مطالعه‌ای بر میانگین باقیمانده رکوردها در مدل تصادفی هندسی

رکوردها در نظریه قابلیت اعتماد کاربرد بسیاری دارند که از جمله آن‌ها می‌توان به مدل‌های شوک و تعمیرات مینیمال اشاره کرد. در این راستا پژوهش‌های زیادی بر اساس رکوردها در مدل کلاسیک انجام شده است. در این مقاله، رکوردها در مدل تصادفی هندسی مورد مطالعه قرار می‌گیرند. مفهوم میانگین مانده رکوردها در مدل تصادفی تعریف و برخی خواص آن در مدل تصادفی هندسی بررسی می‌شود. سپس نشان داده می‌شود با استفاده از دنباله میانگین مانده  رکوردها در مدل تصادفی هندسی، می‌توان توزیع جامعه را مشخص کرد. در پایان، به کاربردی از نتایج مشخص‌سازی در مدل‌های کاریابی در اقتصاد اشاره می‌گردد.

on the mean residual of records in the geometric random record model

record values have many applications in reliability theory, such as the shock and minimal repairs models. in this regard, many works have been done based on records in the classical model. in some random phenomena, instead of an infinite sequence of observations, a sequence of random length x1, . . . ,xn is available, where n is a positive integer-valued random variableindependent of xis. the record values of this sequence form a randomrecord model. the random record model arises naturally when the observationsarrive at time points determined by an independent point processobserved over a finite time. when the number of available observations isgeometrically distributed, this model is called the geometric random record(grr) model. mean residual life is an important criterion in reliability andlifetime analysis. in the present paper, based on the concept of the meanresidual life, some definitions for the mean residual of records are presentedin the random record model. also, the characterization of parent distributionby using the mean residual of records is investigated in the grr model.material and methodslet x1, . . . ,xn be a sequence of independent and identically distributedrandom variables with continuous cumulative distribution function f andn be a random variable independent of xis. let m denotes the number ofnontrivial records observed and rj , j ≤ m, is the jth nontrivial record ofthis sequence. if event {m ≥ n} occurs, based on two interpretations, the means residual of the nth record are defined asψrndn (ν) = e(rn − ν|r0 > ν;m ≥ n),ϕrndn (ν) = e(rn − ν|rn > ν;m ≥ n),and if event {m = n} occurs, these concepts are defined ase ψrndn (ν) = e(rn − ν|r0 > ν;m = n),eϕrndn (ν) = e(rn − ν|rn > ν;m = n).results and discussionin this study, it is shown that the four mentioned functions for the meanresidual of records can be obtained from f using an explicit relation. theconditions for the existence of the mean residual of records are investigated.also, it is proved that the mean residual of the nth record, n ≥ 0, is anascending sequence. moreover, it is shown that the parent distributionf can be characterized uniquely by the sequences {ψrndn (ν), n ≥ 0} and{e ψrndn (ν), n ≥ 0} in a grr model. also, ϕrndn (ν) andeϕrndn (ν) can characterizef uniquely for each n. finally, the characterization results are appliedto job search models to identify the wage-offer distributions.conclusionin this paper, the characterization of distributions is considered by usingthe mean residual of records in the grr model. researchers can apply theobtained characterization results for job search models to similar practicalproblems in seismology, industrial stress testing, sporting and athletic events,hydrology and meteorology.