On stochastic comparisons of largest order statistics in the scale model

Let $X_{\lambda _{1}},X_{\lambda _{2}},\ldots ,X_{\lambda _{n}}$ be independent nonnegative random variables with $X_{\lambda _{i}}\sim F(\lambda _{i}t)$, $i=1,\ldots ,n$, where $\lambda _{i}>0$, $i=1,\ldots ,n$ and $F$ is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic $X_{n:n}^{\lambda }$ is smaller than another one $X_{n:n}^{\theta }$ according to likelihood ratio ordering. Furthermore, we apply these results when $F$ is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases

Stochastic order relations among parallel systems from Weibull distributions

In this article, we focus on stochastic orders to compare the magnitudes of two parallel systems from Weibull distributions when one set of scale parameters majorizes the other. The new results obtained here extend some of those proved by Dykstra et al. (1997) and Joo and Mi (2010) from exponential to Weibull distributions. Also, we present some results for parallel systems from multiple-outlier Weibull models.Comment: 14 pages, 3 figures. Journal of Applied Probability (2015

Optimal component-type allocation and replacement time policies for parallel systems having multi-types dependent components

In this work, we discuss some challenging open problems and conjectures recently proposed in the literature for parallel systems with dependent components of multiple types. Specifically, we present necessary conditions for the existence of the unique optimal value which minimizes the mean cost rate for two optimization problems. In the first place, the aim is to find the optimal number of components of each type which minimizes the associated mean cost rate, and secondly, to find the optimal replacement time before system failure. In both cases, we consider copulas to model the dependence structure for components whose lifetimes follow any distribution function. Moreover, in order to illustrate the theoretical results, we provide some numerical studies for specific copulas and marginal distribution functionsThis work was partially supported by Ministerio de Ciencia e Innovación of Spain under grant number PID2019-108079GB-C22/AEI/ 10.13039/50110001103

On the reversed hazard rate of sequential order statistics

Sequential order statistics can be used to describe the lifetime of a system with n components which works as long as k components function assuming that failures possibly affect the lifetimes of remaining units. In this work, the reversed hazard rates of sequential order statistics are examined. Conditions for the reversed hazard rate ordering and the decreasing reversed hazard rate property of sequential order statistics are given

On stochastic properties between some ordered random variables

A great number of articles have dealt with stochastic comparisons of ordered random variables in the last decades. In particular, distributional and stochastic properties of ordinary order statistics have been studied extensively in the literature. Sequential order statistics are proposed as an extension of ordinary order statistics. Since sequential order statistics models unify various models of ordered random variables, it is interesting to study their distributional and stochastic properties. In this work, we consider the problem of comparing sequential order statistics according to magnitude and location orders.Stochastic orderings, Reliability, Order statistics

On the Conjecture of Kochar and Korwar

In this paper, we solve for some cases a conjecture by Kochar and Korwar (1996) in relation with the normalized spacings of the order statistics related to a sample of independent exponential random variables with different scale parameter. In the case of a sample of size n=3, they proved the ordering of the normalized spacings and conjectured that result holds for all n. We give the proof of this conjecture for n=4 and for both spacing and normalized spacings. We also generalize some results to n>4Heterogeneous exponential distribution, Hazard rate order, Normalized

On the Conjecture of Kochar and Korwar

In this paper, we solve for some cases a conjecture by Kochar and Korwar (1996) in relation with the normalized spacings of the order statistics related to a sample of independent exponential random variables with different scale parameter. In the case of a sample of size n=3, they proved the ordering of the normalized spacings and conjectured that result holds for all n. We give the proof of this conjecture for n=4 and for both spacing and normalized spacings. We also generalize some results to n>

A study on multi-level redundancy allocation in coherent systems formed by modules

The present work studies the effect of redundancies on the reliability of coherent systems formed by modules. Different redundancies at components’ level versus redundancies at modules’ level are investigated, including active and standby redundancies. For that, a new model is presented. This model takes into account the dependence among the components, as well as, the dependence among the modules of the system. In both cases, the dependence structure is modeled by copula functions. Several results are provided to compare systems consisting of heterogeneous components. The comparisons are distribution-free with respect to the components. In particular, we consider the cases when the components in the modules are independent and connected (or not) in series, and when the components are dependent within the modules. In both cases, it is assumed that the modules can be dependent. Furthermore, the case in which the components in each module are identically distributed (dependent or independent) is also considered. We illustrate the theoretical results with several examplesNT is partially supported by Ministerio de Ciencia e Innovación of Spain under grant PID2019-108079GB-C22/AEI/10.13039/501100011033. AA was supported by Ministerio de Economía y Competitividad of Spain under grant MTM2017-89577-P. Finally, JN is partially supported by Ministerio de Ciencia e Innovación of Spain under grant PID2019-103971GBI00/ AEI/10.13039/50110001103

Comparisons among spacings from two populations

In this work, we obtain some new results in the area of stochastic comparisons of simple and normalized spacings from two heterogeneous populations. We also show some applications of our results to multiple-outlier models

Ordenaciones estocásticas e inferencia bayesiana en fiabilidad de software

Within the last decade of the 20th century and the first few years of the 21st century, the demand for complex software systems has increased, and therefore, the reliability of software systems has become a major concern for our modern society. Software reliability is defined as the probability of failure free software operations for a specified period of time in a specified environment. Many current software reliability techniques and practices are detailed by Lyu and Pham. From a statistical point of view, the random variables that characterize software reliability are the epoch times in which a failure of software takes place or the times between failures. Most of the well known models for software reliability are centered around the interfailure times or the point processes that they generate. A software reliability model specifies the general form of the dependence of the failure process on the principal factors that affect it: fault introduction, fault removal, and the operational environment. The purpose of this thesis is threefold: (1) to study stochastic properties of times between failures relative to independent but not identically distributed random variables; (2) to investigate properties of the epoch times of nonhomogeneous pure birth processes as an extension of nonhomogeneous Poisson processes used in the literature in software reliability modelling and, (3) to develop a software reliability model based on the use of covariate information such as software metrics. Firstly, properties of statistics based on heterogeneous samples will be investigated with the aid of stochastic orders. Stochastic orders between probability distributions is a widely studied concept. There are several kinds of stochastic orders that are used to compare different aspects of probability distributions like location, variability, skewness, dependence, etc. Secondly, ageing notions and stochastic orderings of the epoch times of nonhomogeneous pure birth processes are studied. Ageing notions are another important concepts in reliability theory. Many classes of life distributions are characterized or defined according to their aging properties in the literature. Finally, we exhibit a non-parametric model based on Gaussian processes to predict number of software failures and times between failures. Gaussian processes are a flexible and attractive method for a wide variety of supervised learning problems, such as regression and classification in machine learning. This thesis is organized as follows. In Chapter 1, we present some basic software reliability measures. After providing a brief review of stochastic point processes and models of ordered random variables, it discusses the relationship between these kind of models and types of failure data. This is then followed by a brief review of some stochastic orderings and ageing notions. The chapter concludes with a review of some well known software reliability models. The results of Chapter 2 concern stochastic orders for spacings of the order statistics of independent exponential random variables with different scale parameters. These results on stochastic orderings and spacings are based on the relation between the spacings and the times between successive software failures. Due to the complicated expression of the distribution in the non-iid case, only limited results are found in the literature. In the first part of this chapter, we investigate the hazard rate ordering of simple spacings and normalized spacings of a sample of heterogeneous exponential random variables. In the second part of this chapter, we study the two sample problem. Specifically, we compare both simple spacings and normalized spacings from two samples of heterogeneous exponential random variables according to the likelihood ratio ordering. We also show applications of these results to multiple-outlier models. In Chapter 3, motivated by the equality in distribution between sequential order statistics and the first n epoch times of a nonhomogeneous pure birth process, we consider the problem of comparing the components of sequential k-out-of-n systems according to magnitude and location orders. In particular, this chapter discusses conditions on the underlying distribution functions on which the sequential order statistics are based, to obtain ageing notions and stochastic comparisons of sequential order statistics. We also present a nonhomogeneous pure birth process approach to software reliability modelling. A large number of models have been proposed in the literature to predict software failures, but a few incorporate some significant metrics data observed in software testing. In Chapter 4, we develop a new procedure to predict both interfailure times and numbers of software failures using metrics information, from a Bayesian perspective. In particular, we develop a hierarchical non-parametric regression model based on exponential interfailure times or Poisson failure counts, where the rates are modeled as Gaussian processes with software metrics data as inputs, together with some illustrative concrete examples. In Chapter 5 we show some general conclusions and describe the most significant contributions of this thesis. -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------En la última década del siglo 20 y en los primeros años del siglo 21, la demanda de sistemas informáticos ha aumentado considerablemente, muestra de ello es su presencia en satélites espaciales, aviones, cadenas de montaje automatizadas, incluso cada vez están más cercanos a nuestra vida cotidiana como en automóviles, electrodomésticos o teléfonos móviles. Un sistema informático consta de dos tipos de componentes: el hardware y el software. Entre ellos la principal diferencia es que el software no se desgasta. Así, un programa informático podría funcionar al cabo de años con la misma corrección con que lo hizo el primer día sin necesidad de modificación alguna. En general, la calidad de un producto puede valorarse desde diversos puntos de vista. El software no es una excepción, y existen por tanto diferentes enfoques para la valoración de su calidad. Aquí nos centraremos en uno de dichos enfoques: la fiabilidad. Por fiabilidad se entiende la probabilidad de ausencia de fallos durante la operación de un producto de software. Existen diferentes técnicas estadísticas para medir la fiabilidad de un programa informático, algunas de ellas son detalladas en Lyu y Pham. Desde un punto de vista estadístico, las variables aleatorias que caracterizan la fiabilidad del software son los instantes de tiempo en los que se produce un fallo de software, así como, los tiempos entre fallos. Uno de los objetivos principales de esta tesis es modelizar el comportamiento de dichas variables aleatorias. Resulta interesante estudiar el comportamiento estocástico de dichas variables, ya que, de este modo, podemos conocer propiedades de las mismas relacionadas con sus funciones de supervivencia o con sus funciones de tasa de fallo. En este sentido, en el Capítulo 2, presentamos resultados referidos con ordenaciones estocásticas de los tiempos entre fallos de software, relativos a variables aleatorias independientes no idénticamente distribuidas. Estos resultados se basan en la relación que liga dichos tiempos con los espaciamientos (spacings). Tanto los estadísticos de orden como los espaciamientos tienen un gran interés en el contexto del Análisis de Supervivencia, así como en la Teoría de Fiabilidad. En la mayoría de los trabajos existentes, se asume que las variables implicadas son independientes e idénticamente distribuidas (iid). Debido a la complejidad analítica que conlleva relajar alguna de estas dos hipótesis, no hay demasiadas referencias para el caso en el que las variables no sean iid. Kochar y Korwar comprobaron que, cuando el número de exponenciales que se contemplan son tres, los espaciamientos normalizados cumplen la ordenación de tasa de fallo y conjeturaron lo mismo para el caso general de n variables aleatorias exponenciales heterogéneas. En la Sección 2.2, se presentan avances relacionados con dicha conjetura, así como, resultados relativos a la ordenación de tasa de fallo de espaciamientos sin normalizar. También han sido estudiados en este capítulo problemas asociados con espaciamientos obtenidos a partir de muestras aleatorias de dos poblaciones. En particular, hemos obtenido condiciones suficientes para que se verifique la ordenación de razón de verosimilitud entre espaciamientos de dos muestras de exponenciales heterogéneas. Por otra parte, hemos trabajado con estadísticos de orden secuenciales, ya que incluyen un gran número de variables aleatorias ordenadas. Además, este tipo de estadísticos de orden son interesantes porque están ligados con los tiempos en los que ocurre un fallo de procesos no homogéneos de nacimiento puro. Cabe destacar, que este tipo de variables son dependientes y no idénticamente distribuidas, lo que aumenta la complejidad del problema. Nuestro objetivo aquí, es estudiar qué condiciones deben verificar las distribuciones subyacentes a partir de las cuales se definen los estadísticos de orden secuenciales para que éstos cumplan algún tipo de ordenación estocástica. Los resultados obtenidos en este sentido se presentan en el Capítulo 3. En este capítulo, también estudiamos otro concepto importante en fiabilidad, la noción de envejecimiento. Los diferentes conceptos de envejecimiento describen como una componente o un sistema mejora o empeora con la edad. En este sentido, el envejecimiento positivo significa que las componentes tienden a empeorar debido al desgaste. Exactamente esto es lo que le ocurre al hardware. Mientras que, cuando un sistema supera ciertos tests y mejora, diremos que el envejecimiento es negativo, como le sucede al software. En el segundo capítulo de la tesis, estudiamos condiciones bajo las cuales algunas propiedades de envejecimiento verificadas por las distribuciones subyacentes, a partir de las cuales se definen los estadísticos de orden secuenciales, se cumplen también para los estadísticos de orden secuenciales. Si bien es cierto que se han desarrollado en los últimos cuarenta años un gran número de modelos de fiabilidad de software, la mayoría de ellos no tienen en consideración la información proporcionada por covariables. Otra aportación de esta tesis, la cual se encuentra en el Capítulo 4, consiste en la utilización de métricas del software como variables independientes para predecir o bien el número de fallos de un programa informático o bien los tiempos entre sucesivos fallos del software. Una métrica de un programa informático sirve para medir la complejidad y la calidad del mismo, así como, la productividad de los programadores con respecto a su eficiencia y competencia. En esta tesis, hacemos uso de métricas para medir la complejidad de un programa informático a través del volumen del mismo contabilizando el número de líneas de código. En la literatura existen algunos modelos lineales para predecir datos de fallos del software mediante métodos de inferencia clásicos. Sin embargo, nosotros optamos por utilizar procesos gaussianos que relajan la linealidad y que han sido ampliamente usados en problemas de aprendizaje automático, tanto en regresión como en clasificación. Por último, en el Capítulo 5, resumimos las principales aportaciones de esta tesis