
Baha-Eldin KhalediRazi University | razi · Department of Statistics
Baha-Eldin Khaledi
PhD
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72
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Introduction
Skills and Expertise
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December 1992 - present
September 1992 - January 2014
Publications
Publications (72)
The transformer network is a deep learning architecture that uses selfattention mechanisms to capture the long-term dependencies of a sequential data. The Poisson-Lee-Carter model, introduced to predict mortality rate, includes the factors of age andthecalendaryear, whichisatime-dependentcomponent. Inthispaper, weuse the transformer to predict the...
This article presents a Poisson common factor model with an overdispersion factor to predict some multiple populations’ mortality rates. We use Bayesian data analysis and an extension of the Hamiltonian Monte Carlo sampler to compute the estimation of the model parameters and mortality rates prediction. We apply the proposed model to the real morta...
In the usual shock models, the shocks arrive from a single source. Bozbulut and Eryilmaz [(2020). Generalized extreme shock models and their applications. Communications in Statistics – Simulation and Computation 49 (1): 110–120] introduced two types of extreme shock models when the shocks arrive from one of $m\geq 1$ possible sources. In Model 1,...
In this paper, we present an extension of the Poisson Lee-Carter model for predicting mortality rates that includes the cohort effect and an overdispersion component. To compute the estimation of the model parameters and mortality rate predictions, we use Bayesian analysis and the No-U-Turn sampler (NUTS), which is an extension of Hamiltonian Monte...
Prediction of loss reserves corresponding to dependent lines of business is one of the most important problems in the actuarial sciences. In this paper, we propose a class of copula based multivariate distributions to model the losses with the heavy tailed distribution in the run-off triangles to predict unpaid losses. We set up ANOVA, ANCOVA, and...
Suppose that a policyholder faces n risks X 1 ,. .. , X n which are insured under the policy limit with the total limit of l. Usually, the policyholder is asked to protect each X i with an arbitrary limit of l i such that n i=1 l i = l. If the risks are independent and identically distributed with log-concave cumulative distribution function, using...
One of the challenges for decision-makers in insurance and finance is choosing the appropriate criteria for making decisions. Mathematical expectation, expected utility, and distorted expectation are the three most common measures in this area. In this article, we study these three criteria, and by providing some examples, we review and compare the...
We consider n risks X 1 , X 2 , … , X n insured by a layer coverage with deductibles and limits given by ( d 1 , l 1 ) , … , ( d n , l n ) , respectively. We investigate the optimal allocation of insurance layers from the viewpoint of the insurer. We derive lower and upper bounds for the survival function of the smallest and largest claim amounts u...
We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderin...
In this paper, the worst allocation of deductibles and limits in layer policies
are discussed from the viewpoint of the insurer. It is shown that if n independent and identically distributed exponential risks are covered by the layer policies and the policy limits are equal, then the worst allocation of deductibles from the viewpoint of the insurer...
The mean time to failure (MTTF) function in age replacement is used to evaluate the performance and effectiveness of the age replacement policy. In this paper, based on the MTTF function, we introduce two new nonparametric classes of lifetime distributions with nonmonotonic mean time to failure in age replacement; increasing then decreasing MTTF (I...
In this paper, we study the problem of optimal allocation of insurance layers for a portfolio of i.i.d exponential risks. Using the first stochastic dominance criterion, we obtain an optimal allocation for the total retain risks faced by a policyholder. This result partially generalizes the known result in the literature for deductible as well as p...
In this paper, we study the problem of optimal allocation of insurance layers for a portfolio of i.i.d exponential risks. Using the first stochastic dominance criterion, we obtain an optimal allocation for the total retain risks faced by a policyholder. This result partially generalizes the known result in the literature for deductible as well as p...
Stochastic ordering problems in weighted- -out-of- systems are considered. The influence of component lifetimes and weights on the total capacity is studied. When it is allowed to allocate lifetimes to the weights, an optimal allocation to attain maximum total capacity is obtained.
Let X1, . . ., Xn be a set of n risks, with decreasing joint density function f, faced by a policyholder who is insured for this n risks, with upper limit coverage for each risk. Let l=(l1, . . .ln) and l*=(l1*,. . .ln*) be two vectors of policy limits such that l* is majorized by l. It is shown that ∑i=1n(Xi-li)+ is larger than ∑i=1n(Xi-li*)+ acco...
Let X be a continuous random variable with distribution function F. If F is symmetric about 0, then for , , where is the pth quantile of X. Using this well-known observation, based on a random sample from F, a new test of symmetry against right skewness is proposed and its exact null distribution is obtained. It is shown that the proposed test is r...
Let , , be a sequence of independent and identically distributed -dimensional random vectors. Furthermore, let be the order statistics based on the first coordinates of the first n elements in the sequence, and let be the corresponding k-dimensional concomitants. Several results that compare and , with respect to various multivariate stochastic ord...
Let X1,…,Xn be a random sample from a distribution function F that denote lifetimes of $$n$$n components of a coherent system. Suppose the system fails at Xk:n, the kth order statistic of X’s, since we are not aware of the exact time at which the system has been failed, the residual lifetimes of the remaining n-k components, denoted by X1(k),…,Xn-k...
The most of the results obtained about stochastic properties of generalized order statistics and their spacings in the literature are based on equal model parameters. In this paper, with less restrictive conditions on the model parameters, we prove some new multivariate likelihood ratio ordering results between two sub-vectors of GOS's as well as t...
Let X1,X2,ΘX1,X2,Θ and Θ′Θ′ be independent non-negative random variables. The residual life of XiXi at random time ΘΘ, that is, XiΘ=Xi−Θ∣Xi>Θ is considered. Some sufficient conditions which lead to the likelihood ratio ordering, the failure rate ordering, the reverse failure rate ordering and the mean residual life ordering between X1Θ and X2Θ are...
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...
Suppose λ, x, ζ traverse the ordered sets Λ, X and Z, respectively and consider the functions f(λ, x, ζ) and g(λ, ζ) satisfying the following conditions, (a) f(λ, x, ζ) > 0 and f is TP2 in each pairs of variables when the third variable is held fixed; and (b) g(λ, ζ) is TP2. Then the function [Display Equation], defined on Λ × X is TP2 in (λ, x). T...
This work is a comment on the paper of N. Balakrishnan and P. Zhao [Probab. Eng. Inf. Sci. 27, No. 4, 403–443 (2013; Zbl 1288.60023)].
In this paper we consider the problem of testing H 0 : F is an exponential against H 1 : F is NWBUE and not exponential. We propose a new test statistic based on an inequality of moment and obtain its asymptotic distribution to compare our new test with another well known test given by Klefsjö (1989).
Due to its wide range of applications, the distribution theory of convolutions of gamma random variables has attracted the attention from many researchers. In this paper, we review some of the latest developments on this problem.
In this paper we study convolution residuals, that is, if
$X_1,X_2,\ldots ,X_n$
are independent random variables, we study the distributions, and the properties, of the sums
$\sum _{i=1}^lX_i-t$
given that
$\sum _{i=1}^kX_i>t$
, where
$t\in \mathbb R $
, and
$1\le k\le l\le n$
. Various stochastic orders, among convolution residuals based...
In this paper, first we consider the problem of testing that two unknown distributions are identical against the alternative that one is more IFRA than the other and propose a new test that is asymptotically normal and consistent. Next, we prove that beta family of distributions is ordered according to more IFRA ordering. The empirical power of the...
Suppose F and G are two life distribution functions. It is said that F is
more IFRA than G (written by F<_* G) if G^(-1) F(x) is starshaped on (0,infty).
In this paper, the problem of testing H_0:F=_* G against H_1:F<_* G and F
\neq_* G is considered in both cases when G is known and when G is unknown. We
propose a new test based on U-statistics an...
Some distribution-free tests have been discussed in the literature with regard to the comparison of hazard rates of two distributions when the available samples are complete. We generalize here Kochar's [S.C. Kochar, A new distribution-free test for the equality of two failure rates, Biometrika 68 (1981), pp. 423-426] test statistic to the case whe...
Let be n independent random variables such that Xλi has uniform distribution over the interval . It is proved that if (λ1,…,λn) is larger than according to reciprocal order, then is larger than according to mean residual life order as well as increasing convex order. This result gives convenient bounds for mean residual life function of in terms of...
Independent random variables Xλ1,…,Xλn are said to belong to the scale family of distributions if Xλi∼F(λix), for i=1,…,n, where F is an absolutely continuous distribution function with hazard rate r and reverse hazard rate . We show that the hazard rate (reverse hazard rate) of a series (parallel) system consisting of components with lifetimes Xλ1...
Let {x(1)≤···≤x(n)} denote the increasing arrangement of the components of a vector x=(x1, …, xn). A vector x∈Rn majorizes another vector y (written ) if for j = 1, …, n−1 and . A vector x∈R+n majorizes reciprocally another vector y∈R+n (written ) if for j = 1, …, n. Let , be n independent random variables such that is a gamma random variable with...
Let \(X_1,\ldots ,X_n\) be a random sample from a distribution function \(F\) that denote lifetimes of \(n\) components of a coherent system. Suppose the system fails at \(X_{k:n}\) , the \(k\) th order statistic of \(X\) ’s, since we are not aware of the exact time at which the system has been failed, the residual lifetimes of the remaining \(n-k\...
In this article, we establish some results concerning the univariate and multivariate likelihood ratio order of generalized order statistics and the special case of m-generalized order statistics and their associated conditional variables. These results, in addition to being new, also generalizes some of the known results in the literature. Finally...
In this article, we study the stochastic orderings among residual record values as well as inactive record values in two-sample problems. Some of the results obtained here are extensions of the main result in Khaledi and Shojaei (200710.
Khaledi , B.-E. , Shojaei , R. ( 2007 ). On stochastic ordering between residual record values . Statist. Probab...
Let X-lambda 1, ... , X-lambda n be independent random variables such that X-lambda i, i = 1, ... , n has probability density function f(nu,sigma,lambda i)(x) = 2 lambda(nu)(i)/Gamma(nu/2)(2 sigma(2))nu/2 x(nu-1) exp (-(lambda(i)x)(2)/2 sigma(2)), nu > 0, sigma > 0, lambda(i) > 0, known as a generalized Rayleigh random variable. We show that for nu...
Let Xλ1,…,Xλn be nonnegative independent random variables with Xλi having survival function F¯(.,λi), i=1,…,n, where λi>0. Let Ip1,…,Ipn be independent Bernoulli random variables independent of Xλi with E(Ipi)=pi , i=1,…,n. Further, assume that F¯(.,λi) is a decreasing and convex function with respect to λi, i=1,…,n and that the survival function o...
In this paper we establish some stochastic ordering results among residual record values in two sample problems. We also discuss some applications.
The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to establish the usual stochastic and the likelihood ratio orderings of conditional distributions of generalized order statistics from one sample or two samples, strengthening and genera...
This is a survey paper on recent results on stochastic comparisons of order statistics of n independent random variables differing in their scale parameters. Most of the results obtained so far are for the Weibull and the Gamma distributions.
Consider a system of n components that has the property that there exists a number r(r<n), such that if it is known that at most r components have failed, the system is still functioning with probability 1. Suppose that such a system is equipped with a warning light that comes up at the time of the failure of the rth component. The system is still...
In this paper we prove some stochastic comparisons re-sults for progressive type II censored order statistics. The problem of stochastically comparing concomitants of the two progressive type II censored order statistics with possibly different schemes, under dif-ferent kinds of dependence between X and Y is considered and it is proved that if Y is...
Let X1,
,Xn be independent random variables such that Xi has Weibull distribution with shape parameter a and scale parameter ?i, i=3D1,
,n. Let X1*,
,Xn* be another set of independent Weibull random variables with the same common shape parameter a, but with scale parameters as ?*=3D(?1*,
,?n*). Suppose that ??m?*. We prove that when 0<a<1,(X(1),
,X...
To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Mliltivariate Analysis, 2002). It is shown th...
To compare two multivariate random vectors of the same dimension, we
define a new stochastic order called upper
orthant
dispersive
ordering and study its properties. We study its relationship with
positive dependence and multivariate hazard rate ordering as defined by
Hu, Khaledi, and Shaked (Journal
of
Multivariate
Analysis, 2002).
It is show...
Generalized order statistics (gOSs) unify the study of order statistics, record values, k-records, Pfeifer's records and several other cases of ordered random variables. In this paper we consider the problem of comparing the degree of dependence between a pair of gOSs thus extending the recent work of Avérous et al. [2005. J. Multivariate Anal. 94,...
In this paper we specify the conditions on the parameters of pairs of gOS's under which the corresponding generalized order statistics are ordered according to usual stochastic ordering, hazard rate ordering, likelihood ratio ordering and dispersive ordering. We consider this problem in one-sample as well as two-sample problems. We show that some o...
Dispersion-type orders are introduced and studied. The new orders can be used to compare the variability of the underlying random variables, among which are the usual dispersive order and the right spread order. Connections among the new orders and other common stochastic orders are examined and investigated. Some closure properties of the new orde...
Two multivariate hazard rate stochastic orders are introduced and studied. Their meaning, properties, and relationship to other common stochastic orders are examined and investigated. Some examples that illustrate the theory are detailed. Finally, some applications of the new orders in reliability theory and in actuarial science are described.
Consider a mixture $G(\cdot)=\int _{S}F_{\theta}(\cdot)\,dH(\theta)$ . In this paper we derive some bounds on the uniform (Kolmogorov) distance $\Delta (F_{\theta _{0}},G)\equiv {\rm sup}_{x}\,|F_{\theta _{0}}(x)-G(x)|$ for some convenient choices of $\theta _{0}$ . In particular, we identify an optimal $\theta _{0}$ . We illustrate the results by...
In this paper we review some of the results obtained recently in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as non-identically distributed. But most of the time we shall be assuming that the observations are independent. The case of independe...
Lot a((i)) and b((i)) be the ith smallest components of a = (a(1), . . . , a(n)) and b = (b(1), . . . , b(n)) respectively, where a,b is an element of R+n. The vector a is said to be p-larger than b (denoted by a greater than or equal tob(p)) if Pi (k)(i-1) a((i)) less than or equal to Pi (k)(i=1) b((i)), for k = 1, . . . , n. Let U-1, . . . , U-n...
Let (Xi, Yi) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X(r) denotes the rth ordered X-variate then the Y-variate, Y[r], paired with X(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and depe...
Let X1,..., Xn
be independent exponential random variables with possibly
different scale parameters. Kochar and Korwar [J.
Multivar.
Anal. 57 (1996)] conjectured that,
in this case, the successive normalized spacings are increasing
according to hazard rate ordering. In this article, we
prove this conjecture in the case of a single-outlier exp...
Consider a multivariate mixture model where the random variables X
1, ..., X
n
given (Θ1, ..., Θn
), are conditionally independent. Conditions are obtained under which different kinds of positive dependence hold among X
i
's. The results obtained are applied to a variety of problems including the concomitants of order statistics and of record value...
Let X 1 ,…, X n be independent exponential random variables with X i having hazard rate . Let Y 1 ,…, Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i =1 n λ i ) 1/ n , the geometric mean of the λ i s. Let X n : n = max{ X 1 ,…, X n }. It is shown that X n : n is greater than Y n : n according to d...
Let X<sub>1</sub>,...,X<sub>n</sub> be independent exponential random variables with X<sub>i</sub> having hazard rate λ<sub>i</sub>, i = 1,...,n. Let Y<sub>1</sub>,...,Y<sub>n</sub> be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏<sub>i=1</sub><sup>n</sup>λ<sub>i</sub>)<sup>1/n</sup>, the geometric mean...
In this paper, we study the dependence properties of spacings.
It is proved that if X1,...,
Xn are exchangeable random variables
which are TP2 in pairs and their joint density is
log-convex in each argument, then the spacings are MTP2
dependent. Next, we consider the case of independent but
nonhomogeneous exponential random variables. It is s...
[Formula: see text]
AMS (2000) Subject classification: 60E15, 62N05, 62D05.
Let X-i : n denote the ith-order statistic of a random sample of size n from a continuous distribution with distribution function F. It is shown that if F is a decreasing failure rate (DFR) distribution, then X-i : n is less dispersed than X-j : m, for i less than or equal to j and n - i greater than or equal to m - j. Let Y-j : m denote the jth-or...
Let X1:n[less-than-or-equals, slant]X2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn:n denote the order statistics of a random sample of size n from a probability distribution with distribution function F. Similarly, let Y1:m[less-than-or-equals, slant]Y2:m[less-than-or-equals, slant]...[less-than-or-equals, slant]Ym:m denote the o...
Generalized order statistics unify the study of order statistics, record values, k-records, Pfeifer's records and several other cases of ordered random variables. We study the conditions under which a pair of generalized order statistics are ordered according to various stochastic orders like stochastic ordering, hazard rate ordering, likelihood ra...