Generalized function

In this paper, we present control designs for single-input single-output (SISO) nonlinear systems in strict feedback form with an output constraint. To prevent constraint violation, we employ a Barrier Lyapunov Function, which grows to infinity when its arguments approach some limits. By ensuring boundedness of the Barrier Lyapunov Function in the closed loop, we ensure that those limits are not transgressed. Besides the nominal case where full knowledge of the plant is available, we also tackle scenarios wherein parametric uncertainties are present. Asymptotic tracking is achieved without violation of the constraint, and all closed loop signals remain bounded, under a mild condition on the initial output. Furthermore, we explore the use of an Asymmetric Barrier Lyapunov Function as a generalized approach that relaxes the requirements on the initial conditions. We also compare our control with one that is based on a Quadratic Lyapunov Function, and we show that our control requires less restrictive initial conditions. A numerical example is provided to illustrate the performance of the proposed control.

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Based on a generalized Mexican-hat function proposed, the paper presents an algorithm for the design of a new class of continuous wavelets matched to arbitrary transient signals. While the generalized Mexican-hat wavelets are constructed from the standard Mexican-hat function by utilizing a polynomial series with the admissible condition imposed, the signal matching characteristics are achieved by minimizing the spectral differences between the reference signal and the generalized Mexican-hat wavelet via even and odd symmetry decomposition. Detail derivations are given to yield a set of linear simultaneous equations for determination of the appropriate polynomial coefficient values for the generalized Mexican-hat wavelet, thereby shaping its waveform to resemble the reference transient signal. The solution for the equations is shown to be given by a combination of analytical expressions and numerical approximation. Also included in the paper are computer simulations and practical examples to demonstrate the signal matching quality and the influence of various design parameters. Comparisons are made with the standard Mexican-hat wavelet and correlation methods to reveal the superior performance of the generalized and matched Mexican-hat wavelet, in terms of enhancement of signal features in the timescale domain, and reliable detection of transient signals and their variants embedded in noise.

Many different robust estimation approaches for the covariance or shape matrix of multivariate data have been established until today. Tyler's M-estimator has been recognized as the 'most robust' M-estimator for the shape matrix of elliptically symmetric distributed data. Tyler's Mestimators for location and shape are generalized by taking account of incomplete data. It is shown that the shape matrix estimator remains distribution-free under the class of generalized elliptical distributions. Its asymptotic distribution is also derived and a fast algorithm, which works well even for high-dimensional data, is presented. A simulation study with clean and contaminated data covers the complete-data as well as the incomplete-data case, where the missing data are assumed to be MCAR, MAR, and NMAR.

We introduce an autoregressive process called generalized normal-Laplace autoregressive process with generalized normal-Laplace distribution [Reed, W. J., 2007. Brownian-Laplace motion and its use in financial modelling. Comm. Statist. Theory Methods, 36, 473-484], as stationary marginal distribution. Various properties of the distribution and the processes are discussed. The innovation structure is derived and estimation of parameters is addressed. Sample path behaviour, distribution of sums and the joint distribution of contiguous observations, etc. are studied. An algorithm for the generation of the process is also given as appendix.

(1.1) where is the generalized hypergeometric series (cf. Mathai and Saxena, 1973; and Slater, 1966), in which the as, bs and z are assumed to be appropriate reals and the domain of Q(t) is an open interval containing the region of convergence of . Dacey (1972) has ...

Weibull mixtures have been considered in many applied problems, and they have also been generalized by allowing negative mixing weights. In this paper, we study the classification of the aging properties of generalized mixtures of two or three Weibull distributions in terms of the mixing weights, scale parameters and a common shape parameter, which extends the cases of exponential or Rayleigh distributions. We apply these general results to classify the aging properties of the minimum and maximum lifetimes of two-component systems whose component lifetimes follow one of the known bivariate Weibull distributions.

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain nearexact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.

We consider the generalized additive model when responses from the same cluster are correlated. Incorporating correlation in the estimation of nonparametric components for the generalized additive model is important since it improves estimation efficiency and increases statistical power for model selection. In our setting, there is no specified likelihood function for the generalized additive model since the outcomes could be non-normal and discrete, which makes estimation and model selection very challenging problems. We propose consistent estimation and model selection which incorporate the correlation structure. We establish an asymptotic property with L 2-norm consistency for the nonparametric components, which achieves the optimal rate of convergence. In addition, the proposed model selection strategy is able to select the correct generalized additive model consistently. That is, with probability approaching to 1, the estimators for the zero function components converge to 0 almost surely. We will illustrate our method using numerical studies with both continuous and binary responses, and a real data application of binary periodontal data. Supplemental materials including technical details are available online.

In this paper, an anisotropic diffusion model with a generalized diffusion coefficient function is presented for defect detection in low-contrast surface images and, especially, aims at material surfaces found in liquid crystal display (LCD) manufacturing. A defect embedded in a low-contrast surface image is extremely difficult to detect, because the intensity difference between the unevenly illuminated background and the defective region is hardly observable and no clear edges are present between the defect and its surroundings. The proposed anisotropic diffusion model provides a generalized diffusion mechanism that can flexibly change the curve of the diffusion coefficient function. It adaptively carries out a smoothing process for faultless areas and performs a sharpening process for defect areas in an image. An entropy criterion is proposed as the performance measure of the diffused image and then a stochastic evolutionary computation algorithm, particle swarm optimization (PSO), is applied to automatically determine the best parameter values of the generalized diffusion coefficient function. Experimental results have shown that the proposed method can effectively and efficiently detect small defects in various low-contrast surface images.

Some of the most powerful techniques currently available to test the goodness of fit of a hypothesized continuous cumulative distribution function (CDF) use statistics based on the empirical distribution function (EDF), such as those of Kolmogorov, Cramer-von Mises and Anderson-Darling, among others. The use of EDF statistics was analyzed for estimation purposes. In this approach, maximum goodness-of-fit estimators (also called minimum distance estimators) of the parameters of the CDF can be obtained by minimizing any of the EDF statistics with respect to the unknown parameters. The results showed that there is no unique EDF statistic that can be considered most efficient for all situations. Consequently, the possibility of defining new EDF statistics is entertained; in particular, an Anderson-Darling statistic of degree two and one-sided Anderson-Darling statistics of degree one and two appear to be notable in some situations. The procedure is shown to be able to deal successfully with the estimation of the parameters of homogeneous and heterogeneous generalized Pareto distributions, even when maximum likelihood and other estimation methods fail.

The objective of this paper is to integrate the generalized gamma ðGGÞ distribution into the information theoretic literature. We study information properties of the GG distribution and provide an assortment of information measures for the GG family, which includes the exponential, gamma, Weibull, and generalized normal distributions as its subfamilies. The measures include entropy representations of the log-likelihood ratio, AIC, and BIC, discriminating information between GG and its subfamilies, a minimum discriminating information function, power transformation information, and a maximum entropy index of fit to histogram. We provide the full parametric Bayesian inference for the discrimination information measures. We also provide Bayesian inference for the fit of GG model to histogram, using a semi-parametric Bayesian procedure, referred to as the maximum entropy Dirichlet (MED). The GG information measures are computed for duration of unemployment and duration of CEO tenure.

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^∞-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.

The two-parameter linear failure rate distribution has been used quite successfully to analyze lifetime data. Recently, a new three-parameter distribution, known as the generalized linear failure rate distribution has been introduced by exponentiating the linear failure rate distribution. The generalized linear failure rate distribution is a very flexible lifetime distribution, and the probability density function of the generalized linear failure rate distribution can take different shapes. Its hazard function also can be increasing, decreasing and bathtub shaped. The main aim of this paper is to introduce a bivariate generalized linear failure rate distribution, whose marginals are generalized linear failure rate distributions. It is obtained using the same approach as the Marshall-Olkin bivariate exponential distribution. Different properties of this new distribution are established. The bivariate generalized linear failure rate distribution has five parameters and the maximum likelihood estimators are obtained using the EM algorithm. A data set is analyzed for illustrative purposes. Finally, some generalizations to the multivariate case are proposed.

The modeling and analysis of lifetime data is an important aspect of statistical work in a wide variety of scientific and technological fields. Good (1953) introduced a probability distribution which is commonly used in the analysis of lifetime data. For the first time, based on this distribution, we propose the so-called exponentiated generalized inverse Gaussian distribution, which extends the exponentiated standard gamma distribution (Nadarajah and Kotz, 2006). Various structural properties of the new distribution are derived, including expansions for its moments, moment generating function, moments of the order statistics, and so forth. We discuss maximum likelihood estimation of the model parameters. The usefulness of the new model is illustrated by means of a real data set.

A critical issue in modelling binary response data is the choice of the links. We introduce a new link based on the generalized t-distribution. There are two parameters in the generalized t-link: one parameter purely controls the heaviness of the tails of the link and the second parameter controls the scale of the link. Two major advantages are offered by the generalized t-links. First, a symmetric generalized t-link with an unknown shape parameter is much more identifiable than a Student t-link with unknown degrees of freedom and a known scale parameter. Secondly, skewed generalized t-links with both unknown shape and scale parameters provide much more flexible and improved skewed link regression models than the existing skewed links. Various theoretical properties and attractive features of the proposed links are examined and explored in detail. An efficient Markov chain Monte Carlo algorithm is developed for sampling from the posterior distribution. The deviance information criterion measure is used for guiding the choice of links. The proposed methodology is motivated and illustrated by prostate cancer data.

Situations are considered in which a physical quantity varies rapidly in space from positive values to negative values down to some cutoff scale. Typically the cutoff scale is determined by some diffusive process (e.g., viscosity), and, as the cutoff scale decreases (e.g., viscosity decreases), the physical quantity as a function of the spatial coordinate varies more and more rapidly, approaching a generalized function as the cutoff scale approaches zero. As one examines such a generalized function on smaller and smaller scale, it is shown that a certain suitably defined quantity sensitive to the intensity of spatial oscillations from positive to negative can exhibit an algebraic divergence. The exponent of this algebraic divergence with scale size is one characterization of this situation, and is called the cancellation exponent. In the spirit of the multifractal formalism, we introduce a spectrum of cancellation exponents yielding a more complete characterization. Partition function and thermodynamic formalisms are employed. We also discuss how to associate Dq spectra with these signed generalized functions and discuss the relationship of Dq to the cancellation exponent spectrum. Illustrative examples are presented.

Weibull mixtures have been considered in many applied problems, and they have also been generalized by allowing negative mixing weights. In this paper, we study the classification of the aging properties of generalized mixtures of two or three Weibull distributions in terms of the mixing weights, scale parameters and a common shape parameter, which extends the cases of exponential or Rayleigh distributions. We apply these general results to classify the aging properties of the minimum and maximum lifetimes of two-component systems whose component lifetimes follow one of the known bivariate Weibull distributions.

In connection with the classical Schwartz kernel theorem, we show that in the framework of Colombeau generalized functions a large class of linear mappings admit integral kernels. To do this, we need to introduce news spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that in some sense Schwartz' result is contained in our main theorem.

We study the joint distribution of (X, Y, N), where N has a geometric distribution while X and Y are, respectively, the sum and the maximum of N i.i.d. exponential random variables. We present fundamental properties of this class of mixed trivariate distributions, and discuss their applications. Our results include explicit formulas for the marginal and conditional distributions, joint integral transforms, moments and related parameters, stability properties, and stochastic representations. We also derive maximum likelihood estimators for the parameters of this distribution, along with their asymptotic properties, and briefly discuss certain generalizations of this model. An example from finance, where N represents the duration of the growth period of the daily log-returns of currency exchange rates, illustrates the modeling potential of this model. & 2011 Elsevier B.V. All rights reserved. , ð1:3Þ where 3 denotes the maximum of the fE i g. We shall refer to this distribution as the TETLG distribution with parameters b40 and p 2 ð0,1Þ, which stands for t rivariate distribution with e xponential, t runcated l ogistic, and g eometric marginals. Note that the geometric sum X of the IID exponential variables fE i g has an exponential distribution with Contents lists available at ScienceDirect

The generalized Waring distribution is a discrete distribution with a wide spectrum of applications in areas such as accident statistics, income analysis, environmental statistics, etc. It has been used as a model that better describes such practical situations as opposed to the Poisson distribution or the negative binomial distribution. Associated to both the Poisson and the negative binomial distributions are the well-known Poisson and Pólya processes. In this paper, the generalized Waring process is defined. Two models have been shown to lead to the generalized Waring process. One is related to a Cox process, while the other is a compound Poisson process. The defined generalized Waring process is shown to be a stationary, but non-homogenous Markov process. Several properties are studied and the intensity, the individual intensity and the Chapman-Kolmogorov differential equations of it are obtained. Moreover, the Poisson and the Pólya processes are shown to arise as special cases of the generalized Waring process. Using this fact, some known results and some properties of them are obtained.

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.

Inuence diagrams have become a popular tool for representing and solving complex decision-making problems under uncertainty. In this paper, we focus on the task of building probability models from expert knowledge, and also on the challenging and less known task of constructing utility models in inuence diagrams. Our goal is to review the state of the art and list some challenges. Similarly to probability models, which are embedded in inuence diagrams as a Bayesian network, preferential/utility independence conditions can be used to factor the joint utility function into small factors and reduce the number of parameters needed to fully dene the joint function. A number of graphical models have been recently proposed to factor the joint utility function, including the generalized additive independence networks, ceteris paribus networks, utility ceteris paribus networks, expected utility networks, and utility diagrams. Similarly to probability models, utility models can also be engineered from a domain expert or induced from data.

Inuence diagrams have become a popular tool for representing and solving complex decision-making problems under uncertainty. In this paper, we focus on the task of building probability models from expert knowledge, and also on the challenging and less known task of constructing utility models in inuence diagrams. Our goal is to review the state of the art and list some challenges. Similarly to probability models, which are embedded in inuence diagrams as a Bayesian network, preferential/utility independence conditions can be used to factor the joint utility function into small factors and reduce the number of parameters needed to fully dene the joint function. A number of graphical models have been recently proposed to factor the joint utility function, including the generalized additive independence networks, ceteris paribus networks, utility ceteris paribus networks, expected utility networks, and utility diagrams. Similarly to probability models, utility models can also be engineered from a domain expert or induced from data.

Independent random variables X l1 ,. .. ,X ln are said to belong to the scale family of distributions if X l i $ Fðl i xÞ, for i =1,y,n, where F is an absolutely continuous distribution function with hazard rate r and reverse hazard rater. We show that the hazard rate (reverse hazard rate) of a series (parallel) system consisting of components with lifetimes X l1 ,. .. ,X ln is Schur concave (convex) with respect to the vector k, if x 2 r 0 ðxÞ ðx 2r 0 ðxÞÞ is decreasing (increasing). We also show that if xr(x) is increasing in x, then the survival function of the parallel system is increasing in the vector k with respect to p-larger order, an order weaker than majorization. We prove that all these new results hold for the scaled generalized gamma family as well as the power-generalized Weibull family of distributions. We also show that in the case of generalized gamma and power generalized Weibull distribution, under some conditions on the shape parameters, the vector of order statistics corresponding to X l i 's is stochastically increasing in the vector k with respect to majorization thus generalizing the main results in Sun and Zhang (2005) and Khaledi and Kochar (2006).

The history of problems of evaluation of series associated with the Riemann Zeta function can be traced back to Christian Goldbach (1690-1764) and Leonhard Euler (1707-1783). Many different techniques to evaluate various series involving the Zeta and related functions have since then been developed. The authors show how elegantly certain families of series involving the Hurwitz-Lerch Zeta function can be evaluated by starting with a single known identity for the Hurwitz-Lerch Zeta function. Some of the special cases of these series identities are also shown to lead to certain known families of summation formulas involving the Hurwitz Zeta function.

Inference on the largest mean of a multivariate normal distribution is a surprisingly difficult and unexplored topic. Difficulties arise when two or more of the means are simultaneously the largest mean. Our proposed solution is based on an extension of R.A. Fisher's fiducial inference methods termed generalized fiducial inference. We use a model selection technique along with the generalized fiducial distribution to allow for equal largest means and alleviate the overestimation that commonly occurs. Our proposed confidence intervals for the largest mean have asymptotically correct frequentist coverage and simulation results suggest that they possess promising small sample empirical properties. In addition to the theoretical calculations and simulations we also applied this approach to the air quality index of the four largest cities in the northeastern United States

A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of exponential weights. Such an algebra with embedded Dirac's delta distribution induces discrete topology on the basic space. This result is in analogy to Schwartz' impossibility result concerning multiplication of distributions.

... 208, 336–346. New York: Marcel Dekker, Inc.. View all references, it is often assumed that the base survival function is of the Weibull type, thus the model becomes: ... Cantú, M., Arnold, BC and Villaseñor, JA 1998. A distributional study of lifetime data from longitudinal specimens. ...

Recently two-parameter generalized exponential distribution has been introduced by the authors. In this paper we consider the Bayes estimators of the unknown param- eters under the assumptions of gamma priors on both the shape and scale parameters. The Bayes estimators can not be obtained in explicit forms. Approximate Bayes es- timators are computed using the idea of Lindley. We also

Time to failure due to fatigue is one of the common quality characteristics in material engineering applications. In this article, acceptance sampling plans are developed for the Birnbaum–Saunders distribution percentiles when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the specified life percentile is obtained under a given customer's risk. The operating

A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of exponential weights. Such an algebra with embedded Dirac's delta distribution induces discrete topology on the basic space. This result is in analogy to Schwartz' impossibility result concerning multiplication of distributions.

The authors present a systematic investigation of the following log-gamma integral: ∫0zlogΓ(t+1)dt and of its several related integral formulas. Relevant connections among the various mathematical constants involved naturally in the evaluation of the proposed integral are pointed out. Some approximate numerical values of the derivative ζ′(−1,a) of the Hurwitz zeta function are also considered. Importance of such derivatives as ζ′(−1,a)

The statistical meaning of higher D ðpÞ N ðp ¼ 1; 2;. . .Þ and fractional D ðpÞ N ð0opo1Þ moments for an arbitrary random sequence of the length N has been found. The higher moments help to reduce the sequence analyzed to a finite set of k statistically stable parameters, keeping invariant the values of the first kth moments D ðpÞ k ðp ¼ 1; 2;. .. ; kÞ. The conditions of statistical stability and proximity expressed in terms of higher moments D ðpÞ N ¼ D ðpÞ Nþk ðp ¼ 1; 2;. .. ; kÞ help to find k unknown stable points and predict possible future behavior of the random sequence analyzed. The generalized mean value (GMV)-function defined as G ðpÞ N ¼ ðD ðpÞ N Þ 1=p is turned to be very effective in analysis of statistically close random sequences or containing large numbers of measured points ðNb1Þ. The approximate analytical expression for an arbitrary p value from the range (À1opo1) entering into G ðpÞ N has been found. It gives a possibility to transform any random sequence to the determined GMV curve and express quantitatively the reduced characteristics of any random sequence in terms of a 'universal' set of the fitting parameters defined by the determined GMV-function. Statistical proximity factor can be used for construction of calibration curves, when it is necessary to compare one random sequence with another one to respect of variations of some given external factor (small signal). The higher moments are easily generalized for the fractional and even complex moments. In turn, the GMV-function can be also generalized and then calculated for 2D and 3D random sequences. The approach developed in this paper is free from any model assumption and can be extremely helpful in comparison of different random sequences using for these purposes the 'unified' quantitative language based on introduction of the given set of fractional moments. The relationship between the value of the fractional moment and nonextensive parameter q entering into the definition of the nonextensive Tsallis entropy has been found. A possible model of statistical protection of plastic cards and other valuable documents demonstrating the effectiveness of the statistics of fractional moments has been considered. Some instructive examples in detection of superweak signals embedded into the basic random sequence (S=N ¼ 10 À2 ; 10 À3) based on model and real data confirm the effectiveness of new approach and can serve a new basis for numerous practical applications. Analysis of dielectric spectroscopy data by means of fractional moments gives unique possibility to compare quantitatively each measurement with each other and express the influence of a neutral additive in terms of calibration curve without concrete knowledge of the corresponding fitting function.

In the study of the stochastic behaviour of the lifetime of an element as a function of its length, it is often observed that the failure time (or lifetime) decreases as the length increases. In probabilistic terms, such an idea can be expressed as follows. Let T be the lifetime of a specimen of length x, so the survival function,

The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is extended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong's technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the K and the Hankel transforms.

We study the regularity of the viscosity solution of a quasilinear parabolic partial differential equation with Lipschitz coefficients by using its connection with a forward backward stochastic differential equation (in short FBSDE) and we give a probabilistic representation of the generalized gradient (derivative in the distribution sense) of the viscosity solution. This representation is a kind of nonlinear Feynman-Kac formula. The main idea is to show that the FBSDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. If the diffusion coefficient of the forward equation is uniformly elliptic, we approximate the FBSDE by smooth ones and use Krylov's estimate to prove the convergence of the derivatives. In the degenerate case, we use techniques of Bouleau-Hirsch on absolute continuity of probability measures.

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non-increasing hazard function also. This paper introduces a new distribution, which generalizes the well known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this paper. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well known distributions.

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.

ABSTRACT The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and StatisticsMedhi Festschrift, A. J., on the Occasion of his70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated.

We consider estimation of the ratio of arbitrary powers of two normal generalized variances based on two correlated random samples. First, the result of Iliopoulos [Decision theoretic estimation of the ratio of variances in a bivariate normal distribution, Ann. Inst. Statist. Math. 53 -446] on UMVU estimation of the ratio of variances in a bivariate normal distribution is extended to the case of the ratio of any powers of the two variances. Motivated by these estimators' forms we derive the UMVU estimator in the multivariate case. We show that it is proportional to the ratio of the corresponding powers of the two sample generalized variances multiplied by a function of the sample canonical correlations. The mean squared errors of the derived UMVU estimator and the maximum likelihood estimator are compared via simulation for some special cases.

The modeling and analysis of lifetime data is an important aspect of statistical work in a wide variety of scientific and technological fields. Good (1953) introduced a probability distribution which is commonly used in the analysis of lifetime data. For the first time, based on this distribution, we propose the so-called exponentiated generalized inverse Gaussian distribution, which extends the exponentiated standard gamma distribution . Various structural properties of the new distribution are derived, including expansions for its moments, moment generating function, moments of the order statistics, and so forth. We discuss maximum likelihood estimation of the model parameters. The usefulness of the new model is illustrated by means of a real data set.

Many processes can be represented in a simple form as infinite-order linear series. In such cases, an approximate model is often derived as a truncation of the infinite-order process, for estimation on the finite sample. The literature contains a number of asymptotic distributional results for least squares estimation of such finite truncations, but for quantile estimation, results are not available at a level of generality that accommodates time series models used as finite approximations to processes of potentially unbounded order. Here we establish consistency and asymptotic normality for conditional quantile estimation of truncations of such infinite-order linear models, with the truncation order increasing in sample size. We focus on estimation of the model at a given quantile. The proofs use the generalized functions approach and allow for a wide range of time series models as well as other forms of regression model. The results are illustrated with both analytical and simulation examples.