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Bu çalışmada, Türkiye’de 2007 – 2020 yılları arasında mikro, küçük, orta ve büyük ölçekli işletmelere (KOBİ) bankalar tarafından kullandırılan kredilerin Gayri safi Yurtiçi Hasıla (GSYİH) ile istihdam (IST) üzerinde yarattığı etkilerin ortaya çıkarılması hedeflenmiştir. KOBİ kredileri ile GSYİH ve istihdam arasındaki kısa ve uzun vadeli ilişkinin incelendiği çalışmada, Maki eşbütünleşme testi (2012) ve ARDL sınır testi (1995) kullanılmıştır. Değişkenler arasındaki nedenselliğin tespiti için ise, Hacker ve Hatemi-J (2006) tarafından geliştirilen Bootstrap nedensellik testinden faydalanılmıştır. Yapılan analizler neticesinde, mikro ve küçük işletme kredileriyle GSYİH arasında bir eşbütünleşme ilişkisine ulaşılamazken, orta ve büyük ölçekli işletmelere kullandırılan krediler ile GSYİH arasında hem kısa hem de uzun vadede eşbütünleşme ilişkisi tespit edilmiştir. Öte yandan küçük işletme kredileri ile istihdam arasında bir eşbütünleşme ilişkisi tespit edilememiş olup, mikro, orta ve büyük ölçekli işletme kredileri ile istihdam arasında eşbütünleşme ilişkisi ortaya çıkarılmıştır. Son olarak nedensellik testleri neticesinde ise, GSYİH’den mikro işletme kredilerine doğru tek yönlü bir nedensellik tespit edilirken, küçük, orta ve büyük ölçekli işletme kredileri ile GSYİH arasında çift yönlü nedensellik tespit edilmiştir. Bununla birlikte KOBİ kredileri ile istihdam arasında herhangi bir nedensellik tespit edilememiştir., In this paper, it is aimed that the impact of credits provided by banks to micro, small, medium and large-sized enterprises (SMEs) is aimed uncovering on Gross Domestic's Product (GDP) and employment (EMP) between 2007 - 2020 in Turkey. In the manuscript examining the short and long-term relationship between these variables, the Maki cointegration test (2012) and the ARDL boundary test (1995), the Hacker Hatemi-J (2006) bootstrap causality test were used. As a result of the analysis, a cointegration relationship between micro and small enterprise credits and GDP could not be reached, while the relationship between credits extended to medium and large-scale enterprises and GDP was determined in both short and long terms. On the other hand, no cointegration relationship was found between small enterprise credits and employment, and the relationship between micro, medium and large-scale enterprise loans and employment was revealed. Finally, as a result of the causality tests, a unidirectional causality from GDP to micro enterprise credits was determined, while bidirectional causality between small, medium and large- sized enterprise credits and GDP was determined. However, no causality has been identified between SME credits and employment.

For a system (or unit) whose lifetime is measured by the number cycles, according to the discrete time age replacement policy, it is replaced preventively after n cycles or correctively at failure, whichever occurs first. In this paper, discrete time age replacement policy is revisited when the lifetime of the system is modeled by a discrete phase-type distribution. In particular, the necessary conditions for the unique and finite replacement cycle which minimizes the expected cost per unit of time are obtained. The necessary conditions are mainly based on the behavior of the hazard rate. The results are illustrated for some special discrete phase-type lifetime distributions. Computational results are also presented for the optimal replacement cycle under specific real life setups.

The reliability of a weighted-[Formula: see text]-out-of-[Formula: see text] system that consists of three-state components is studied. The system is assumed to comprise [Formula: see text] three-state components, namely, perfect functioning, partial working, and complete failure and functions if the total weight of all the working components is at least [Formula: see text]. Reliability expressions are presented when the times spent by components in perfect functioning and partial working states are dependent with a given joint distribution. Sufficient conditions are also provided to compare the expected total weights of two systems.

A new mixed shock model is introduced and studied. According to the model, for two fixed critical values d 1 and d 2 such that d 1 d 2 , the system under concern fails upon the occurrence of k consecutive shocks of size at least d 1 or a single large shock of size at least d 2 . The new model combines run and extreme shock models. Reliability properties of the system are studied under two cases: when the interarrival time X i between the ( i − 1 ) th and i th shock, and the magnitude of the i th shock Y i are independent for all i , and when the interarrival time between the ( i − 1 ) th and i th shock, and the magnitude of the i th shock are dependent for all i .

In this article, we develop a general statistical inference procedure for the probability of successful startup p in the case of startup demonstration tests when only the number of trials until termination of the experiment are observed. In particular, we define a class of startup demonstration tests and present expectation-maximization (EM) algorithm to get the maximum likelihood estimate of p for this class. Most of well-known startup testing procedures are involved in this class. Extension of the results to Markovian startups is also presented.

In this paper, a new shock model called Marshall–Olkin run shock model is defined and studied. According to the model, two components are subject to shocks that may arrive from three different sources, and component i fails when it is subject to k consecutive critical shocks from source i or k consecutive critical shocks from source 3, i = 1 , 2 . Reliability and mean residual life functions of such components are studied when the times between shocks follow phase-type distribution.

This paper is concerned with a system consisting of multiple types of components and having ( k 1 , k 2 , … , k m ) -out-of- n structure. The ( k 1 , k 2 , … , k m ) -out-of- n system is a system consisting of n i components of type i , i = 1 , 2 , … , m , and functions if at least k 1 components of type 1, k 2 components of type 2, …, k m components of type m work, n = ∑ i = 1 n n i . The exact and approximate expressions are obtained for the survival function of the system under concern. The weighted- ( k 1 , k 2 , … , k m ) -out-of- n system is also defined and studied. This weighted model is applied to evaluate the wind power system that consists of two wind plants.

This paper investigates a discrete-time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite-time interval. Specifically the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition comparisons are made involving the corresponding results of the classical discrete-time compound binomial risk model for which claim occurrences are independent and identically distributed. © 2018 John Wiley & Sons Ltd.

In this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

Shock models are of great interest in engineering reliability. Among the others, the δ-shock model has been widely studied in the literature. In this model, the system breaks down due to the arrivals of two successive shocks which are too close to each other. That is, the system fails when the time between two consecutive shocks falls below a fixed threshold δ. In the literature, the δ-shock model has been mostly studied by assuming that shocks arrive according to a renewal process so that the interarrival times between shocks are independent and identically distributed. In the current paper, we consider the case when the shock arrival process is described by a Polya process which has dependent interarrival times. In particular, we obtain survival function and mean lifetime of the system and study the optimal replacement policy for the δ-shock model based on Polya process.

In this paper we study a discrete time risk model based on exchangeable dependent claim occurrences. In particular we obtain expressions for the finite time non-ruin probability and the joint distribution of the time to ruin the surplus immediately before ruin and the deficit at ruin. An illustration of the results is given and some implications of the results are provided. Comparisons are made with the corresponding results for the classical compound binomial model of independent and identically distributed claim occurrences. (C) 2016 Elsevier E.V. All rights reserved.

Particular class of reliability shock models is studied.Phase-type modeling is proposed.Optimal replacement time and mean residual life are computed.Matrix-based efficient formulae are obtained. In this paper, matrix-based methods are presented to compute the optimal replacement time and mean residual lifetime of a system under particular class of reliability shock models. The times between successive shocks are assumed to have a common continuous phase-type distribution. The system's lifetime is represented as a compound random variable and some properties of phase-type distributions are utilized. Extreme shock model, run shock model, and generalized extreme shock model are shown to be the members of this class. Graphical illustrations and numerical examples are presented for the run shock model when the interarrival times between shocks follow Erlang distribution.

Phase-type distributions have been found to be very useful in reliability, queueing theory, and some other operational research applications. In this work, we de_ne a new mixed shock model and study lifetime properties of the corresponding system using phase-type distributions. The system under concern fails upon the occurrence of k consecutive shocks of size at least d1 or a single large shock of size at least d2, d1 < d2. In particular, using closure properties of phase-type distributions, we obtain survival function and mean time to failure of the system. Optimal replacement time is also studied.

In this paper, a general formula for computing the joint reliability importance of two components is obtained for a binary coherent system that consists of exchangeable dependent components. Using the new formula, the joint reliability importance can be easily calculated if the path sets of the system are known. As a special case, an expression for the joint reliability importance of two components is also obtained for a system consisting of independent and identical components. Illustrative numerical results are presented to compare the joint reliability importance of two components in the bridge system for the two cases when the components are exchangeable dependent and when the components are independent and identical.

Most practical systems consist of multiple types of components although the components perform the same task within the system. The analysis of such systems is more challenging than the systems with single type of components. In this paper, we present expressions for the survival function of the failure time, and mean time to failure of the system under the general case when the random failure times of components of the same type are exchangeable dependent and the random failure times of components of different type are dependent. As a case study, we apply our results to linear consecutive- $k$ -out-of- $n$ :F systems consisting of three different types of dependent components.

In this paper, we study a three-state ${\bf k}$ -out-of- $n$ system with $n$ independent components ( $ {\bf k} = ({k_{1}},{k_{2}}) $ ). Each component can be in a perfect functioning state (state “2”), partially working (state “1”), or failed (state “0”). We assume that, at time $ t = 0 $ , $ {n_{1}} $ components are in a partially working state while the rest $ {n_{2}} $ components are fully functioning ( $ n = {n_{1}} + {n_{2}} $ ). The system is considered to be at state “1” or above if at least $ {k_{1}} $ components are working (fully or partially). If at least $ {k_{1}} $ components are working and at least $ {k_{2}} $ components are in a perfect functioning state, we shall say that the system is at state “2”. In this paper, we develop formulae for the survival functions corresponding to the two different system's states described above. For illustration purposes, a numerical example which assumes that the degradation occurs according to a Markov process is presented.

This paper deals with two different shock models in a Markovian environment. We study a system from a reliability point of view under these two shock models. According to the first model, the system fails if the cumulative shock magnitude exceeds a critical level, while in the second model the failure occurs when the cumulative effect of the shocks in consecutive periods is above a critical level. The shock occurrences over discrete time periods are assumed to be Markovian. We obtain expressions for the failure time distributions of the system under the two model. Illustrative computational results are presented for the survival probabilities and mean time to failure values of the system.

This paper is concerned with the computation of the Barlow-Proschan importance measure for systems involving two common failure criteria, and consisting of statistically independent and identical components. The failure or survival of these systems generally depends on the number of consecutively failed or working components, or the total number of failed or working components in the whole system. The results are applied to $(n,f,k):{\rm F}$ and $ :{\rm F}$ systems.

Let { Y i } i ≥ 1 be a sequence of { 0 , 1 } variables which forms a Markov chain with a given initial probability distribution and one-step transition probability matrix. Define N n to be the number of trials until the n th success (“1”) in { Y i } i ≥ 1 . In this paper, we study the distribution of the random variable T = ∑ i = 1 N n X i , where { X i } i ≥ 1 is a sequence of independent and identically distributed random variables having a common phase-type distribution. The distribution of T is obtained by means of phase-type distributions.

The number of components that are failed at the time of system failure is a useful quantity since it gives an idea of how many spares should be available to replace all failed components upon the system failure. In this paper, the number of failed components is considered at subsystem and system levels for the series–parallel system that consists of K subsystems. In particular, the joint behavior of the number of failed components in each subsystem is studied when each subsystem has identical components and different subsystems have different types of components. The results are then used to find the optimal number of components in each subsystem by minimizing an expected cost per unit of time upon the system failure.

According to the well-known age replacement policy, the system is replaced preventively at time t or correctively at system failure, whichever occurs first. For a coherent system consisting of components having common failure time distribution which has increasing failure rate, we present necessary conditions for the existence of the unique optimal value which minimizes the mean cost rate. The conditions are mainly based on the signature which only depends on the system’s structure. The results are illustrated for linear and circular consecutive type systems.

The study of compound sums have always been very popular in the literature. Many models in insurance and engineering have been represented and solved by compound sums. In this paper, two different bivariate compound sums are proposed and studied. The phase-type distribution is applied to obtain the probability generating function of the bivariate sum.

Marginal and joint reliability importance measures have been found to be useful in optimal system design. Various importance measures have been defined and studied for a variety of system models. The results in the literature are mostly based on the assumption that the components within the system are independent or identical. The present paper is concerned with computation of marginal and joint reliability importance for a coherent system that consists of multiple types of dependent components. In particular, by utilizing the concept of survival signature, expressions for marginal and joint reliability importance measures are presented. We also introduce reliability importance for a system of which only the survival signature is known, which therefore can be regarded to be a black box system. (C) 2017 Elsevier Ltd. All rights reserved.

Mean residual life is a useful dynamic characteristic to study reliability of a system. It has been widely considered in the literature not only for single unit systems but also for coherent systems. This article is concerned with the study of mean residual life for a coherent system that consists of multiple types of dependent components. In particular, the survival signature based generalized mixture representation is obtained for the survival function of a coherent system and it is used to evaluate the mean residual life function. Furthermore, two mean residual life functions under different conditional events on components’ lifetimes are also defined and studied.

We consider two different sets of exchangeable samples which are assumed to be dependent. A single set of observations is obtained from these two dependent samples. The distribution of single order statistic, and the joint distribution of the minimum and an arbitrary order statistic are derived. The results are illustrated in the context of reliability problem.

This paper investigates the stress-strength reliability in the presence of fuzziness. The fuzzy membership function is defined as a function of the difference between stress and strength values, and the fuzzy reliability of single unit and multicomponent systems are calculated. The inclusion of fuzziness in the stress-strength interference enables the user to make more sensitive analysis. Illustrations are presented for various stress and strength distributions.

This paper is concerned with dynamic reliability modeling of three-state systems consisting of three-state $s$ -independent components. The components and the systems are assumed to be in three states: perfect functioning, partial performance, and complete failure. Survival functions of such systems are studied in different state subsets. It is shown that the survival function of a three-state system with a general structure can be represented as a mixture of the survival functions of the three-state $ {\bf k}$ -out-of- $n$ :G systems. The results are illustrated for the three-state consecutive- $ {\bf k}$ -out-of- $n$ :G systems whose components degrade according to a Markov process.

A generalized $k$ -out-of- $n$ system which is denoted by $((n_{1},\ldots,n_{N}),f,k)$ consists of $N$ modules ordered in a line or a circle, and the $i$ th module is composed of $n_{i}$ components in parallel $(n_{i}\geq 1,i=1,\ldots,N)$ . The system fails iff there exist at least $f$ failed components or at least $k$ consecutive failed modules. In this paper, we compute the signature of this system when $n_{1}=\ldots=n_{N}=n$ , and present illustrative examples to demonstrate its application. Simulation based computation of the signature is provided when the modules have different numbers of components.

This paper is concerned with dynamic reliability analysis of three-state k-out-of-n:G systems. It is assumed that the components and the systems can be in three states: perfect functioning, partial performance and complete failure. Using the concept of permanent, we study marginal and joint survival functions for the lifetime of two different three-state k-out-of-n:G systems that consist of independent and nonidentical components. Illustrative examples are also provided for the components which follow the Markov degradation process.

In this paper, we generalize geometric and binomial distributions of order k to q -geometric and q -binomial distributions of order k using Bernoulli trials with a geometrically varying success probability. In particular, we derive expressions for the probability mass functions of these distributions. For q = 1 , these distributions reduce to geometric and binomial distributions of order k which have been extensively studied in the literature.

Modeling statistical dependence between two systems or components is an important problem in reliability theory. Such a problem has been well studied for binary systems and components. In the present paper, we provide a way for modeling s-dependence between two multi-state components. Our method is based on the use of copulas which are very popular for modeling s-dependence. We obtain expressions for the joint state probabilities of the two components, and illustrate the results for the case when the degradation in both components follows a Markov process.

A weighted- k -out-of- n :G system is a system that consists of n binary components, each with its own positive weight, and operates only when the total weight of working components is at least k . Such a structure is useful when the components have different contributions to the performance of the entire system. This paper is concerned with both marginal and joint Birnbaum, and Barlow–Proschan (BP) importances of the components in weighted- k -out-of- n :G systems. The method of universal generating function is used for computing marginal and joint Birnbaum importances. The method for computing BP-importance is based on a direct probabilistic approach. Extensive numerical calculations are presented. By the help of these calculations and illustrations, it is possible to observe how the marginal and joint importances change with respect to the weights of components.

The purpose of this paper is to show the usefulness of system signature for computing some important reliability indices of repairable systems. In particular, we obtain signature-based expressions for stationary availability, rate of occurrence of failure, and mean time to the first failure of repairable systems. Using these expressions we compute corresponding reliability indices of all systems with three and four components. Computational results are also presented for consecutive-k-within-m-out-of-n:F and m-consecutive-k-out-of-n:F systems. (C) 2013 Elsevier B.V. All rights reserved.

In this paper, we obtain an expression between the sums of the marginal distributions of the order statistics and the common marginal distribution of an exchangeable random sequence. We also derive an expression between the sums of the joint distribution of two order statistics and the two dimensional joint distribution of an exchangeable random sequence.

We study the joint reliability importance (JRI) of two components in Lin/m/Con/k/n:F systems. A Lin/m/Con/k/n:F system consists of n linearly ordered binary components, and the system fails iff there are at least m nonoverlapping runs of k consecutive failed components (n ≥ mk). In particular, we obtain expressions for the JRI in Lin/m/Con/k/n:F systems when the components are s-independent & identical, when the components are s-independent & nonidentical, and when the components are exchangeable & s-dependent. We present extensive numerical illustrations.

The study of stress-strength reliability in a time-dependent context needs to model at least one of the stress or strength quantities as dynamic. We study the stress-strength reliability for the case in which the strength of the system is decreasing in time and the stress remains fixed over time; that is, the strength of the system is modeled as a stochastic process and the stress is considered to be a usual random variable. We present stochastic ordering results among the lifetimes of the systems which have the same strength but are subjected to different stresses. Multicomponent form of the aforementioned stress-strength interference is also considered. We illustrate the results for the special case when the strength is modeled by a Weibull process.

A k-out-of-n:G system consists of n components, and operates if at least k of its components operate. Its reliability properties have been widely studied in the literature from different perspectives. This paper is concerned with the reliability analysis of a k-out-of-n:G system equipped with a single warm standby unit. We obtain an explicit expression for the reliability function of the system for arbitrary lifetime distributions. Two different mean residual life functions are also studied for the system.

The concept of mean residual life is one of the most important characteristics that has been widely used in dynamic reliability analysis. It is a useful tool for investigating ageing properties of technical systems. In this paper, we define and study three different mean residual life functions for k -out-of- n :G system with a single cold standby component. In particular, we obtain explicit expressions for the corresponding functions using distributions of order statistics. We also provide some stochastic ordering results associated with the lifetime of a system. We illustrate the results for various lifetime distributions.

We propose a new model which generalizes the linear m-consecutive-k-out-of-n:F system to the case of the m-consecutive-k-out-of-n:F system with l-overlapping runs. The new system is called the m-consecutive-k, l-out-of-n:F system, and consists of linearly ordered components such that the system fails iff there are at least m l-overlapping runs of k consecutive failed components. The number of path sets including a specified number of working components is obtained via a combinatorial method, and this quantity is used to obtain the reliability and the signature of the system.

The consecutive k -out-of-n systems are important structures in reliability engineering due to their applications in various real life situations. In this paper, we study the residual lifetime of these systems under the condition that the total number of failed components at time t1 is r , and at time t2 (t2 >; t1) the system is still working. We obtain explicit expressions for the survival function of the corresponding residual life when the components are exchangeably dependent. We also obtain a signature-based mixture representation for a coherent system satisfying a certain property.

This paper is concerned with the number of components that are failed at the time of system failure. We study the corresponding quantity for a coherent structure via the system signature. Furthermore, we study the distribution of the number of failures after a specified time until the system failure. We illustrate the results for well-known general classes of coherent systems such as linear consecutive -within- -out-of- :F, and -consecutive- -out-of- :F.

A combined m-consecutive-k-out-of-n:F & consecutive k ;c ;-out-of-n:F systems consists of linearly ordered components, and fails iff there exist at least k ;c ; consecutive failed components, or at least m nonoverlapping runs of k consecutive failed components. This structure has applications for modeling systems such as infrared detecting and signal processing, and bank automatic payment systems. In this paper, we derive a combinatorial equation for the number of path sets of this structure including a specified number of working components. This number is used to derive a reliability function, and a signature based survival function formulae, for the system consisting of i.i.d. components. We also obtain a combinatorial equation for the reliability of a system with Markov dependent components.

Start-up demonstration testing is an effective method for illustrating the reliability of a unit before purchasing it. The test consists of starting-up the unit, and observing the outcomes, either success or failure. According to the total successes total failures (TSTF) test procedure, a unit under test is accepted when a specified number of successes is observed before a specified number of failures; otherwise, the unit is rejected. We study the TSTF procedure for dependent start-ups where the outcome of the present start-up depends on the total number of successful start-ups so far. The main characteristics of the TSTF test are obtained under this previous-sum dependent model, and numerical illustrations are presented.

The concept of residual lifetime has attracted considerable research interest in reliability theory. It is useful for evaluating the dynamic behavior of a system. In this paper, we study the extreme residual lives, that is, the minimum and maximum residual lives of the remaining components after the failure of the system. The system is assumed to have an arbitrary structure. We obtain signature-based distributional and ordering results for the extreme residual lives.

This paper is concerned with the mean, minimum and maximum distances between two successive failures in a binary sequence consisting of Markov dependent elements. These random variables are potentially useful for the analysis of the frequency of critical events occurring in certain stochastic processes. Exact distributions of these random variables are derived via combinatorial techniques and illustrative numerical results are presented.

Let {X i } i=1 n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable S n,m to be the maximum number of 1s within any m consecutive trials in {X i } i=1 n . The random variable S n,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {X i } i=1 n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of S n,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

Mean residual, and mean past lifetime functions are important reliability measures for evaluating systems' performance. These functions have been successfully used in binary state reliability analysis. This paper studies mean residual, and mean past lifetime concepts for multi-state systems. We evaluate mean residual, and mean past lifetime functions of one unit multi-state systems, and multi-state -out-of- :G systems under the assumption that the degradations in systems and components follow an acyclic Markov process which has a discrete state space.

The signature of a system is a useful tool in a variety of applications including the evaluation of the reliability characteristics of systems, and the comparison of the performance of competing systems. We study the evaluation and application of signatures of systems involving two common failure criteria which are common in real life applications. The failure or survival of these systems generally depends on the number of consecutively failed or working components, or total number of failed or working components in the whole system. We provide a method for obtaining the signatures of such systems. Applications of the results are also presented.