Rth Moment Research Articles

On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics

Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r < p(α) or E|X1|r < ∞ when r ≥ p(α). As an application we show that for every α < 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p < ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.

0,1] Truncated Lomax – Uniform distribution with properties

In this paper, we produce a new of continuous distribution based on [0,1] Truncated Lomax distribution, is called [0,1] Truncated Lomax – Uniform distribution. The cumulative distribution function, the rth moment, the mean, the variance, the skewness, the kurtosis, the median, the characteristic function, the reliability function, the hazared rate function, the quantile function, the Shannon entropy function and Relative entropy function has been derived for the distribution. We know that the an element fails when it exceeds the stress of the corresponding strength. therefor, the [0,1] Truncated Lomax-Uniform strength – stress model with different parameters has also been derived here. Finally, we proposed a study in Estimation for parameters of [0,1] Truncated Lomax – Uniform distribution by using maximum likelihood estimation method and we used simulation to generate random variables to take four experiments for the default values of three real parameters with samples sizes (n = 30, 60, 90, 120) and sample iteration (S = 1000).

Power Length-Biased Suja Distribution: Properties and Application

In this paper, a new distribution called weighted size biased two-parameterAkash distribution (WSBTPAD) is proposed. The WSBTPAD is a newmodication of the size biased two-parameter Akash distribution. The mainstatistical properties of the WSBTPAD are derived and proved. These prop-erties include the moments, particularly the rth moment, moment generatingfunction, harmonic mean, Bonferroni and Lorenz curves as well as the Giniindex. Also, the mean deviations of the population mean and median andthe Renyi entropy are presented. The reliability analysis of the random vari-able following WSBTPAD random variable are discussed. The method ofmaximum likelihood estimation is considered for estimating the parametersof the distribution. The distribution of order statistics from the WSBTPADare provided.

Generalized Modified Inverse Weibull Distribution: Its Properties and Applications

In this paper, we introduce a new useful continuous distribution called generalized modified inverse Weibull distribution. This distribution is a four-parameter extension of the modified inverse Weibull which generalizes some well-known distributions. Various statistical and probabilistic properties are derived such as rth moment, moment generating function, Renyi and Shannon entropies and hazard rate function. We also discuss estimation of the parameters by maximum likelihood and provide the information matrix. The likelihood ratio order (which implies the hazard rate and usual stochastic orders) between smallest order statistics from two independent heterogeneous samples of this new family are discussed. Finally, a real numerical example is also considered for illustrative purposes.

Weak Laws of Large Numbers for sequences of random variables with infinite rth moments

Let $${(X_n;n \geq 1)}$$ be a sequence of independent random variables with infinite rth absolute moments for some $${0 < r < 2}$$ . We investigate weak laws of large numbers for the weighted sum $${S_n = \sum_{j=1}^{m_n}c_{nj}X_j}$$ , where $${(c_{nj};1 \leq j \leq m_n,n \geq 1)}$$ is an array of real numbers. As illustrative examples, we obtain a weak law of large numbers of extended Pareto–Zipf distributions and generalized Feller Game. Furthermore, these results are applied to study weak law of large numbers of moving average sums of a sequence of i.i.d. random variables.

Moments about the mean of the size of a self-conjugate [formula omitted]-core partition

Johnson proved that if s,t are coprime integers, then the rth moment of the size of an (s,t)-core is a polynomial of degree 2r in t for fixed s. After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of S˜t∕St and t-cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an (s,t)-core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an (s,t)-core and the first nine moments about the mean of the size of an (s,s+1)-core using Maple. To get the moments about the mean of the size of a self-conjugate (s,t)-core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if s,t are coprime integers, then the rth moment about the mean of the size of a self-conjugate (s,t)-core is a quasipolynomial of period 2 and degree 2r in t for fixed odd s. Then, based on a bijection of Ford, Mai and Sze between self-conjugate (s,t)-cores and lattice paths in s2×t2 rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate (s,t)-cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate (s,t)-core.

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

ABSTRACTUndergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.

General distribution function for the superstatistics and new superstatistics with non-vanishing rth moments for even r

In this paper, the general procedure for obtaining the distribution for the superstatistics is presented. Besides, the new type of effective Boltzmann factor with non-vanishing [Formula: see text]th moments for even [Formula: see text] is presented and some examples are discussed.

0,1] Truncated Fréchet-Weibull and Fréchet Distributions

In this paper, we introduce a new family of continuous distributions based on [0, 1] Truncated Frechet distribution. [0, 1] Truncated Frechet Weibull ([0, 1] ) and [0, 1] Truncated Frechet ([0, 1] ) distributions are discussed as special cases. The cumulative distribution function, the rth moment, the mean, the variance, the skewness, the kurtosis, the mode, the median, the characteristic function, the reliability function and the hazard rate function are obtained for the distributions under consideration. It is well known that an item fails when a stress to which it is subjected exceeds the corresponding strength. In this sense, strength can be viewed as “resistance to failure.” Good design practice is such that the strength is always greater than the expected stress. The safety factor can be defined in terms of strength and stress as strength/stress. So, the [0, 1] strength-stress and the [0, 1] strength-stress models with different parameters will be derived here. The Shannon entropy and Relative entropy will be derived also.

Transmuted Kumaraswamy Quasi Lindley Distribution with Applications

The Lindley distribution is one of the widely used models for studying most of reliability modeling. Besides, several of researchers have motivated new classes of distributions based on modifications of the quasi Lindley distribution. In this article, a new version of generalized distributions named as the transmuted Kumaraswamy quasi Lindley (TKQL) is introduced. Various statistical properties of the TKQL distribution are provided. The rth moment of the TKQL distribution and its moment generating function are explored. Moreover, estimation of the model parameters is discussed via the method of maximum likelihood. Applications to real data are performed to clarify the flexibility of the TKQL distribution in comparison with some sub-models.

Kumaraswamy odd Burr G family of distributions with applications to reliability data

We exhibit a general family of distributions named “Kumaraswamy odd Burr G family of distributions” with four additional parameters to generalize any existing baseline distribution. Some statistical properties of the family are derived, including rth moments, mth incomplete moments, moment generating function and entropies. The parameters of the family are estimated by the maximum likelihood (ML) method for complete sam- ples as well as censored samples. Some sub-models of the family are considered and it is noted that their density functions can be symmetric, left-skewed, right-skewed, unimodal, bimodal and their hazard rate functions can be increasing, decreasing, bathtub, upside- down bathtub and J-shaped. Simulation is carried out for one of the sub-models to check the asymptotic behavior of the ML estimates. Applications to reliability (complete and censored) data are carried out to check the usefulness of some sub-models of the family.

The Marshall-Olkin Kappa distribution: Properties and applications

This study proposes a new competitive model, the Marshall-Olkin Kappa distribution and presents its various properties. These properties include the derivation of probability density function, quantiles, rth moment, mode, mean deviation and survival function. Results for the various inequality indices are obtained. Expression regarding order statistics is given. The unknown parameters of the proposed distribution are estimated using the maximum likelihood estimation method. Empirical illustration is provided using two real life data sets.

Modification of two Parameters Rayleigh into three Parameters one Through Exponentiated

The continuous probability Rayleigh distribution is one of the important Is an important distribution that can be used To analyze signal data and statistical error data as well as time to failure, so we work on modifying two parameters Rayleigh ( ) into three parameters one’s ( ) through exponentiated, the new obtained, also the formula for rth moments about origin is derived, to be used in estimating of parameters and also of reliability function. All derivation required were explained and the estimators by maximum likelihood and moments were obtained using different sets of initial values and the replicate of each experiment is ( ), the results are compared by using mean squared error (MSE).

0, 1] truncated fréchet-gamma and inverted gam-ma distributions

In this paper, we introduce a new family of continuous distributions based on [0, 1]] truncated Fréchet distribution. [0, 1]] Truncated Fréchet Gamma ([0, 1]] TFG) and truncated Fréchet inverted Gamma ([0, 1]] TFIG) distributions are discussed as special cases. The cumulative distribution function, the rth moment, the mean, the variance, the skewness, the kurtosis, the mode, the median, the characteristic function, the reliability function and the hazard rate function are obtained for the distributions under consideration. It is well known that an item fails when a stress to which it is subjected exceeds the corresponding strength. In this sense, strength can be viewed as "resistance to failure." Good design practice is such that the strength is always greater than the expected stress. The safety factor can be defined in terms of strength and stress as strength/stress. So, the [0, 1]] TFG strength-stress and the [0, 1]] TFIG strength-stress models with different parameters will be derived here. The Shannon entropy and Relative entropy will be derived also.

On the rates of convergence for moments convergence in regression models

In one-dimensional regression models, we establish a rate for the rth moment convergence (r⩾1) of the ordinary least-squares estimator involving explicitly the regressors, answering to an open question raised lately by Afendras and Markatou (Test 25:775–784, 2016). An extension of the classic Theorem 2.6.1 of Anderson (The statistical analysis of time series, Wiley, New York, 1971) is also presented.

Nonparametric and Gaussian bivariate transvariation theory: its applications to economics

This paper is concerned with the bivariate transvariation theory. It presents an historical account of the subject and a development of the theory. It deals with the probability of transvariation and its related concepts, namely, transvariability and its maximum; the rth moment of transvariation, its maximum and the rth intensity of transvariation; area of transvariation and discriminative value. These concepts are developed without parametric constraint and under the assumption of Gaussian distribution. The sample variances and covariances of the transvariation parameters estimators are herein deduced. The applications, performed on economic variables, underline the fruitfulness of transvariation theory as a quantitative method to deal with comparative statics analysis.

A note on the generalized linear exponential distribution

A paper on generalized linear exponential distribution (GLED) recently published by Mahmoud and Alam (2010) was found to contain several typos in their mathematical equations as well as an incorrect proof of Lemma 1. In this note, we provide corrections along with discussions to those mistakes and a numerical study to demonstrate that the formula for our proposed rth moment is accurate.

A generalized binomial exponential 2 distribution: modeling and applications to hydrologic events

ABSTRACTDeveloping statistical methods to model hydrologic events is always interesting for both statisticians and hydrologists, because of its importance in hydraulic structures design and water resource planning. Because of this, a flexible 3-parameter generalization of the exponential distribution is introduced based on the binomial exponential 2 (BE2) distribution [2]. The proposed distribution involving the exponential, gamma and BE2 distributions as submodels; and it exhibits decreasing, increasing and bathtub-shaped hazard rates, so it turns out to be quite flexible for analyzing non-negative real life data. Some statistical properties, parameters estimation and information matrix of the distribution are investigated. The proposed distribution, Gumbel, generalized Logistic and other distributions are utilized to model and fit two hydrologic data sets. The distribution is shown to be more appropriate to the data than the compared distributions using the selection criteria: average scaled absolute error, Akaike information criterion, Bayesian information criterion and Kolmogorov–Smirnov statistics. As a result, some hydrologic parameters of the data are obtained such as return level, conditional mean, mean deviation about the return level and the rth moments of order statistics.

The beta Burr type X distribution properties with application.

We develop a new continuous distribution called the beta-Burr type X distribution that extends the Burr type X distribution. The properties provide a comprehensive mathematical treatment of this distribution. Further more, various structural properties of the new distribution are derived, that includes moment generating function and the rth moment thus generalizing some results in the literature. We also obtain expressions for the density, moment generating function and rth moment of the order statistics. We consider the maximum likelihood estimation to estimate the parameters. Additionally, the asymptotic confidence intervals for the parameters are derived from the Fisher information matrix. Finally, simulation study is carried at under varying sample size to assess the performance of this model. Illustration the real dataset indicates that this new distribution can serve as a good alternative model to model positive real data in many areas.

A New Bivariate Distribution with Generalized Quadratic Hazard Rate Marginals

Recently, Sarhan (2009) introduced a new distribution named generalized quadratic hazard rate distribution. In this study, we define a bivariate Generalized Quadratic Hazard Rate Distribution (BGQHRD). The joint cumulative distribution function and joint survival function are derived in compact forms. Several properties of BGQFRD have been discussed. The conditional probability density function, rth moments and joint and marginal moment generating functions are obtained. Parameters estimators using the maximum likelihood method are obtained. A numerical illustration is used to obtain maximum likelihood estimators (MLEs). Moreover, we study the behavior of the estimators numerically.