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??In this paper, the gamma distribution has been extended by adding an extra shape parameter, we refer to the new distribution as alpha power gamma distribution. It is found that the distribution has a relatively flexible hazard rate function. The properties of the new distribution are studied, including explicit expressions for the $s^{\text{th}}$ raw moments, moment generating function and distributions of order statistics are derived. Also, the integral expressions for the entropy, mean residual life and mean waiting time are obtained. The maximum likelihood estimators of the distribution parameters under complete sample are discussed, the Fisher information matrix is derived. Then, the estimation of the parameters under the general progressive type-II censoring is studied. Finally, the real data set is used to illustrate the practicality of the proposed distribution.     

The beta generalized Gompertz distribution

A new five-parameter continuous model called the beta generalized Gompertz distribution is introduced and studied. This distribution contains the Gompertz, generalized Gompertz, beta Gompertz, generalized exponential, beta generalized exponential, exponential and beta exponential distributions as special sub-models. Some mathematical properties of the new model are derived. We show that the density function of the new distribution can be expressed as a linear combination of Gompertz densities. We obtain explicit expressions for the moments, moment generating function, quantile function, density function of the order statistics and their moments, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. The model parameters are estimated by using the maximum likelihood method of estimation and the observed information matrix is determined. Finally, an application to real data set is given to illustrate the usefulness of the proposed model.     

A perturbed gamma degradation process with degradation dependent non‐Gaussian measurement errors

This paper proposes and illustrates a new perturbed gamma degradation process where the measurement error is modeled as a non‐Gaussian random variable that depends stochastically on the actual degradation level. The expression of the likelihood function for a generic set of noisy degradation measurements is derived, and the expression of the remaining useful life distribution of a degrading unit that fails when its degradation level exceeds a given threshold limit is formulated. A particle filter method is suggested, which allows one to compute the likelihood function and to estimate the remaining useful life distribution in a quick yet efficient manner. In addition, a closed‐form approximation of the perturbed gamma process is proposed to use in the special, yet meaningful, case where the standard deviation of the measurement error depends linearly on the actual degradation level. Finally, an applicative example is discussed, where the parameters of the perturbed gamma process, the remaining useful life distribution, and the mean remaining useful life of the degrading units are estimated from a set of noisy real degradation data.     

启动延迟N-策略批到达多重休假排队

目前N-策略批到达排队系统稳态队长分布难以给出解析解.提出一种新的递归算法研究顾客批到达,服务台延迟启动且多重休假的N-策略休假排队系统稳态队长分布.首先采用条件随机分解的方法得到稳态队长分布的概率母函数;然后采用递归算法推导附加队长分布的解析表达式;最后推导出稳态队长分布的递推关系式.     

Representations for the derivative at zero and finite parts of the Barnes zeta function

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes gamma function, including the particular case also known as multiple gamma function.     

Estimations from the exponentiated rayleigh distribution based on generalized Type-II hybrid censored data

In this article, unknown parameters of exponentiated Rayleigh distribution based on generalized Type II Hybrid censored data, survival function, failure rate function and coefficient of variation are derived by applying the maximum likelihood, Bayes and percentile bootstrap methods. Approximate confidence intervals for the unknown parameters, survival function, failure rate function and coefficient of variation are obtained. We study Bayes estimates under gamma priors distributions depending on symmetric and asymmetric loss functions via the Gibbs within Metropolis-Hasting samplers procedure. Finally, the proposed methods can be understood through illustrating the results of the real data analysis.     

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??The Bayes estimators of variance components are derived under weighted square loss function for the balanced one-way classification random effects model with the assumption that variance component has the conjugate prior distribution. The superiorities of the Bayes estimators for variance components to traditional ANOVA estimators are studied in terms of the mean square error (MSE) criterion. Finally, a remark for main results is given.     

Distributional Properties of Exceedance Statistics

Behaviour of a sequence of independent identically distributed random variables with respect to a random threshold is investigated. Three statistics connected with exceeding the threshold are introduced, their exact and asymptotic distributions are derived. Also distribution-free properties, leading to some common and some new discrete distributions, are considered. Identification of equidistribution of observations and the threshold are discussed. In this context relations between the exponential and gamma distributions are studied and a new derivation of the celebrated Laplace expansion for the standard normal distribution function is given.     

两个广义伽玛分布之间新的Kullback-Leibler距离

在Kullback-Leibler距离的基础上,对Kullback-Leibler距离进行改进,给出了新的Kullback-Leibler距离,并讨论了它的性质.计算了两个不同广义伽玛分布之间新的Kullback-Leibler距离.推导出伽玛分布、Weibull分布、Rayleigh分布、正态分布、指数分布新的Kullback-Leibler距离.另外在新的KullbackLeibler距离下,还得到digamma函数Ψ(x)=(Γ'(x)/(Γ(x))为单调递增函数.     

The exponentiated generalized inverse Gaussian distribution

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.     

Optimality and robustness of combinations of moving averages

A combination of moving averages has been shown previously to be more accurate than simple moving averages, under certain conditions, and to be more robust to non-optimal parameter specification. However, the use of the method depends on specification of three parameters: length of greater moving average, length of shorter moving average, and the weighting given to the former. In this paper, expressions are derived for the optimal values of the three parameters, under the conditions of a steady state model. These expressions reduce a three-parameter search to a single-parameter search. An expression is given for the variance of the sampling error of the optimal combination of moving averages and this is shown to be marginally greater than that for exponentially weighted moving averages (EWMA). Similar expressions for optimal parameters and the resultant variance are derived for equally weighted combinations. The sampling variance of the mean of such combinations is shown to be almost identical to the optimal general combination, thus simplifying the use of combinations further. It is demonstrated that equal weight combinations are more robust than EWMA to noise to signal ratios lower than expected, but less robust to noise to signal ratios higher than expected.     

Temperature distribution in a parallelopiped under the influence of heat source with boundary conditions of second kind

In the present paper, the general expressions for temperature distribution in an orthotropic plate of finite dimensions have been derived under the influence of an arbitrary volume heat source and arbitrary initial temperature distribution. The boundary conditions prescribed on the faces are of second kind. The solution is obtained by constructing an eigen value problem. The expression for unsteady temperature distribution consists of two parts—steady and pseudo-steady. The pseudo-steady parts of the solution have been obtained with the help of finite integral transform technique. The general expressions are further studied under different cases. The whole analysis is presented in the dimensionless form with the help of various dimensionless parameters,viz., F _o , P _o , K _i , etc.     

Reservoir storage with skew inflows

The storage problem for infinite reservoirs where annual inflows are distributed as independent gamma variables is treated in this paper. After having determined the distribution of the water content in the reservoir under consideration for a given period, an attempt was made to derive the expressions for the first four moments of the surplus and deficit, and then to approximate their distributions by the Type I curve using the Pearson criterion. The expected value of the range was also derived, and its variance was approximated by a linear equation of the reservoir life. Finally, the distribution of the range was approximated by the Type III curve.     

On progressively first failure censored Lindley distribution

This article deals with the progressively first failure censored Lindley distribution. Maximum likelihood and Bayes estimators of the parameter and reliability characteristics of Lindley distribution based on progressively first failure censored samples are derived. Asymptotic confidence intervals based on observed Fisher information and bootstrap confidence intervals of the parameter are constructed. Bayes estimators using non-informative and gamma informative priors are derived using importance sampling procedure and Metropolis–Hastings (MH) algorithm under squared error loss function. Also, HPD credible intervals based on importance sampling procedure and MH algorithm for the parameter are constructed. To study the performance of various estimators discussed in this article, a Monte Carlo simulation study is conducted. Finally, a real data set is studied for illustration purposes.     

Partial functional linear quantile regression

This paper studies estimation in partial functional linear quantile regression in which the dependent variable is related to both a vector of finite length and a function-valued random variable as predictor variables. The slope function is estimated by the functional principal component basis. The asymptotic distribution of the estimator of the vector of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. It is showed that this rate is optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function. The convergence rate of the mean squared prediction error for the proposed estimators is also be established. Finite sample properties of our procedures are studied through Monte Carlo simulations. A real data example about Berkeley growth data is used to illustrate our proposed methodology.     

A lifetime model with increasing failure rate

This paper deals with a new two-parameter lifetime distribution with increasing failure rate. This distribution is constructed as a distribution of a random sum of independent exponential random variables when the sample size has a zero truncated binomial distribution. Various statistical properties of the distribution are derived. We estimate the parameters by maximum likelihood and obtain the Fisher information matrix. Simulation studies show the performance of the estimators. Also, estimation of the parameters is considered in the presence of censoring. A real data set is analyzed for illustrative purposes and it is noted that the distribution is a good competitor to the gamma, Weibull, exponentiated exponential, weighted exponential and Poisson-exponential distributions for this data set.     

不同先验分布下的后验分布确定土力学参数

岩土工程中各土层参数的取值是根据现场及室内试验数据,采用经典统计学方法进行确定的,但这往往忽略了先验信息的作用.与经典统计学方法不同的是,Bayes法能从考虑先验分布的角度结合样本分布去推导后验分布,为岩土参数的取值提供一种新的分析方法.岩土工程勘察可视为对总体地层的随机抽样,当抽样完成时,样本分布密度函数是确定的,故Bayes法中的后验分布取决于先验分布,因此推导出两套不同的先验分布:利用先验信息确定先验分布及共轭先验分布.通过对先验及后验分布中超参数的计算,当样本总体符合N(μ,σ²)正态分布时,对所要研究的未知参数μ和σ展开分析,综合对比不同先验分布下后验分布的区间长度,给出岩土参数Bayes推断中最佳后验分布所要选择的先验分布.结果表明:共轭情况下的后验分布总是比无信息情况下的后验区间短,概率密度函数分布更集中,取值更方便.在正态总体情形下,根据未知参数μ和σ的联合后验分布求极值方法,确定样本总体中最大概率均值μ_max和方差σ_max作为工程设计采用值,为岩土参数取值方法提供了一条新的路径,有较好的工程意义.     

Comparative analysis of availability between three systems with general repair times, reboot delay and switching failures

This paper compares the availability characteristics between three different systems with reboot delay and standby switching failures. Three systems are studied under the assumption that the time-to-failure and the time-to-repair of the primary and standby units are exponentially and generally distributed, respectively. The reboot times are assumed to be exponentially distributed with parameter β. It is assumed that there is a significant probability q of a switching failure. Using the supplementary variable technique, we develop the explicit expressions for the steady-state availability, Av, for three configurations and perform comparative analysis for three various repair time distributions, such as exponential, gamma, and uniform. Under the cost/benefit criterion, comparisons are made based on assumed numerical values given to the distribution parameters, and to the cost of the primary and standby units.     

Assessment of the Maintenance of the Effectiveness of Safety Schemes by Inspection and Age-replacement of Protective Devices

Expressions are derived for quantities measuring deficienciesoccurring in the protective capabilities of safety schemes comprisedof devices which are subject to random failure. The devicesare assumed to be prone to two modes of failure, one which isreadily detectable in service, and one which is undetectablein service. The expressions take into account the effects ofpreventive maintenance procedures in which devices are inspectedat regularly spaced times to discover and renew devices whichhave failed "detectably", and also devices are replaced aftera specified age in service to mitigate the effects of "undetectable"failures. These expressions are evaluated for devices havingconstant failure rate (i.e. exponential failure distributions)for both modes of failure; and also for a constant "detectable"-failurerate and an increasing "undetectable"-failure rate (i.e. representedby a second order gamma function failure distribution).     

Exact fill rates for (<Emphasis Type="Italic">R,s, S</Emphasis>) inventory control with gamma distributed demand

For the familiar (R, s, S) inventory control system only approximate expressions exist for the fill rate, ie the fraction of demand that can be satisfied from stock. Best-known are the approximations derived from renewal theory, holding under specific conditions; in particular, S–s should be reasonably large. Here, an exact expression for the fill rate is derived, holding generally in the situation that demand has a gamma distribution with known integer-valued shape parameter, while lead time is constant. These exact results allow a check of the renewal theory based approximations. In addition, an extremely fast simulation program was written, obviously holding for general shape parameter values.