/abs/2501.14485v1 |
2 4 4 4 2 2 2 2 2 2 22 2 2 222 2 2, 2j i i j j i j i i j i j i j j i j i i j j i j i i j i j j i j i i j j i j i i j j i i j i j i j i j j i i j j i j ixx xx x x x x x x x x x x x x xx 2 2 22 22 2 2... | https://arxiv.org/abs/2501.14485v1 |
( , ) ( , ) ( , )ni j i j i j i j k x k x k x x k x x dx . Corollary 1. For 1n formula (16) becomes 2 1 2 2 2 12 221( , ), ( , ) exp2( ) 2ijij ij ijijxx k x k x , 0, 0ij . Thus for an anisotropic kernel 2 22 1 1( ) ( ) 1( , ) exp(2 ) 2in l i l i n l i in i lxxk x x... | https://arxiv.org/abs/2501.14485v1 |
difference b e- tween obtained approximations. Fig. 1. The original function ( , )f a b Fig. 2. 100m random sample points within feasible set 2( , ) 2, 2 :ab within the feasible set. Fig. 3. NW -regression with 10k Fig. 4. NW -regression with 3k summands and absolute accuracy summands and absolute accuracy 0.71... | https://arxiv.org/abs/2501.14485v1 |
b the accuracy estimate. We illustrate this approach on the Example. The following figures 5-7 illustrate the proposed procedure of calculating 11, for 410 . Figure 5 shows that the regularized losses have a unique minimum at some 1 . This observation is the main experimental finding of this section. Figures... | https://arxiv.org/abs/2501.14485v1 |
)m ii ixy as in (11). The test data set can be used for selection of optimal regular i- zation parameter by minimizing the test erro r over (for some s ), 0 , , ,( ) inf max ( ) ( ) m s s jjms opt x xxf x f x . We illustrate the proposed adaptation procedure on Example from Section 2 for 50, 10... | https://arxiv.org/abs/2501.14485v1 |
Editors. 2001, Springer. p. 269 -292. 21 3. Tsybakov, A.B., Introduction to Nonparametric Estimation . 2009, New York, NY: Springer. 214. 4. Nadaraya, E.A., On estimating regression. Theory of Probability & Its Applicati ons, 1964. 9(1): p. 141 -142. 5. Nadaraya, E.A., Nonparametric estimation of probability densities ... | https://arxiv.org/abs/2501.14485v1 |
M achine Learning Research, 2009. 10(12). 25. Aronszajn, N., Theory of reproducing kernels. Transactions of the American mathematical society, 1950. 68(3): p. 337 -404. 26. Berlinet, A. and C. Thomas -Agnan, Reproducing kernel Hilbert spaces in probability and statistics . 2011: Springer Science & Business Media. 27. E... | https://arxiv.org/abs/2501.14485v1 |
arXiv:2501.14594v1 [math.PR] 24 Jan 2025ON THE MULTIDIMENSIONAL ELEPHANT RANDOM WALK WITH STOPS BERNARD BERCU University of Bordeaux, France Abstract. The goalofthis paper is to investigatethe asymptotic behaviorof t he multidimensional elephant random walk with stops (MERWS). In contr ast with the standard elephant ra... | https://arxiv.org/abs/2501.14594v1 |
0··· ··· 0 1 and Jd= 0 1 0 ···0 0 0 1...... ............0 0···0 0 1 1 0···0 0 and where p+(2d−1)q+r= 1. Therefore, the position of the MERWS at time n+1 is given by (1.4) Sn+1=Sn+Xn+1. Inallthesequel, weassumethat0 < r <1inasmuchasthecase r= 0waspreviously investigated by Bercu and Laulin [4], whil... | https://arxiv.org/abs/2501.14594v1 |
long history in complex an alysis and probability. It is defined, for all z∈C, by Eα(z) =∞/summationdisplay n=0zn Γ(1+nα) whereαis a positive real parameter and Γ stands for the Euler Gamma funct ion. We refer the reader to [ ?] for a monograph devoted to the main properties of Mittag- Leffler functions. Definition 2.1. We... | https://arxiv.org/abs/2501.14594v1 |
the almost sure convergence (3.8) lim n→∞1 nSn= 0 a.s. More precisely, (3.9)/bardblSn/bardbl2 n2=O/parenleftbigglognloglogn n1+r/parenrightbigg a.s. Hereafter, we focus our attention on the law of iterated logarithm for the MERWS. Theorem 3.5. We have the law of the iterated logarithm (3.10) limsup n→∞/bardblSn/bardbl2... | https://arxiv.org/abs/2501.14594v1 |
nn/summationdisplay k=1d/summationdisplay i=1Ji dXkXT k(Ji d)Ta.s. which implies via the definition of the permutation matrix Jdgiven by (1.3) that (4.6) E[Xn+1XT n+1|Fn] =a nΣn+2q nσ2 nIda.s. where (4.7) σ2 n= Tr(Σ n) =n/summationdisplay k=1/bardblXk/bardbl2. Therefore, we obtain from (1.6) and (4.6) that E[Sn+1ST n+1|... | https://arxiv.org/abs/2501.14594v1 |
the Mittag-Lefflerdistributionischaracterizedbyitsmoments, itmeans thattherandom variable Σ has a ML(b) distribution. Therefore, as b= 1−r, convergence (4.21) immediately leads to (2.5). Moreover, (2.6) follows from (4.20) and to gether with the fact that ( Nn) converges in Lmfor any integer m≥1. We now carry on with ON ... | https://arxiv.org/abs/2501.14594v1 |
=∞/summationdisplay k=0(a)(k)(b)(k)(c)(k)(d)(k) (e)(k)(f)(k)(g)(k)k!zk. However, we already saw in Lemma 4.1 that the sequence ( Nn) is a martingale such that for all n≥1,E[Nn] =E[bnσ2 n] = 1. It implies that for all n≥1,E[wn] =vn. Therefore, we deduce from the monotone convergence theorem that sup n≥1E[wn]<∞. Hence, w... | https://arxiv.org/abs/2501.14594v1 |
is easy to see from (1.5) that E[Xn+1/bardblXn+1/bardbl2|Fn] =E[Xn+1|Fn] and E[/bardblXn+1/bardbl4|Fn] =E[/bardblXn+1/bardbl2|Fn] almost surely. Therefore, it follows from (4.6), (4.8), (A.6), (A.7) and (A.8) together with tedious but straightfor ward calculations that E[/bardblSn+1/bardbl2Sn+1|Fn] =/parenleftbigg 1+3a... | https://arxiv.org/abs/2501.14594v1 |
n→∞1 nb−2an/summationdisplay k=1E/bracketleftig ∆M2 k(u)I/braceleftbig |∆Mk(u)|>η√ nb−2a/bracerightbig/bracketrightig = 0. We have for any η >0, 1 nb−2an/summationdisplay k=1E/bracketleftig ∆M2 k(u)I/braceleftbig |∆Mk(u)|>η√ nb−2a/bracerightbig/bracketrightig ≤1 η2n2(b−2a)n/summationdisplay k=1E/bracketleftig ∆M4 ... | https://arxiv.org/abs/2501.14594v1 |
for any vector u∈Rd, limsup n→∞/parenleftbigg1 2wnloglogwn/parenrightbigg1/2 Mn(u) =−liminf n→∞/parenleftbigg1 2wnloglogwn/parenrightbigg1/2 Mn(u) =/parenleftbiggb d/parenrightbigg1/2 /bardblu/bardbla.s. which leads via (4.3) and (4.21) to (3.10). Finally, (3.11) follows from (3 .10) and (4.21) which achieves the proof... | https://arxiv.org/abs/2501.14594v1 |
One can observe from (4.15) and (C.8) that E[σ2 n] =d/summationdisplay i=1E[σ2 n(i)] =d/summationdisplay i=11 dbn=1 bn, which is consistent with (C.5). Hereafter, we immediately obtain from (4.14) and (C.8) that for all n≥1, (C.9) E[Σn] =1 dbnId=(b)(n) bd(n−1)!Id. Hence, it follows from the conjunction of (C.4), (C.5) ... | https://arxiv.org/abs/2501.14594v1 |
σ2 nL−→ n→+∞N/parenleftbig 0,ϑ2/bardblu/bardbl2/parenrightbig . Consequently, we deduce the Gaussian fluctuation (3.18) from (C.2 1) together with the Cram´ er-Wold theorem. Moreover, we also obtain from Theore m 1 in [17] that for any vector u∈Rd, (C.22)√ n2a−b/parenleftbig Mn(u)−M(u)/parenrightbigL−→ n→+∞Γ(a+1)√ Σ′N/p... | https://arxiv.org/abs/2501.14594v1 |
arXiv:2501.14797v1 [math.ST] 12 Jan 2025A characterization of uniform distribution using varextro py with application in testing uniformity Santosh Kumar Chaudhary1and Nitin Gupta2 1Department of Statistics, Central University of Jharkhand , Cheri-Manatu, Ranchi, Jharkhand, 835222, India. 2Department of Mathematics, In... | https://arxiv.org/abs/2501.14797v1 |
non parametric estimator is given in Section 4. In Section 5, a test of uniformity is proposed. Section 7 concludes this paper. 2 Some properties Suppose that X1,...,X nare independent and identically distributed observations with cdf F and pdf f.An observation Xjwill be called an upper record value if its value exceed... | https://arxiv.org/abs/2501.14797v1 |
[0,1]. Th enVJ(X) = 0 if and only if X has a uniform distribution on the interval [0, 1]. 4 ProofLet random variable X have a uniform distribution on the inte rval [0,1], then f(x) = 1,0≤x≤1 and VJ(X) =1 4/integraldisplay1 0f3(x)dx−1 4/bracketleftbigg/integraldisplay1 0f2(x)dx/bracketrightbigg2 = 0 Conversely, VJ(X) = ... | https://arxiv.org/abs/2501.14797v1 |
the test statistics as the critical value at levelα. Critical value for α= 0.05 are give in Table 1 for different value of mandn. Table 1.Critical values at significance level α= 0.05 m\n10 20 30 40 50 80 100 2 4.7570 3.1388 2.2838 1.9478 1.6402 1.1355 1.0451 3 1.4909 1.2126 0.7925 0.6502 0.5729 0.4235 0.3559 4 0.7064 0.... | https://arxiv.org/abs/2501.14797v1 |
The consistency of the proposed es timators assures their robustness and reliability in estimating varextropy. Building upon th e characterization of the uniform 7 distribution using varextropy, we developed a statistical test for assessing the uniformity of data. By deriving critical values and evaluating the test’s p... | https://arxiv.org/abs/2501.14797v1 |
, 58(4), 827–841. [14] Noughabi, H. A., & Jarrahiferiz, J. (2018). On the estim ation of extropy. Journal of Nonparametric Statistics , 31(1), 88–99. [15] Park, S. (1999). A goodness-of-fit test for normality ba sed on the sample entropy of order statistics. Statistics and Probability Letters , 44, 359–363. [16] Qiu, G.... | https://arxiv.org/abs/2501.14797v1 |
Assessing Skew-Normality in Marks Distribution: A Comparative Analysis of Shapiro–Wilk Tests Himadri Mukherjee1and Pratham Bhonge1 1Department of Mathematics, BITS Pilani K. K. Birla Goa Campus, Goa, India. Abstract This paper investigates the distribution of marks obtained by students across multiple courses to explor... | https://arxiv.org/abs/2501.14845v1 |
at BITS Pilani, Goa Campus. Through this analysis, we aim to demonstrate the importance of selecting appropriate statistical methods for evaluating the distributional properties of educational data. By incorporat- ing tests tailored for skew-normal distributions, we provide a framework for robust and accurate analysis ... | https://arxiv.org/abs/2501.14845v1 |
data into a form suitable for classical normality testing, the modified Shapiro–Wilk test accounts for the asymmetry inherent in many real-world datasets. It provides a robust framework for assessing skew-normality, outperforming traditional tests that are sensitive to skewness. This methodology is particularly valuabl... | https://arxiv.org/abs/2501.14845v1 |
the snpackage for skew- normal parameter estimation and transformation. The implementation of the modified Shapiro–Wilk test was carried out using the following R func- tion: # Define the function for the skew-normal test sn.test <- function(x) { n <- length(x) estim <- sn.mple(y = x, penalty = "Qpenalty", opt.method =... | https://arxiv.org/abs/2501.14845v1 |
educational datasets due to factors such as grading distributions and outlier performance. The application of the modified Shapiro–Wilk test revealed that both datasets are consistent with a skew-normal distribution. This result aligns with the intuition that student marks often exhibit slight asymmetry, with a concent... | https://arxiv.org/abs/2501.14845v1 |
and robust- ness of the modified test under varying degrees of skewness and sample sizes. 7 •Exploration of Skewness Sources: Investigate the factors contributing to skewness in ed- ucational datasets, such as grading schemes, assessment difficulty, and student demographics. •Applications in Other Domains: Explore the ... | https://arxiv.org/abs/2501.14845v1 |
The typicality principle and its implications for statistics and data science Yiran Jiang1, Zeyu Zhang2, Ryan Martin3, and Chuanhai Liu2 1Department of Biostatistics, Yale University 2Department of Statistics, Purdue University 3Department of Statistics, NC State University January 28, 2025 Abstract A central focus of ... | https://arxiv.org/abs/2501.14860v1 |
challenge in modern empirical sciences, not found in the “all swans are white” style examples often considered in philosophy texts, is that the empirical data cannot logically contradict any legitimate theory, so there will inevitably be uncertainty in drawing inferences. This necessitates the reliable quantification o... | https://arxiv.org/abs/2501.14860v1 |
performance of our proposed typicality-focused regularization by applying it to several challenging problems that have historically served as points of contention in the foundations of statistics. Our results demonstrate its efficiency in point estima- tion and uncertainty quantification more broadly, highlighting its ... | https://arxiv.org/abs/2501.14860v1 |
does not belong to the posited model can also be considered, but we will not consider such cases here; see Jiang and Liu (2025) for further discussion along these lines. Of course, data X=x is observed, and our objective is to make inference on the uncertain Θ, relative to the posited statistical model. The most basic ... | https://arxiv.org/abs/2501.14860v1 |
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