NORMALITY AND THE NUMBER OF ZEROS

In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential polynomial,which extends the related result of Li,and Chen et al..An example is given to show that the hypothesis on the zeros of a(z)is necessary.

THE CONSISTENCY AND ASYMPTOTIC NORMALITY OF NEAREST NEIGHBOR DENSITY ESTIMATOR UNDER α-MIXING CONDITION

We investigate the consistency and asymptotic normality of nearest-neighbor density estimator of a sample data process based on α-mixing assumption. We extend the correspondent result under independent identical cases.

ASYMPTOTIC NORMALITY OF PARAMETERSESTIMATION IN EV MODEL WITH REPLICATEDOBSERVATIONS

This paper based on the essay [1], studies in case that replicated observations are available in some experimental points., the parameters estimation of one dimensional linear errors-in-variables (EV) models. Asymptotic normality is established.

NORMALITY CRITERIA FOR FAMILIES OF MEROMORPHIC ALGEBROID FUNCTIONS

By using the definition of Hausdorff distance, we prove some normality criteria for families of meromorphic algebroid functions. Some examples are given to complement the theory in this article.

Consistency and Asymptotic Normality of MLE of Parameter Vector in a Randomly Censored GLM with Incomplete Information

The generalized linear model（GLM） based on the observed data with incomplete information in the case of random censorship is defined. Under the given conditions, the existence and uniqueness of the solution on the likelihood equations with respect to the parameter vector β of the model are discussed, and the consistency and asymptotic normality of the maximum likelihood estimator（MLE） βn^-, are proved.

Applications of Normality Test in Statistical Analysis

In this study, to power comparison test, different univariate normality testing procedures are compared by using new algorithm. Different univariate and multivariate test are also analyzed here. And also review efficient algorithm for calculating the size corrected power of the test which can be used to compare the efficiency of the test. Also to test the randomness of generated random numbers. For this purpose, 1000 data sets with combinations of sample size n = 10, 20, 25, 30, 40, 50, 100, 200, 300 were generated from uniform distribution and tested by using different tests for randomness. The assessment of normality using statistical tests is sensitive to the sample size. Observed that with the increase of n, overall powers are increased but Shapiro Wilk (SW) test, Shapiro Francia (SF) test and Andeson Darling (AD) test are the most powerful test among other tests. Cramer-Von-Mises (CVM) test performs better than Pearson chi-square, Lilliefors test has better power than Jarque Bera (JB) Test. Jarque Bera (JB) Test is less powerful test among other tests.

ASYMPTOTIC NORMALITY OF WAVELET ESTIMATOR IN HETEROSCEDASTIC REGRESSION MODEL

The following heteroscedastic regression model Yi = g（xi） ＋σiei （1 ≤i ≤ n） is 2 considered, where it is assumed that σi^2 = f（ui）, the design points （xi,ui） are known and nonrandom, g and f are unknown functions. Under the unobservable disturbance ei form martingale differences, the asymptotic normality of wavelet estimators of g with f being known or unknown function is studied.

Normality and Shared Value Concerning Differential Polynomials

Let F be a family of functions meromorphic in a domain D, let m, n k , k be three positive integers and b be a finite nonzero complex number. Suppose that, (1) for eachf∈F, all zeros of f have multiplicities at least k ; (2) for each pair of functions f, g ∈F,P(f)H(f) and P(g)H(g) share b, where P(f) and H(f) were defined as (1.1) and (1.2) and nk ≥ max 1≤i≤k-1 {n i }; (3) m ≥ 2 or nk ≥ 2, k ≥ 2, then F is normal in D.

Normality Criteria of Zero-free Meromorphic Functions

Let k be a positive integer,let h(z)■0 be a holomorphic functions in a domain D,and let F be a family of zero-free meromorphic functions in D,all of whose poles have order at least l.If,for each f∈P(f)(z)-h(z) has at most k+l-1 distinct zeros(ignoring multiplicity) in D,where P(f)(z)=f(k)(z)+a1(z)f((k-1)(z)+…+ak(z)f(z) is a differential polynomial of f and aj(z)(j=1,2,···,k) are holomorphic functions in D,then F is normal in D.

Testing for Normality from the Parametric Seven-Number Summary

The objective of this study is to propose the Parametric Seven-Number Summary (PSNS) as a significance test for normality and to verify its accuracy and power in comparison with two well-known tests, such as Royston’s W test and D’Agostino-Belanger-D’Agostino K-squared test. An experiment with 384 conditions was simulated. The conditions were generated by crossing 24 sample sizes and 16 types of continuous distributions: one normal and 15 non-normal. The percentage of success in maintaining the null hypothesis of normality against normal samples and in rejecting the null hypothesis against non-normal samples (accuracy) was calculated. In addition, the type II error against normal samples and the statistical power against normal samples were computed. Comparisons of percentage and means were performed using Cochran’s Q-test, Friedman’s test, and repeated measures analysis of variance. With sample sizes of 150 or greater, high accuracy and mean power or type II error (≥0.70 and ≥0.80, respectively) were achieved. All three normality tests were similarly accurate;however, the PSNS-based test showed lower mean power than K-squared and W tests, especially against non-normal samples of symmetrical-platykurtic distributions, such as the uniform, semicircle, and arcsine distributions. It is concluded that the PSNS-based omnibus test is accurate and powerful for testing normality with samples of at least 150 observations.

Wigner-Matrix-Based Normality Test and Application to Weak Signal Detection in SISO/SIMO Systems

Based on the asymptotic spectral distribution of Wigner matrices, a new normality test method is proposed via reforming the white noise sequence. In this work, the asymptotic cumulative distribution function （CDF） of eigenvalues of the Wigner matrix is deduced. A numerical Kullback-Leibler divergence of the empiric-d spectral CDF based on test samples from the deduced asymptotic CDF is established, which is treated as the test statistic. For validating the superiority of our proposed normality test, we apply the method to weak SIPSK signal detection in the single-input single-output （SISO） system and the single-input multiple-output （SIMO） system. By comparing with other common normality tests and the existing signal detection methods, simulation results show that the proposed method is superior and robust.

PERIODIC POINTS AND NORMALITY CONCERNING MEROMORPHIC FUNCTIONS WITH MULTIPLICITY

In this article,two results concerning the periodic points and normality of meromorphic functions are obtained:(i)the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by let ting R(z)be a nonpolynomial rational function,and if all zeros and poles of R(z)—z are multiple,then Rk(z)has at least k+1 fixed points in the complex plane for each integer k≥2;(ii)a complete solution to the problem of normality of meromorphic functions with periodic points is given by let ting F be a family of meromorphic functions in a domain D,and let ting k≥2 be a positive integer.If,for each f∈F,all zeros and poles of f(z)-z are multiple,and its iteration fk has at most k distinct fixed points in D,then F is normal in D.Examples show that all of the conditions are the best possible.

Asymptotic Normality of Pseudo-LS Estimator of Error Variance in Partly Linear Autoregressive Models

Consider the model Yt = βYt-1＋g（Yt-2）＋εt for 3 〈 t 〈 T. Hereg is anunknown function, β is an unknown parameter, εt are i.i.d, random errors with mean 0 andvariance σ2 and the fourth moment α4, and α4 are independent of Y8 for all t ≥ 3 and s = 1, 2.Pseudo-LS estimators σ, σ2T α4τ and D2T of σ^2,α4 and Var（ε2↑3） are respectively constructedbased on piecewise polynomial approximator of g. The weak consistency of α4T and D2T are proved. The asymptotic normality of σ2T is given, i.e., √T（σ2T -σ^2）/DT converges indistribution to N（0, 1）. The result can be used to establish large sample interval estimatesof σ^2 or to make large sample tests for σ^2.

Asymptotic Normality of Multi-Dimension Quasi Maximum Likelihood Estimate in Generalized Linear Models withAdaptive Design

We study the quasi likelihood equation in Generalized Linear Models（GLM） with adaptive design ∑（i=1）^n xi（yi-h（x＇iβ））=0, where yi is a q=vector, and xi is a p×q random matrix. Under some assumptions, it is shown that the Quasi- Likelihood equation for the GLM has a solution which is asymptotic normal.

Normality and Its Variants on Fuzzy Isotone Spaces

The study of fuzzy sets is specifically designed to mathematically represent uncertainty and vagueness by assigning values of membership to objects that belong to a particular set. This notion has been broadly extended to other areas of topology where various topological concepts have been shown to hold on fuzzy topology. Some notions naturally extend to closure spaces without requiring a lot of modification of the underlying topological ideas. This work investigates the variants of normality on fuzzy isotone spaces.

Test of Normality of Waist Measurement Data of Young Male and Female Adults based on the Quantile - Quantile Plot

This paper investigates the normality of some real data set obtained from waist measurements of a group of 49 young adults. The quantile - quantile （Q-Q） plot and the analysis of correlation coefficients for the Q-Q plot is used to determine the normality or otherwise of the data set. In this regards, the probabilities of the quantiles were computed, modified and plotted. Thereafter the correlation coefficients for the quantile - quantile plots were obtained. Results indicate that at 0.1 level of significance, the data for young adult males of the sample were not normally distributed, and had a mean value that is within the range of low risk, healthwise, whereas the distribution of the data for young female adults showed reasonable normality, but also with a mean value that is within the range of low risk in terms of health condition.

On Rotational Robustness of Shapiro-Wilk Type Tests for Multivariate Normality

The Shapiro-Wilk test (SWT) for normality is well known for its competitive power against numerous one-dimensional alternatives. Several extensions of the SWT to multi-dimensions have also been proposed. This paper investigates the relative strength and rotational robustness of some SWT-based normality tests. In particular, the Royston’s H-test and the SWT-based test proposed by Villase?or-Alva and González-Estrada have R packages available for testing multivariate normality;thus they are user friendly but lack of rotational robustness compared to the test proposed by Fattorini. Numerical power comparison is provided for illustration along with some practical guidelines on the choice of these SWT-type tests in practice.

The Rate of Asymptotic Normality of Frequency Polygon Density Estimation for Spatial Random Fields

This paper is to investigate the convergence rate of asymptotic normality of frequency polygon estimation for density function under mixing random fields, which include strongly mixing condition and some weaker mixing conditions. A Berry-Esseen bound of frequency polygon is established and the convergence rates of asymptotic normality are derived. In particularly, for the optimal bin width , it is showed that the convergence rate of asymptotic normality reaches to ?when mixing coefficient tends to zero exponentially fast.

The Behavior of Normality when Iteratively Finding the Normal to a Line in an _lpGeometry

The normal direction to the normal direction to a line in Minkowski geometries generally does not give the original line. We show that in lp geometries with p>1?repeatedly finding the normal line through the origin gives sequences of lines that monotonically approach specific lines of symmetry of the unit circle. Which lines of symmetry that are approached depends upon the value of p and the slope of the initial line.

Emotions and Stressing Situations Adaptation of Nursing Students in the New Normality

Background: New normality is uncertain in every sense, specifically in education and for many health disciplines. Being immersed in COVID-19 pandemics brought serious consequences for mental health, and is very important to handle emotions and stress coping strategies to obtain positive outcomes. Objective: To identify the most frequent emotions, as well as the adaptation strategies to the new normality faced by the students of nursing. Methods: Qualitative and phenomenological research, with the participation of 20 students from both genders in the middle term of nursing career at Faculty of Higher Studies Iztacala, National Autonomous University of Mexico, from August to November 2021. Information was collected from a focal group for ten sessions;analysis was according to De Souza Minayo, and there was a signed informed consent letter from participants. Results: Four categories emerged with sub-categories. Category I Maximized emotions. Sub-categories: 1) Frustration, anger, disappointment;2) Personal disappointment, hopelessness, uncertainty;3) Depression. Category II Support elements close to the new normality. Sub-categories: 1) Family communication;2) Education for mental and physical health. Category III Stressing situations that exceeded the student. Sub-category: Disease in lovely ones. Category IV Stress coping strategies. Sub-categories: 1) Friends and relatives that help to get better;2) Family values. Informers pointed out to have maximized emotion, and having no self-control on its negative outcomes occurred;in addition, the situation was not favorable at home with several losses of loved ones, as well as a poor economy that threatened students to give up studies. Conclusion: Emotions facing this new normality are very important and should be attended to, their proper handling will result in a new learning of socio-emotional abilities, stress coping strategies development, better adaptation and informed decisions taken.