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System signatures have transitioned from an emerging topic to a growing research area. It has become essential to classify different research directions and examine the general trends of this promi...

In the literature on Statistical Reliability Theory and Stochastic Orders, several results based on theory of TP2/RR2 functions have been extensively used in establishing various properties. In this paper, we provide a review of some useful results in this direction and highlight connections between them

Copula models have become one of the most popular tools, especially in finance and insurance, for modeling multivariate distributions in the past few decades, and they have recently received increa...

Copula models have attracted significant attention in the recent literature for modeling multivariate observations. An essential feature of copulas is that they enable us to specify the univariate marginal distributions and their joint behaviors separately. This paper provides asymptotic results for misspecification of copula models, and examines the consequences and detection of misspecification in copula models under constant-stress accelerated life-tests for one-shot devices. The one-shot device is considered as a two-component system. The reliability is an important factor in lifetime data applications, and we focus on the effect of misspecification on the estimation of the reliability of one-shot devices. Moreover, the Akaike information criterion is used as a specification test for copula model validation. A simulation study is carried out to evaluate the effect of misspecification under the Gumbel-Hougaard, the Frank and the Clayton copulas incorporated with Weibull and gamma distributions as marginal distributions, in terms of asymptotic bias, asymptotic relative bias, and root mean square error.

In this paper, we discuss the optimal allocation problem in a multi-level accelerated life testing experiment under time (Type-I) censoring when an extreme-value regression model is used for statistical analysis. We derive the expected Fisher information and the asymptotic variance-covariance matrix of the maximum likelihood estimators. Three optimality criteria are considered and the corresponding optimal allocations are determined. Under Type-I censoring, because the optimal allocations are depending on the model parameters, the sensitivity of the optimal allocations due to mis-specification of the model parameters is studied. A numerical example is used to illustrate the methodologies developed in this paper.

Computing the system signature is an attractive but challenging problem in system reliability. In this article, we propose a novel algorithm to compute the signature of a system with exchangeable c...

The computation of the signature of a complex system is often challenging, as it may involve a large number of components and a complex architecture. In this study, we propose a novel approach to computing the signature of a system consisting of subsystems with shared components. The proposed approach relies on a new concept called decomposed survival signatures. This new concept can be efficiently used to compute the signature of a modular system with an arbitrary organizing structure and to provide explicit formulas for computation. Applications in cyber systems, transportation systems, and branched oil pipeline systems are highlighted.

In the design of constant-stress life-testing experiments, the optimal allocation in a multi-level stress test with Type-I or Type-II censoring based on the Weibull regression model has been studied in the literature. Conventional Type-I and Type-II censoring schemes restrict our ability to observe extreme failures in the experiment and these extreme failures are important in the estimation of upper quantiles and understanding of the tail behaviors of the lifetime distribution. For this reason, we propose the use of progressive extremal censoring at each stress level, whereas the conventional Type-II censoring is a special case. The proposed experimental scheme allows some extreme failures to be observed. The maximum likelihood estimators of the model parameters, the Fisher information, and asymptotic variance–covariance matrices of the maximum likelihood estimates are derived. We consider the optimal experimental planning problem by looking at four different optimality criteria. To avoid the computational burden in searching for the optimal allocation, a simple search procedure is suggested. Optimal allocation of units for two- and four-stress-level situations is determined numerically. The asymptotic Fisher information matrix and the asymptotic optimal allocation problem are also studied and the results are compared with optimal allocations with specified sample sizes. Finally, conclusions and some practical recommendations are provided.

The failure of any component within a series system results in the failure of the system. Incorporating active redundancy into a system generally improves its reliability and availability. This type of system does not belong to the category of coherent systems, and thus those results on coherent systems are not applicable. Hence, it is of great interest to develop statistical methodologies that perform well for inference on reliability of redundancy systems. In a life-test for multicomponent systems, component lifetimes may not be observable, but one may observe a set of components that failed along with the system. The data available in this form are called autopsy data, and, in this case, lifetime information on some of the components is missing. In this paper, we consider an equal load-sharing model and develop an expectation-maximization algorithm for estimating system reliability characteristics based on such an autopsy data. The load-sharing parameter indicates whether incorporating active redundancy improves system reliability or induces rapid failure of the system. The performance of the proposed methodology is then evaluated through Monte Carlo simulations and then illustrated with two numerical examples.

In this article, we discuss the optimal allocation problem in a multi-level accelerated life-testing experiment under Type-I censoring when an exponential regression model is used for the analysis. We derive the expected Fisher information matrix and use it to obtain the asymptotic variance–covariance matrix of the maximum likelihood estimators (MLEs). We then consider the optimal allocation under the D-optimality criterion, and present an algorithm for determining the optimal allocation. A numerical example is presented for the purpose of illustration. The optimal allocation depends on the model parameters and so a sensitivity analysis of the optimal allocation to misspecification of the model parameters is carried out as well.

We provide simple computational formulas of both expected termination time and Fisher information of the flexible progressive censoring scheme proposed by Bairamov and Parsi (2011). Then, the design and planning of the flexible progressive censoring schemes are discussed with illustrative examples.

Hybrid censoring schemes are commonly used in life-testing experiments to reduce the experimental time and the cost. A Type-II progressive hybrid censoring scheme (PHCS) was introduced by Kundu and Joarder (Comput Stat Data Anal 50:2509–2528, 2006) that combines progressive Type-II censoring and Type-I censoring. In this paper, we consider the statistical inference of a two-parameter exponential distribution under the Type-II PHCS. The conditional maximum likelihood estimates (MLEs) of the model parameters and their joint and marginal conditional moment generating functions are derived. Based on these exact conditional moments, bias-reduced estimators are proposed and their distributions are discussed. Confidence intervals of the model parameters based on exact and asymptotic distributions of the MLEs and bias-reduced estimators are developed. The performances of the point and interval estimation procedures are evaluated and compared through exact calculations and Monte Carlo simulations. Recommendations are made based on these results and an illustrative example is presented.

In this article, we discuss the optimal allocation problem in an experiment when a regression model is used for statistical analysis. Monotonic convergence for a general class of multiplicative algorithms for $D$-optimality has been discussed in the literature. Here, we provide an alternate proof of the monotonic convergence for $D$-criterion with a simple computational algorithm and furthermore show it converges to the $D$-optimality. We also discuss an algorithm as well as a conjecture of the monotonic convergence for $A$-criterion. Monte Carlo simulations are used to demonstrate the reliability, efficiency and usefulness of the proposed algorithms., Comment: 17 pages and 2 tables, accepted by Journal of Computational and Applied Mathematics in 2013 Journal of Computational and Applied Mathematics 2013

Allocation of active standby] redundancies in a system is a topic of great interest in reliability engineering and system safety because optimal configurations can significantly increase the reliability of a system. In this paper, we study the problem of allocating two exponentially distributed active standby] redundancies in a two-component series system using the tools of stochastic ordering. We establish two interesting results on likelihood ratio ordering which have no restriction on the parameters.

The allocation of redundancies in a system to optimize the reliability of system performance is an interesting problem in reliability engineering and system security. In this article, we focus on the optimal allocation of two exponentially distributed active (standby) redundancies in a two-component series system using the tool of stochastic ordering. For the case of active redundancy, stochastic comparisons are carried out in terms of the likelihood ratio and reversed hazard rate orders. For the case of standby redundancy, a likelihood ratio ordering result is developed. The results established here generalize and strengthen corresponding results in the recent literature. In addition, several numerical examples are used to explicate the results. (c) 2013 Wiley Periodicals, Inc.

Adaptive Type-II progressive censoring schemes have been shown to be useful in striking a balance between statistical estimation efficiency and the time spent on a life-testing experiment. In this article, some general statistical properties of an adaptive Type-II progressive censoring scheme are first investigated. A bias correction procedure is proposed to reduce the bias of the maximum likelihood estimators (MLEs). We then focus on the extreme value distributed lifetimes and derive the Fisher information matrix for the MLEs based on these properties. Four different approaches are proposed to construct confidence intervals for the parameters of the extreme value distribution. Performance of these methods is compared through an extensive Monte Carlo simulation.

It is of great interest for the problem of how to allocate redundancies in a system so as to optimize the system performance in reliability engineering and system security. In this paper, we consider the problems of optimal allocation of both active and standby redundancies in series systems in the sense of various stochastic orderings. For the case of allocating one redundancy to a series system with two exponential components, we establish two likelihood ratio order results for active redundancy case and standby redundancy case, respectively. We also discuss the case of allocating K active redundancies to a series system and establish some new results. The results developed here strengthen and generalize some of the existing results in the literature. Specifically, we give an answer to an open problem mentioned in Hu and Wang [T. Hu, Y. Wang, Optimal allocation of active redundancies in r-out-of-n systems, Journal of Statistical Planning and Inference 139 (2009) 3733–3737]. Numerical examples are provided to illustrate the theoretic results established here.

In this paper, we provide sufficient conditions on weighted coefficients under which the peakedness comparison between weighted sums of independent random variables can be carried out. These results extend and enrich the existing peakedness results in the literature including those presented by Proschan (1965) and Ma (1998) .

To examine the effect of individual job coaching and use of health threat in a job-specific occupational health education program in preventing work-related musculoskeletal back injuries during manual materials handling in construction laborers.Two hundred five laborers were recruited by 3-stage cluster sampling process and randomly assigned to receive different education programs on manual materials handling. Control group was given a conventional program, whereas experimental group received a job-specific program.A statistically significant group difference was found in the knowledge and practical skills on manual materials handling after training (P0.0001). Furthermore, the results from 1-year cumulative incidence revealed a significantly lesser number of first-time reports of work-related musculoskeletal back injuries in the experimental group.These results indicate that job-specific education is an effective preventive strategy for work-related musculoskeletal back injury.

Shapiro and Wilk's [Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance test for normality (complete samples). Biometrika 52, 591-611] W-statistic was found to have competitive power performance in testing univariate normality. Generalizations of the W-statistic to the multivariate case have been proposed by many researchers. In this paper, we propose a family of generalized W-statistics for testing high-dimensional normality by using the theory of spherical distributions. The proposed statistics apply to the case that the sample size is smaller than the dimension. Monte Carlo studies demonstrate feasible performance of the proposed tests in controlling type I error rates and power against some non-normal data. It is concluded that the proposed statistics are superior to existing generalizedW-statistics and show competitive benefits in testing high-dimensional normality with small sample size.

In this article, a mixture of Type-I censoring and Type-II progressive censoring schemes, called an adaptive Type-II progressive censoring scheme, is introduced for life testing or reliability experiments. For this censoring scheme, the effective sample size m is fixed in advance, and the progressive censoring scheme is provided but the number of items progressively removed from the experiment upon failure may change during the experiment. If the experimental time exceeds a prefixed time T but the number of observed failures does not reach m, we terminate the experiment as soon as possible by adjusting the number of items progressively removed from the experiment upon failure. Computational formulae for the expected total test time are provided. Point and interval estimation of the failure rate for exponentially distributed failure times are discussed for this censoring scheme. The various methods are compared using Monte Carlo simulation. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009

In this article, we discuss the maximum likelihood estimators and approximate maximum likelihood estimators of the parameters of the Weibull distribution with two different progressively hybrid censoring schemes. We also present the associated expressions of the expected total test time and the expected effective sample size which will be useful for experimental planning purpose. Finally, the efficiency of the point estimation of the parameters based on the two progressive hybrid censoring schemes are compared and the merits of each censoring scheme are discussed.

The classic hierarchical linear model formulation provides a considerable flexibility for modelling the random effects structure and a powerful tool for analyzing nested data that arise in various areas such as biology, economics and education. However, it assumes the within-group errors to be independently and identically distributed (i.i.d.) and models at all levels to be linear. Most importantly, traditional hierarchical models (just like other ordinary mean regression methods) cannot characterize the entire conditional distribution of a dependent variable given a set of covariates and fail to yield robust estimators. In this article, we relax the aforementioned and normality assumptions, and develop a so-called Hierarchical Semiparametric Quantile Regression Models in which the within-group errors could be heteroscedastic and models at some levels are allowed to be nonparametric. We present the ideas with a 2-level model. The level-1 model is specified as a nonparametric model whereas level-2 model is set as a parametric model. Under the proposed semiparametric setting the vector of partial derivatives of the nonparametric function in level-1 becomes the response variable vector in level 2. The proposed method allows us to model the fixed effects in the innermost level (i.e., level 2) as a function of the covariates instead of a constant effect. We outline some mild regularity conditions required for convergence and asymptotic normality for our estimators. We illustrate our methodology with a real hierarchical data set from a laboratory study and some simulation studies.

In this paper, we investigate four existing and three new confidence interval estimators for the negative binomial proportion (i.e., proportion under inverse/negative binomial sampling). An extensive and systematic comparative study among these confidence interval estimators through Monte Carlo simulations is presented. The performance of these confidence intervals are evaluated in terms of their coverage probabilities and expected interval widths. Our simulation studies suggest that the confidence interval estimator based on saddlepoint approximation is more appealing for large coverage levels (e.g., nominal level≤1% ) whereas the score confidence interval estimator is more desirable for those commonly used coverage levels (e.g., nominal level>1% ). We illustrate these confidence interval construction methods with a real data set from a maternal congenital heart disease study.

In this paper, we investigate various confidence intervals for the risk ratio under inverse sampling (also known as negative binomial sampling). Three existing confidence intervals (namely, the confidence intervals that are based on Fieller's theorem, the delta method and the F-statistic) are reviewed and three new confidence intervals (namely, the score, likelihood ratio and saddlepoint approximation (SA)-based confidence intervals) are developed. Comparative studies among these confidence intervals through Monte Carlo simulations are evaluated in terms of their coverage probabilities and expected interval widths under different settings. Our simulation results suggest that the SA-based confidence interval is generally more appealing. We illustrate these confidence interval construction methods with real data sets from a drug comparison study and a congenital heart disease study.

Inference for the extreme-value regression model under Type-II censoring is discussed. The likelihood function and the score functions of the unknown parameters are presented. The asymptotic variance-covariance matrix is derived through the inverse of the expected Fisher information matrix. Since the maximum likelihood estimators (MLE) cannot be solved analytically, an approximation to these MLE are proposed. The variance-covariance matrix of these approximate estimators is also derived. Next, confidence intervals are proposed based on the MLE and the approximate estimators. An extensive simulation study is carried out in order to study the bias and variance of all these estimators. We also examine the coverage probabilities as well as the expected widths of the confidence intervals. Finally, all the inferential procedures discussed here are illustrated with practical data.

Let {X 1, …, X n } and {Y 1, …, Y m } be two samples of independent and identically distributed observations with common continuous cumulative distribution functions F(x)=P(X≤x) and G(y)=P(Y≤y), respectively. In this article, we would like to test the no quantile treatment effect hypothesis H 0: F=G. We develop a bootstrap quantile-treatment-effect test procedure for testing H 0 under the location-scale shift model. Our test procedure avoids the calculation of the check function (which is non-differentiable at the origin and makes solving the quantile effects difficult in typical quantile regression analysis). The limiting null distribution of the test procedure is derived and the procedure is shown to be consistent against a broad family of alternatives. Simulation studies show that our proposed test procedure attains its type I error rate close to the pre-chosen significance level even for small sample sizes. Our test procedure is illustrated with two real data sets on the lifetimes of guinea pigs from ...

In this article, we discuss the optimal allocation problem in a multiple stress levels life-testing experiment when an extreme value regression model is used for statistical analysis. We derive the maximum likelihood estimators, the Fisher information, and the asymptotic variance–covariance matrix of the maximum likelihood estimators. Three optimality criteria are defined and the optimal allocation of units for two- and k-stress level situations are determined. We demonstrate the efficiency of the optimal allocation of units in a multiple stress levels life-testing experiment by using real experimental situations discussed earlier by McCool and Nelson and Meeker. Monte Carlo simulations are used to show that the optimality results hold for small sample sizes as well. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007

Different hybrid progressive censoring schemes , which are mixtures of Type-I censoring and Type-II progressively censoring schemes , have been proposed in the literature. These censoring schemes impose a time constraint on the life-testing experiment and the number of progressively censored order statistics observed before this time constraint is recorded. Conditional on the number of progressively censored order statistics being observed before the time constraint, a computational method for the conditional moments of progressively censored order statistics is discussed. Simple computational formulae are presented and these formulae are illustrated with examples when the underlying distributions are uniform and exponential. These results will be useful for the development of estimation methods such as the least squares estimation , best linear unbiased estimation and approximate maximum likelihood estimation methods and for deriving asymptotic distributions of the estimates of model parameters for Type-I hybrid progressively censored data.

Exact methods for small-sample dose–response analyses with binary outcome have been rapidly developed in recent years. For exact conditional approach, nuisance parameters (e.g. the intercept) are eliminated by conditioning on their reduced sufficient statistics (e.g. marginal row totals for the intercept under the logit link). For exact unconditional approach, nuisance parameters, on the other hand, are eliminated by using the ‘worst-case’ scenario. For instance, the p-value is the tail probability maximized overall possible values for the nuisance parameters. The performance of exact conditional and unconditional approaches on more general models including logit, probit, one-hit, and extreme-value are investigated here. Comparisons are conducted using the well-known Cochran–Armitage (C–A) trend test under different model specifications. Our simple empirical studies clearly show that the exact conditional approach is generally inferior to the exact unconditional approach with respect to actual significanc...

The prediction intervals proposed by J. F. Lawless (1971) and G. S. Lingappaiah (1973) for the exponential distribution are both simple to use. In this note, we make a comparison of these two prediction intervals based on the expected width of the prediction interval, as well as by means of the probability of the width of one being smaller than the other. For the computation of the latter, we use an algorithm, which is described briefly in the Appendix. Numerical results of these comparisons are presented for different choices of the parameters involved. Both these comparisons reveal that the prediction interval in is better than that in in that it has smaller expected width, as well as higher probability of having smaller width. Finally, we present an example to illustrate the results discussed in this paper.

In this article we compute the expected Fisher information and the asymptotic variance–covariance matrix of the maximum likelihood estimates based on a progressively type II censored sample from a Weibull distribution by direct calculation as well as the missing-information principle. We then use these values to determine the optimal progressive censoring plans. Three optimality criteria are considered, and some selected optimal progressive censoring plans are presented according to these optimality criteria. We also discuss the construction of progressively censored reliability sampling plans for the Weibull distribution. Three illustrative examples are provided with discussion.

Tester la non-inferiorite est de plus en plus important dans la medecine moderne comme un moyen de comparer une nouvelle procedure de test par rapport a une procedure deja disponible. Des methodes asymptotiques ont ete recemment developpees pour analyser les essais de non-inferiorite avec des rapports de taux dans les plans d'etudes apparies. Dans les petits echantillons cependant, la performance de ces methodes asymptotiques peut ne pas etre fiable et elles ne sont pas recommandees. Dans cet article, nous etudions des methodes alternatives qui sont adaptees pour evaluer la non-inferiorite en utilisant la mesure de rapport de taux dans les plans d'etude apparies avec des petits echantillons. En particulier nous proposons un test inconditionnnel exact et exact approxime et les intervalles de confiance correspondants bases sur la statistique du score. La methode inconditionnelle exacte garantit que le taux d'erreur de type I ne depasse pas le niveau nominal et est recommandee si un strict controle de l'erreur de type I (protection contre un risque accru d'accepter des traitements inferieur) est requis. Cependant la methode exacte tend a etre trop conservatrice (donc moins puissante) et numeriquement difficile. Grace a des etudes empiriques, nous demontrons que la methode du score exact approxime, qui est numeriquement plus simple a implementer, controle raisonnablement bien le taux d'erreur de type I et a une puissance elevee pour le test d'hypothese. Finalement. la methode exacte approximee offre une bonne alternative pour analyser les donnees binaires correlees des plans d'etude apparies avec de petits echantillons. Nous illustrons ces methodes avec deux exemples tires d'une etude en crossover sur les lentilles souples et sur la pneumonie a Pneumocystis carinii, et nous contrastons les methodes avec un exemple hypothetique.

Summary. In this article, we propose approximate sample size formulas for establishing equivalence or noninferiority of two treatments in match-pairs design. Using the ratio of two proportions as the equivalence measure, we derive sample size formulas based on a score statistic for two types of analyses: hypothesis testing and confidence interval estimation. Depending on the purpose of a study, these formulas can be used to provide a sample size estimate that guarantees a prespecified power of a hypothesis test at a certain significance level or controls the width of a confidence interval with a certain confidence level. Our empirical results confirm that these score methods are reliable in terms of true size, coverage probability, and skewness. A liver scan detection study is used to illustrate the proposed methods.

EM algorithm is used to determine the maximum likelihood estimates when the data are progressively Type II censored. The method is shown to be feasible and easy to implement. The asymptotic variances and covariances of the ML estimates are computed by means of the missing information principle. The methodology is illustrated with two popular models in lifetime analysis, the lognormal and Weibull lifetime distributions.

The probability generating function of number of trials for the start-up demonstration test with rejection upon d failures is derived. The exact distribution of the number of trials is obtained. Some recurrence relations for the probabilities are also established. The average length of the test is derived. Some illustrative examples are finally presented.

In this note, we discuss the record values arising from a normal N(μ, δ2) distribution. After computing the means, variances and covariances of the record values, we determine the best linear unbiased estimators (BLUEs) of μ and δ based on the first n record values. Using these BLUEs, we then develop a prediction interval for a future record value and also propose a test for spuriosity of a current record value. Finally, we present an illustrative example to explain the inference procedures developed in this note.

A conditional method of inference is used to derive exact confidence intervals for the location, scale and quantiles of location-scale family based on the upper record values. We will also discuss the prediction of the future record based on the current record values. For illustrative purpose, the extreme-value distribution is used to demonstrate the inference procedures. We also present a numerical example.

This paper is concerned with the problem of finding those distributionsminimizing Fisher information for location over Kolmogorov neighbourhoods ofdistribution functions G satisfying certain mild conditions. The case when Gis symmetric has been considered by quite a few authors. The general form ofthe solution is discussed. Furthermore we provide the solution of twoasymmetric distributions, namely, the extreme value distribution and thePearson's type IV distribution.

In this paper some recurrence relations between the moments of record values from the generalized extreme value distribution are established. It is shown that using these recurrence relations, all the single and product moments of all record values can be obtained in a simple recursive manner.

In this paper some recurrence relations between the moments of record values from the Gumbel distribution are established. It is shown that using these recurrence relations, all the single and product moments of all record values can be obtained in a very simple recursive process.

We derive the best linear unbiased estimator (BLUE) based on doubly Type-II censored samples for the scaled half logistic distribution. Next, we derive the best linear unbiased estimator and the asymptotic best linear unbiased estimator based on k optimally selected order statistics and show that the asymptotic result provides very close approximation to the finite sample result even for a sample of size as small as 20. The maximum likelihood estimator (MLE) based on either complete or Type-II censored samples does not exist in explicit form. We determine its unbiasing factor and variance through Monte Carlo simulations employing a numerical iterative procedure. We derive an approximate maximum likelihood estimator (AMLE) which has an explicit form and is almost as efficient as the MLE and the BLUE. We illustrate all these methods of estimation with two examples.

Let Yr+1:n ≤ Y:r+2:n ≤≤…

Let X1, X2, …, Xn be a random sample of size n from an extreme value distribution and X1:n less than or equal X2:n less than or equal … less than or equal Xn:n be the order statistics ob-tained from this sample. Tables of the means, variances, and covariances of the order statistics for samples of size n are given for n = 1(1)15(5)30. The computational formulae and procedure used and some checks employed are explained.

Inverse sampling is considered to be a more appropriate sampling scheme than the usual binomial sampling scheme when subjects arrive sequentially, when the underlying response of interest is acute, and when maximum likelihood estimators of some epidemiologic indices are undefined. In this article, we study various statistics for testing non-unity rate ratios in case-control studies under inverse sampling. These include the Wald, unconditional score, likelihood ratio and conditional score statistics. Three methods (the asymptotic, conditional exact, and Mid-P methods) are adopted for P-value calculation. We evaluate the performance of different combinations of test statistics and P-value calculation methods in terms of their empirical sizes and powers via Monte Carlo simulation. In general, asymptotic score and conditional score tests are preferable for their actual type I error rates are well controlled around the pre-chosen nominal level, and their powers are comparatively the largest. The exact version of Wald test is recommended if one wants to control the actual type I error rate at or below the pre-chosen nominal level. If larger power is expected and fluctuation of sizes around the pre-chosen nominal level are allowed, then the Mid-P version of Wald test is a desirable alternative. We illustrate the methodologies with a real example from a heart disease study.

The density function of a log-gamma distribution is given. Its moment generating function and hence the mean, variance, coefficients of skewness, and kurtosis are derived. Two other forms of the distribution under reparametrization are also given such that the normal distribution is included as a member of the distribution. Its relationship to the generalized gamma distribution will also be discussed Keywords: generalized gamma distribution; log-gamma distribution; moment generating function; reparametrization