
Ismihan Bayramogluİzmir University of Economics · Mathematics
Ismihan Bayramoglu
PhD
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July 1993 - present
Publications
Publications (105)
We propose a consistent test for testing the distribution of bivariate random samples. The probability of type I, type II errors and probability of making no decisions under null and alternative hypotheses are obtained based on copula functions. The consistency of the proposed test is discussed under some null and alternative hypotheses. An unbiase...
This paper is a short review of classical and recent results on Marshall–Olkin shock models and their applications in reliability analysis. The classical Marshall–Olkin shock model was introduced in Marshall and Olkin (J Am Stat Assoc 62:30–44, 1967). The model describes a joint distribution of lifetimes of two components of a system subjected to t...
A new concept of fuzzy improved distribution function based on fuzzy ordering defined on the support set of a given random variable is introduced. A fuzzy improved random variable based on this distribution function is defined. The existence of such a fuzzy improved random variable follows from probability integral transformation. The order statist...
We introduce a max–min model to bivariate random sequences and applying bivariate binomial distribution in fourfold scheme derive the distributions of associated order statistics in a new model. Some examples for special cases are presented and applications of the results in reliability analysis and actuarial sciences are discussed.
This paper presents derivations of asymptotic and exact distributions of bivariate exceedance statistics defined for some general threshold models. The asymptotic distribution of exceedance statistics appear to be marginal free, depending only on copulas of underlying joint distributions. In particular cases, the exact formulas for these new distri...
This paper considers the joint distribution of elements of a random sample and an order statistic of the same sample. The motivation for this work stems from the important problem in reliability analysis, to estimate the number of inspections we need in order to detect failed components in a coherent system. We consider an (n−r+1)-out-of-n system,...
The theory of record values has been extensively studied in the statistical literature. However, there are not many papers devoted to the theory of records for bivariate and multivariate random sequences. This paper presents the marginal record values and record times in extended sequence of bivariate random vectors. The joint distributions of some...
The focus of this paper is the random sequences in the form $\{X_{0},X_{1},$ $X_{n}=X_{n-2}+X_{n-1},n=2,3,..\dot{\}},$ referred to as Fibonacci Random Sequence (FRS). The initial random variables $X_{0}$ and $X_{1}$ are assumed to be absolutely continuous with joint probability density function (pdf) $f_{X_{0},X_{1}}.$ The FRS is completely determi...
This paper considers the joint distribution of elements of a random sample and an order statistic of the same sample. \ The motivation for this work stems from the important problem in reliability analysis, to estimate the number of inspections we need in order to detect failed components in a coherent system. We consider an $(n-r+1)$-out-of-$n$ sy...
We consider simple n-variate distributions, which are the special cases of general elliptically contoured and Kotz type distributions. In particular, we analyze the distributions having joint probability density functions (pdf) defined as functions of (x1² + x2² + ... + xn²) allowing easy calculations of different probabilities when using transform...
We consider new ordering among continuous distribution functions of independent, but not necessarily identically distributed random variables (INID), and introduce new order statistics defined as the observations from these ordered distribution functions. The new order statistics can be used instead of order statistics from INID random variables, a...
For bivariate random sequences we consider random threshold models and derive the finite and asymptotic distributions of exceedance statistics. Some examples with different underlying bivariate distributions are given and the resulting new limit distribution functions are presented.
Two types of records in multivariate sequences are considered in this paper. According to the first definition, a multivariate observation is accepted as a record if it is not dominated in at least one of the coordinates of previous record and the first observation is a record. Some basic straightforward results concerning the distributions of reco...
We consider coherent systems subjected to Marshall-Olkin type shocks coming at random times and destroying components of the system. The paper combines two important models, coherent systems and Marshall-Olkin type shocks and studies the mean residual life (MRL) and the mean inactivity time (MIT) functions of coherent systems that is subjected to r...
In this paper, we consider the mean remaining strength at the system level of two-component parallel and series systems in the case of possibly dependent components which are subject to a common stress. We assume that the strengths of the components have FGM, Freund’s, Marshall–Olkin and Block–Basu’s bivariate exponential models. A numerical study...
Let (Formula presented.) , be independent copies of bivariate random vector (Formula presented.) with joint cumulative distribution function (Formula presented.) and probability density function (Formula presented.). For (Formula presented.) , the vector of order statistics of (Formula presented.) and (Formula presented.) , respectively, is denoted...
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 reliabi...
In designing structures of technical systems, the reliability engineers often deal with the reliability analysis of coherent systems. Coherent system has monotone structure function and all components of the system are relevant. This paper considers some particular models of coherent systems having identical components with independent lifetimes. T...
We consider a (n - κ + 1)-out-of-n system with exchangeable lifetimes of the components. The paper investigates the stochastic ordering properties of the residual lifetime of the system under condition that, at time t, at least (n - r + 1) components are alive. Some results are then extended to the case where the system has a coherent structure wit...
In classical Marshall–Olkin type shock models and their modifications a system of two or more components is subjected to shocks that arrive from different sources at random times and destroy the components of the system. With a distinctive approach to the Marshall–Olkin type shock model, we assume that if the magnitude of the shock exceeds some pre...
In this paper, we consider some results on distribution theory of multivariate progressively Type-II censored order statistics. We also establish some characterizations of Freund's bivariate exponential distribution based on the lack of memory property. © 2014 Australian Statistical Publishing Association Inc. Published by Wiley Publishing Asia Pty...
Let (X, Y) be a bivariate random vector with joint distribution function FX, Y(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (Xi, Yi), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (Xi, Yi) for which Xi ⩽ Yi....
Order statistics and exceedances for some general models of independent but not necessarily identically distributed (INID) random variables are considered. The distributions of order statistics from INID sample are described in terms of symmetric functions. Some exceedance models based on order statistics from INID random variables are considered,...
In the classical Marshall–Olkin model, the system is subjected to two types of shocks coming at random times, and destroying components of the system. In statistics and reliability engineering literature, there are numerous papers dealing with various extensions of this model. However, none of these works takes into account the system structure, i....
Abstract
In this article, we consider the mean remaining strength of a k-out-of-n:F system in the stress–strength setup for the exchangeable components. We provide some results for parallel and series systems under this setup, where X1, X2, …, Xn are the strengths of the components designed under the common stress. An illustrative example is given...
The copula representations for conditionally independent random variables and the distribution properties of order statistics of these random variables are studied.
Let X1:n≤X2:n⋯≤Xn:nX1:n≤X2:n⋯≤Xn:n be the order statistics from some sample, and let Y[1:n],Y[2:n],…,Y[n:n]Y[1:n],Y[2:n],…,Y[n:n] be the corresponding concomitants. One purpose of this paper is to obtain results that stochastically compare, in various senses, the random vector (Xr:n,Y[r:n])(Xr:n,Y[r:n]) to the random vector (Xr+1:n,Y[r+1:n])(Xr+1:n...
This paper presents some novel trivariate discrete distributions that are obtained by modifying the bivariate binomial distribution. These distributions are important probability models for the development of conditional bivariate order statistics. The distributional properties of bivariate order statistics are studied and derived under the conditi...
The reliability and mean residual life of complex systems are discussed. These systems consist of n elements each having two s -dependent subcomponents. The reliability of such systems involves the distributions of bivariate order statistics, and are connected with a bivariate binomial distribution. The mean residual life function of complex system...
Lemma 2 in the paper (page 1803) requires an additional condition and without this condition may not be true. We correct this lemma and consequently the conditions of Corollary 2 and Theorem 3.
For a univariate distribution function (cdf) the upper and lower bounds expressed in terms of mixtures of cdf’s of order statistics are obtained. For a bivariate cdf, similar bounds, expressed in terms of mixtures of joint cdf’s of order statistics and their concomitants, are given. It is shown that these bounds converge to the corresponding cdf.
For practical applications in reliability analysis, the assumption of dependence among lifetimes of components of the system is more realistic than the assumption of independence. This paper investigates the residual lifetimes of coherent systems in situation where there exists a stochastic dependence among the components of the system. A stochasti...
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 h...
Suppose that n independent and identically distributed items have been placed in a life-test with Progressive Type-II censoring scheme (R-1, R-2, ..., R-m). In this paper, we investigate some characterizations and ordering results based on the residual lifetimes of the remaining items following the k(th) failure in the test. Keywords: Mean Residual...
Two different exchangeable samples are considered and these two samples are assumed to be independent of each other. From these two samples a new sample is combined and treated as a single set of observations. The distribution of a single order statistic and the joint distribution of two order statistics for a new mixed sample are derived and expre...
In life testing experiment under the setup of progressive Type II censoring we consider a new flexible scheme, in which the experimenter removes r1∗ (r1∗≤r1) units from the experiment if the first failure occurs before some predefined time t1, and removes r1 units when the first failure occurs after time t1. If the second failure occurs before the...
Let be independent and identically distributed random vectors with continuous distribution. Let L(n) and X(n) denote the nth record time and the nth record value obtained from the sequence of Xs. Let Y(n) denote the concomitant of the nth record value, which relates to the sequence of Ys. We call a near bivariate nth record-concomitant observation...
Baker (20087.
Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99: 23122327.[CrossRef], [Web of Science ®]View all references) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n....
Let Kn(a) be the number of observations in the interval (Mn,−a, Mn), where Mn is the maximum value in a sequence of size n. We study the asymptotic properties of Kn(a) under the F -scheme and discuss the influence of the associated sequence αn on the limit behaviour of this random variable.
Huang and Kotz (1999) [17] considered a modification of the Farlie–Gumbel–Morgenstern (FGM) distribution, introducing additional parameters, and paved the way for many research papers on modifications of FGM distributions allowing high correlation. The first part of the present paper is a review of recent developments on bivariate Huang–Kotz FGM di...
The test scores of students in an exam are usually ordered by the lecturer in order to overview the exam results and to evaluate the class performance. Since the exam results are not completely predictable, we admit the assumption that the test scores of students are independent and identically distributed random variables. In this case the ordered...
Let Δ
k:n
= X
k,n
− X
k-1,n
(k= 1, 2, . . . , n+1) be the spacings based on uniform order statistics, provided X
0,n
= 0 and X
n+1,n
= 1. Obtained from uniform spacings, ordered uniform spacings 0= Δ0,n
< Δ1,n
< . . .<Δ
n+1,n
, are discussed in the present paper. Distributional and limit results for them are in the focus of our attention.
A new class of multivariate discrete distributions with binomial and multinomial marginals is studied. This class of distributions is ob-tained in a natural manner using probabilistic properties of the sam-pling model considered. Some possible applications in game theory, life testing and exceedance models for order statistics are discussed.
Let be independent and identically distributed random vectors with continuous distribution. Let Kn(a,b1,b2) be the number of sample elements that belong to the open rectangle -- numbers of near-maxima in the bivariate case. In the present paper, we discuss asymptotic properties of Kn(a,b1,b2) and Kn([infinity],0,[infinity]).
The model in which joint Type-II progressive censoring is implemented on two samples from different populations in a combined manner is considered and the probabilities of failures are discussed. This model may have relevance for practical applications, when an experimenter need to know the expected values of the number of failures for each populat...
Waiting time distributions associated with a sequence of binary trials have been widely studied and applied in various areas of probability and statistics. In this paper we study simple and compound waiting time random variables associated with random threshold models which make the elements of sequence dependent. The exact distributions are derive...
A system with n independent components which has a k-out-of-n: G structure operates if at least k components operate. Parallel systems are 1-out-of-n: G systems, that is, the system goes out of service when all of its components fail. This paper investigates the mean residual life function of systems with independent and nonidentically distributed...
This paper concerns the joint behaviour of precedence and exceedance statistics in random threshold models. Joint distributions of precedence and exceedance statistics, both exact and asymptotic, are obtained, and the results are illustrated for random thresholds based on order statistics and record values.
Suppose that a system consists of n independent components and that the lifelength of the ith component is a random variable Xi (i = 1, 2, ..., n). For k ∈ {1, 2, ..., n - 1}, denote by X1(k), X2(k), ..., Xn - k(k), the residual lifelengths of the remaining functioning components following the kth failure in the system. We discuss the joint distrib...
Distributions of exceedance statistics based on minimal spacing of generalized order statistics are obtained in a random threshold model. As the special cases of the generalized order statistics the ordinary order statistics, Progressively Type II right censored order statistics and record values are considered. The results obtained in the paper im...
In this study, some continuous distributions through the properties of conditional expectations of generalized order statistics are characterized. Let X_{1:n:m:k},...,X_{n:n:m:k} be the generalized order statistics, where n∈ℕ,k>0,m₁,...,m_{n-1}∈ℝ, M_{r}=∑_{j=r}ⁿ⁻¹m_{j},1≤r≤n-1,γ_{r}=k+n-r+M_{r}>0 for all r∈{1,...,n-1} and let m={m₁,...,m_{n-1}}, if...
Exact, limiting distributions along with their rates of convergence of exceedance statistics for both order statistics and record statistics are pro- vided when the underlying distribution is arbitrary. The exact distribution of record statistics for arbitrary underlying distributions is obtained as well.
Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variet...
For a general class of distributions some characterizations through the properties of conditional expectations of order statistics and progressively Type-II censored order statistics are given. Let X 1:n , X 2:n ,…, X n:n be the order statistics of the sample X 1, X 2,…, X n from a continuous distribution and be the progressively Type-II censored o...
In the present work, we introduce a new local dependence function characterizing dependence structure between two random variables in an ε-neighborhood of a particular point from the domain of underlying bivariate distribution and investigate its properties. As an example the local dependence function for Farlie-Gumbel-Morgenstern distribution is p...
Let Xw(n) be weak record values derived from samples consisting of independent identically distributed discrete random variables. A limit theorem for large deviations for Xw(n) is proposed in the present paper.
In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that, at time t, all the components of the system are working, we propose...
For a sequence of independent and identically distributed random vectors , i=1,2,...,n, we consider the conditional ordering of these random vectors with respect to the magnitudes of , where N is a p-variate continuous function defined on the support set of X1 and satisfying certain regularity conditions. We also consider the Progressive Type II ri...
Distributional properties of spacings and exceedances based on progressive type II censored order statistics are investigated. Recurrence relations between moments of concomitants for the model of progressive type II censoring are given. We unify and extend some results on spacings, exceedances and concomitants for the model of progressive censorin...
This review paper consists of a description of the recent results connected with the exceedance statistics based on ordered random variables. In particular, the distributions of exceedance statistics in general random threshold model based on functions of the independent and identically distributed random variables are given.
A k-out-of-n system is working if at least k of its n components are operating. The system breaks down at the time of the (n-k+1) th component failure. Since all components start working at the same time, this approach leads to a kind of redundancy called active redundancy of n-k components. Important particular cases of k-out-of-n system are paral...
The exponential type is characterized in terms of the regression of a (possibly non-linear) function of a record value with its adjacent record values as covariates. Monotone transformations extend this result to more general settings, and these are illustrated with some specific examples.
Van Dorp and Kotz (2003 Van Dorp , J. R. , Kotz , S. ( 2003 ). Generalizations of two sided power distributions and their convolutions . Commun. Statist. Theor. Meth. 32 ( 9 ): 1703 – 1723 . [CROSSREF] [Google Scholar]) introduced a general class of two-sided power distribution. We provide several examples of this distribution by considering a non...
One of the most important types of system structures is the parallel structure. In the present article, we propose a definition for the mean residual life function of a parallel system and obtain some of its properties. The proposed definition measures the mean residual life function of a parallel system consisting of n identical and independent co...
The set of dependent random variables with multivariate Farlie- Gumbel-Morgenstern (FGM) distribution is considered. For the simple multivariate FGM distributions the admissible range of association parameters are investigated. Distributional properties of some exceedance statistics of finite FGM sequences with respect to a random threshold are stu...
The marginal and joint distributions of the new sample rank of an order statistic and its concomitant are derived.
A new local dependence function based on regression concepts is introduced. This function can characterize the dependence structure of two random variables localized at the fixed point. Some properties of the local dependence function are given. Examples of important bivariate distributions are provided.
Recently Lai and Xie (Statistics and Probability Letters, 46 (2000) 359-364) using the uniform representation of the Farlie-Gumbel-Morgenstern bivariate distribution introduce and study continuous bivariate distributions possessing positive quadrant dependence (PQD) property with the association parameter contained between 0 and 1. We show that the...
An extension of FGM class of bivariate distributions with given marginals is presented. For Huang-Kotz FGM distributions
some theorems characterizing symmetry and conditions for independence are obtained. The new family of distributions allows
us to achieve correlation between the components greater than 0.5.
We define the survival and mean residual life function of system consisting of n identical and independent components having se- ries or parallel structure. Let Xi, i = 1,2,...,n be the survival time of i th component, such that X1,X2,...,Xn are independent, identically dis- tributed random variables with continuous distribution functionF. LetXi:n,...
We consider a generalization of the bivariate Farlie-Gumbel-Morgenstern (FGM) distribution by introducing additional parameters. For the generalized FGM distribution, the admissible range of the association parameter allowing positive quadrant dependence property is shown. Distributional properties of concomitants for this generalized FGM distribut...
Exceedance statistics associated with the order statistics and record values for the generalized Bernoulli model are investigated
for the case of an arbitrary underlying distribution. Distributions of these statistics are studied. Also the distribution
of the second record value is obtained when the underlying distribution function contains atoms.
An additional insight to the distributional properties of statistics based on minimal spacing and record exceedances is provided. Behaviour of a sequence of independent identically distributed random variables with respect to a random threshold constructed by the help of minimal spacing and records is studied.
In this work invariant confidence intervals for parametric class of distribu- tions are considered. The finite sample and asymptotic properties of statistics based on invariant confidence intervals are investigated and their use in statis- tical inference is discussed. Abstract
A general class of bivariate distributions is introduced. This class includes the so-called Sarmanov-Lee class (and consequently the Farlie-Gumbel-Morgenstem class). It is shown that using procedures described in the paper it is possible to construct distributions of the FGM form for which the correlation coefficient between the marginals can achie...
Exact and asymptotic distributions of some statistics based on spacing having minimal length are investigated. The behaviour of a sequence of independent identically distributed random variables with respect to a record threshold is studied.
New characterizations for the exponential distribution are given in terms of record values and the probabilities of finite sums of independent and identically distributed nonnegative random variables provided that the underlying distribution is either new better than used or new worse than used.
We consider a class of probability density functions f with the property: for all 0<x 1 <x 2 <∞ and x>0 eitherf(x 1 +x)f(x 2 )≥f(x 1 )f(x 2 +x)orf(x 1 +x)f(x 2 )≤f(x 1 )f(x 2 +x)· This class includes the Polya frequency functions of order 2 (PF 2 ) and consequently many important densities useful in engineering, industry and biometry. We characteri...
Some new characterizations of continuous distributions expressed in terms of conditional expectations are provided. Characterizations of continuous distributions in terms of regressions of order statistics are given.
Published: Journal of Applied Statistical Science, Volume 10. No 1, pp. 57-68, 2000.
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A new nonparametric test for two sample problem based on order statistics is suggested. This test can be successful in the case of small samples. The probabilities of the statisti...
We introduce a notion "distance sequence of distribution functions "d.s.d.f) defined as a sequence of functional of distribution functions which characterize the proximity of two distribution functions. Some examples of d.s.d.f. are given, the asymptotic properties of statistics obtained with the help of d.s.d.f are investigated and their use in st...
Let X be a random variable with continuous distribution functions (d.f.) F. Suppose X i:n (i=1,2,⋯,n) is the i-th smallest order statistic in a random sample of size n from F. Let {X U(n) , n≥1} be a sequence of record values of a sequence of independent identically distributed (i.i.d.) random variables (r.v.’s) X 1 ,X 2 ,⋯,X n ,⋯ with distribution...
Let Rm,m 1, be the real Euclidean space. Suppose X1,X2,..,.Xn 2 Rm are independent identically distributed (i.i.d.) random variables (m > 1 random vectors) (r.v.'s) with distribution function (d.f.) F. Denote by k.k the norm defined in Rm. It is clear that kX1k,kX2k,...,kXnk are i.i.d. r.v. with d.f. P {kXik x} F (x), x 2 R. If F is assumed to be c...
Let X 1 ,X 2 ,⋯,X n ,⋯ is a sequence of independent and identically distributed (i.i.d.) random variables (r.v.) with continuous distribution function (d.f.) F. Define a sequence of record times U(n) as follows: U(1)=1, U(n)=min{j:j>U(n-1), X j >X U(n-1) , n>1}. Let X U(n) be upper record values, n=1,2,⋯ . Suppose that X 1 ' ,X 2 ' ,⋯,X n ' ,⋯ is a...
We introduce the notion of invariant confidence interval containing the main mass of values for some class of populations, which is independent of the distribution of the population from the given class. It is shown that an invariant one-tail confidence interval for the class of populations with continuous distributions is generated by some order s...
We propose consistent tests of statistical hypotheses based on training samples. Unbiased consistent estimators of the probability of no decision are obtained.