A system consisting of two dissimilar, redundant, repairable units is considered. We shall say that a major breakdown occurs in the system when both units fails. The system fails if one unit under repair is not repaired within a fixed time measured from the instant at which major breakdown occurs, or if the number of major breakdowns during the mission period exceeds a fixed number. As a special case, this number is allowed to be " infinite ". The Laplace transforms of the reliability and the mean time to system failure are derived, and the explicit formulas in the special cases are exhibited. § 1. Model definition. 1. The system consists of two dissimilar redundant units Al and A2. 2. There is only one repair station. When one unit fails, its repair begins at once, and when two units fail simultaneously, unit Ai is sent for repair with a specified constant probability ai, i= 1, 2, where cr1-Fa2=1. Concerning failure and repair we assume the following : 3. When the two units are good, unit-failures occur as three independent Poission processes with failure rates 21, 22 and 212. Events in the process with rate Ai cause failure of unit Ai only, and events in the process with rate 212 cause simultaneous failure of unit Al and A2. 4. When only one unit, Ai is good, failure of the unit is Poisson with rate 2/i 2i. 5. The repair time for each unit, Ai is independently distributed with general probability density function f i(t), but must be well behaved enough for the appropriate analytic operations to be performed. 6. The failure and repair processes for the two units are entirely independent. 7. The repaired unit is considered to be new again. * This research was supported by the Science Research Council under Grant No . 4349/00. ** Sheffield University , Sheffield and Osaka University, Osaka. *** Gifu University , Gifu.
{"title":"RENEWAL THEORETICAL APPROACH TO THE MISSION RELIABILITY OF A REDUNDANT REPAIRABLE SYSTEM WITH TWO DISSIMILAR UNITS","authors":"M. Kodama, J. Fukuta","doi":"10.5109/13098","DOIUrl":"https://doi.org/10.5109/13098","url":null,"abstract":"A system consisting of two dissimilar, redundant, repairable units is considered. We shall say that a major breakdown occurs in the system when both units fails. The system fails if one unit under repair is not repaired within a fixed time measured from the instant at which major breakdown occurs, or if the number of major breakdowns during the mission period exceeds a fixed number. As a special case, this number is allowed to be \" infinite \". The Laplace transforms of the reliability and the mean time to system failure are derived, and the explicit formulas in the special cases are exhibited. § 1. Model definition. 1. The system consists of two dissimilar redundant units Al and A2. 2. There is only one repair station. When one unit fails, its repair begins at once, and when two units fail simultaneously, unit Ai is sent for repair with a specified constant probability ai, i= 1, 2, where cr1-Fa2=1. Concerning failure and repair we assume the following : 3. When the two units are good, unit-failures occur as three independent Poission processes with failure rates 21, 22 and 212. Events in the process with rate Ai cause failure of unit Ai only, and events in the process with rate 212 cause simultaneous failure of unit Al and A2. 4. When only one unit, Ai is good, failure of the unit is Poisson with rate 2/i 2i. 5. The repair time for each unit, Ai is independently distributed with general probability density function f i(t), but must be well behaved enough for the appropriate analytic operations to be performed. 6. The failure and repair processes for the two units are entirely independent. 7. The repaired unit is considered to be new again. * This research was supported by the Science Research Council under Grant No . 4349/00. ** Sheffield University , Sheffield and Osaka University, Osaka. *** Gifu University , Gifu.","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1975-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122222254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ON SLIPPAGE RANK TESTS-(I) : THE DERIVATION OF OPTIMAL PROCEDURES","authors":"Itsuro Kakiuchi, M. Kimura","doi":"10.5109/13095","DOIUrl":"https://doi.org/10.5109/13095","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1975-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122317433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we shall present two types of sufficient conditions under which a switching function can be represented by a threshold gate, i. e., a threshold function, with n variables for an arbitrary positive integer n. In deriving these sufficient conditions, a new concept called an orientating vector is introduced in this paper and it will play an important role in our discussion since it gives an insight into the structure of a threshold gate. We shall begin with notations and preliminaries in Section 2. In Section 3, we shall introduce the notion of orientating vectors in terms of which we give the sufficient conditions for threshold functions. Indeed, an orientating vector can be used to classify the set of all the input vectors Ix"), x(2), ••• , x(2n)} into two subsets where one is a set of true (i. e., on) vectors and the other a set of false (i. e., off) vectors. If we arrange all input vectors in an inverse lexical order (see Kitagawa [3]), then, for any p, 1 p_27', a classification of all the input vectors into the two sets Ix"), x(2) x(1 and fx(p+1) x(p+2), x(271)1 represents a threshold function as stated in Proposition 3.2. In Section 4, it is described that the combination of two sufficient conditions turn out to be necessary so far as p in the above classification is not greater than 4. This is the reason why the combination of these two sufficient conditions amounts to be necessary and sufficient so far as n is not greater than 3 as given in Section 5. Our results can be compared with the notion of 2-asummability due to Elgot [2] and Chow [1] which gives the necessary and sufficient condition that a switching function is a threshold function when n in not greater than 8. It is noted that the results of this paper can be used to get all the possible digraphs associated with dynamical behaviors of the neuronic equation
{"title":"SUFFICIENT CONDITIONS FOR SWITCHING FUNCTIONS TO BE THRESHOLD ONES AND THEIR APPLICATIONS","authors":"Shojiro Tagawa","doi":"10.5109/13099","DOIUrl":"https://doi.org/10.5109/13099","url":null,"abstract":"In this paper we shall present two types of sufficient conditions under which a switching function can be represented by a threshold gate, i. e., a threshold function, with n variables for an arbitrary positive integer n. In deriving these sufficient conditions, a new concept called an orientating vector is introduced in this paper and it will play an important role in our discussion since it gives an insight into the structure of a threshold gate. We shall begin with notations and preliminaries in Section 2. In Section 3, we shall introduce the notion of orientating vectors in terms of which we give the sufficient conditions for threshold functions. Indeed, an orientating vector can be used to classify the set of all the input vectors Ix\"), x(2), ••• , x(2n)} into two subsets where one is a set of true (i. e., on) vectors and the other a set of false (i. e., off) vectors. If we arrange all input vectors in an inverse lexical order (see Kitagawa [3]), then, for any p, 1 p_27', a classification of all the input vectors into the two sets Ix\"), x(2) x(1 and fx(p+1) x(p+2), x(271)1 represents a threshold function as stated in Proposition 3.2. In Section 4, it is described that the combination of two sufficient conditions turn out to be necessary so far as p in the above classification is not greater than 4. This is the reason why the combination of these two sufficient conditions amounts to be necessary and sufficient so far as n is not greater than 3 as given in Section 5. Our results can be compared with the notion of 2-asummability due to Elgot [2] and Chow [1] which gives the necessary and sufficient condition that a switching function is a threshold function when n in not greater than 8. It is noted that the results of this paper can be used to get all the possible digraphs associated with dynamical behaviors of the neuronic equation","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1975-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116561036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"ON TESTS FOR EQUALITY OF COHERENCES OF TWO BIVARIATE STATIONARY TIME SERIESES","authors":"T. Nagai, M. Goto","doi":"10.5109/13097","DOIUrl":"https://doi.org/10.5109/13097","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1975-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116743310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"PARTLY SUFFICIENT STATISTICS AND COMPLETE CLASS THEOREMS FOR STATISTICAL DECISION PROBLEMS WITH NUISANCE PARAMETERS","authors":"Akimichi Okuma","doi":"10.5109/13094","DOIUrl":"https://doi.org/10.5109/13094","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"136 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1975-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121569265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of selecting a subset of fixed size s which includes the t best of k populations (t s < k), based on a pre-determined sample size n from each of the k populations, is studied as a multiple decision problem. It is assumed that the bestness of a population is characterized by its scale parameter ; the best population being the one with the largest scale parameter, and so on. Exact small and large sample methods of finding n are given for the scale parameter problem for ( i) Gamma distributions with known (possibly unequal) shape parameters (ii) Weibull distributions with known shape parameters. Some tables computed by these methods are provided. A dual problem is also discussed.
基于预先确定的k个总体的样本量n，选择包含k个总体(t s < k)中的t个最优的固定大小的子集s的问题，作为一个多重决策问题进行研究。假设种群的最优性由其尺度参数表征;最佳种群是具有最大尺度参数的种群，以此类推。对于(i)具有已知(可能不相等)形状参数的Gamma分布(ii)具有已知形状参数的Weibull分布，给出了确定n的精确小样本和大样本方法。给出了用这些方法计算的一些表。讨论了一个对偶问题。
{"title":"ON SELECTING A SUBSET WHICH INCLUDES THE $ t $ BEST OF $ k $ POPULATIONS : SCALE PARAMETER CASE","authors":"J. B. Ofosu","doi":"10.5109/13090","DOIUrl":"https://doi.org/10.5109/13090","url":null,"abstract":"The problem of selecting a subset of fixed size s which includes the t best of k populations (t s < k), based on a pre-determined sample size n from each of the k populations, is studied as a multiple decision problem. It is assumed that the bestness of a population is characterized by its scale parameter ; the best population being the one with the largest scale parameter, and so on. Exact small and large sample methods of finding n are given for the scale parameter problem for ( i) Gamma distributions with known (possibly unequal) shape parameters (ii) Weibull distributions with known shape parameters. Some tables computed by these methods are provided. A dual problem is also discussed.","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1975-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121377753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let X(t) = (X1(t), X2(t), • •• , Xp(t))', t= •-• , —1, 0, 1, 2, •-• be a real multiple stationary Gaussian process with zero mean and with covariance matrix T(h) = EIX(t)X(t-l-h)'}, where " ' " denote the transposes. We assume that the spectral density matrix f(2)=Ifik(2), j, k= 1, 2, ••• PI, —7r 2 7r, exists with f(2)= riv2F(v). Let {X(1), X(2), , X(N)} be N observables of the process X(t) and X N = (X(I-)' X(2)', ••• , X(N)')' a pN-column vector obtained by rearranging {X(1), X(2), ••• , X(N)}. We denote rN=E{XNX10.
{"title":"THE LIMIT PROCESS OF WEIGHTED SPECTRAL ESTIMATES FOR A MULTIPLE STATIONARY GAUSSIAN PROCESS","authors":"T. Nagai","doi":"10.5109/13092","DOIUrl":"https://doi.org/10.5109/13092","url":null,"abstract":"Let X(t) = (X1(t), X2(t), • •• , Xp(t))', t= •-• , —1, 0, 1, 2, •-• be a real multiple stationary Gaussian process with zero mean and with covariance matrix T(h) = EIX(t)X(t-l-h)'}, where \" ' \" denote the transposes. We assume that the spectral density matrix f(2)=Ifik(2), j, k= 1, 2, ••• PI, —7r 2 7r, exists with f(2)= riv2F(v). Let {X(1), X(2), , X(N)} be N observables of the process X(t) and X N = (X(I-)' X(2)', ••• , X(N)')' a pN-column vector obtained by rearranging {X(1), X(2), ••• , X(N)}. We denote rN=E{XNX10.","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1975-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126451900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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