In the above article (Bull. Math. Statist. Vol. 15, No. 3-4, 79-91), in Theorem 5.1 and its Corollary we need to set Assumption (I) in addition to Assumption (II). The proof of Theorem 5.1 made in the article is incomplete. Line 3 from the bottom of page 85-line 7 from the top of page 86 should read : For T = r/N there exists a Borel measurable map f N such that EN-1ng=7rNqg(sN, aN, sN-Fi, EN7cg) —5 _fNqg(sN, aN, sN+1, ENrg)+T I K1K2 ••• KN-1 with Prig ••• 7N-1q—prob. 1 . That is, EN-17'g Elf-v-v7,1g-Fr IK,K,--• KN_1 with Prig -•• 7N-ig—prob. 1. By Assumptions (I) and (II) we have zN-14g(sN-1, aN_i, sN, EN-17'g) 7r N-igg(sN-1, aN-1, SN, E IfN,N7og+rIK,K,.•• KN_,) with Prig ••• 7N-1g—prob. 1 . . 7rN-1(1g(sN-1, aN-1, SN, E (fN,N,r1g)-Fr IK1K2••• KN-2 with Prig ••• 7rN-2q—prob. 1. Using this procedure N times will produce a Markov policy 7r* = {f1, f2, -} such that E'rg< Elf Df 2,-,f N,Nlogd-Nr = E'g+e with p—prob. 1,
在上面的文章(牛。数学。中央集权。第15卷，第3-4卷，79-91页)，在定理5.1及其推论中，除了假设(II)，我们还需要设置假设(I)。文章中对定理5.1的证明是不完整的。第85页底部第3行-第86页顶部第7行应该是:对于T = r/N，存在一个Borel可测量映射fN，使得EN-1ng=7rNqg(sN, aN, sN- fi, EN7cg) -5 _fNqg(sN, aN, sN+1, ENrg)+ t1 K1K2•••KN-1 with Prig•••7N-1q-prob。1 ． 即EN-17'g Elf-v-v7,1g-Fr IK,K，——•KN_1与Prig -••7N-ig-prob。1. 根据假设(I)和(II)，我们得到zN-14g(sN-1, aN_i, sN, EN-17'g) 7r N-igg(sN-1, aN-1, sN, E - IfN,N7og+rIK,K，。••KN_，)与Prig••••7N-1g-prob。1 .;7rN-1(1g(SN -1, aN-1, SN, E (fN,N,r1g)-Fr IK1K2••••KN-2与Prig••••7rN-2q-prob。1. 使用此过程N次将产生一个Markov策略7r* = {f1, f2， -}，使得E'rg< Elf Df 2，-，f N, nlog - nr = E'g+ E, p - probb。1，
{"title":"CORRECTION TO \"MARKOVIAN DECISION PROCESSES WITH RECURSIVE REWARD FUNCTIONS\"","authors":"N. Furukawa, Seiichi Iwamoto","doi":"10.5109/13088","DOIUrl":"https://doi.org/10.5109/13088","url":null,"abstract":"In the above article (Bull. Math. Statist. Vol. 15, No. 3-4, 79-91), in Theorem 5.1 and its Corollary we need to set Assumption (I) in addition to Assumption (II). The proof of Theorem 5.1 made in the article is incomplete. Line 3 from the bottom of page 85-line 7 from the top of page 86 should read : For T = r/N there exists a Borel measurable map f N such that EN-1ng=7rNqg(sN, aN, sN-Fi, EN7cg) —5 _fNqg(sN, aN, sN+1, ENrg)+T I K1K2 ••• KN-1 with Prig ••• 7N-1q—prob. 1 . That is, EN-17'g Elf-v-v7,1g-Fr IK,K,--• KN_1 with Prig -•• 7N-ig—prob. 1. By Assumptions (I) and (II) we have zN-14g(sN-1, aN_i, sN, EN-17'g) 7r N-igg(sN-1, aN-1, SN, E IfN,N7og+rIK,K,.•• KN_,) with Prig ••• 7N-1g—prob. 1 . . 7rN-1(1g(sN-1, aN-1, SN, E (fN,N,r1g)-Fr IK1K2••• KN-2 with Prig ••• 7rN-2q—prob. 1. Using this procedure N times will produce a Markov policy 7r* = {f1, f2, -} such that E'rg< Elf Df 2,-,f N,Nlogd-Nr = E'g+e with p—prob. 1,","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125873914","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}
§ 1. Based on moments of variations of random elements on CV, 112), a simple sufficient condition for tightness** is considered. § 2. Let C2 C([0, 112) be the set of all real valued continuous functions on [0, 112 with the uniform topology. W. J. Park [2], [3] considered random elements on C2 to prove the existence of Wiener measure and invariance principle on C2. W. J. Park's sufficient condition for tightness (lemma 3 in [3]) is not necessarily easy to apply directly. The object of this paper is to give a handy sufficient condition for tightness of random elements on C2. We consider random elements {Z,i(t, s), t, s 1} n 1 satisfying :
§1。基于随机单元在CV, 112)上的变化矩，考虑了紧性**的一个简单充分条件。§2。设C2 C([0,112)是在[0,112]上具有一致拓扑的所有实值连续函数的集合。W. J. Park[2]，[3]考虑C2上的随机元素，证明了C2上的Wiener测度和不变性原理的存在性。W. J. Park的紧性充分条件([3]中的引理3)不一定容易直接应用。本文的目的是给出C2上随机元素紧密性的一个方便的充分条件。假设随机元素{Z,i(t, s)， t, s 1} n 1满足:
{"title":"A SIMPLE TIGHTNESS CONDITION FOR RANDOM ELEMENTS ON $ C([0,1]^2) $","authors":"T. Nagai","doi":"10.5109/13083","DOIUrl":"https://doi.org/10.5109/13083","url":null,"abstract":"§ 1. Based on moments of variations of random elements on CV, 112), a simple sufficient condition for tightness** is considered. § 2. Let C2 C([0, 112) be the set of all real valued continuous functions on [0, 112 with the uniform topology. W. J. Park [2], [3] considered random elements on C2 to prove the existence of Wiener measure and invariance principle on C2. W. J. Park's sufficient condition for tightness (lemma 3 in [3]) is not necessarily easy to apply directly. The object of this paper is to give a handy sufficient condition for tightness of random elements on C2. We consider random elements {Z,i(t, s), t, s 1} n 1 satisfying :","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123132313","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 A PATTERN CLASSIFICATION PROBLEM ON THE BASIS OF A TRAINING SEQUENCE ASSOCIATED WITH DEPENDENT RANDOM VARIABLES","authors":"Masafumi Watanabe","doi":"10.5109/13081","DOIUrl":"https://doi.org/10.5109/13081","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"95 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132223279","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":"GENERALIZED CONVEXITIES OF CONTINUOUS FUNCTIONS AND THEIR APPLICATION TO MATHEMATICAL PROGRAMMING","authors":"Shojiro Tagawa, 田川 正二郎","doi":"10.5109/13087","DOIUrl":"https://doi.org/10.5109/13087","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129224867","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 the paper [5], N. Furukawa and S. Iwamoto have defined Markovian decision processes with a new broad class of reward systems, that is, recursive reward functions, and have studied the existence and properties of optimal policies. Under some conditions on the reward functions, they have proved that there exists a (p, s)-optimal stationary policy and that in the case of a finite action space there exists an optimal stationary policy. These are some generalizations of results by D. Blackwell [3]. In this paper the author defines a dynamic programming problem with a recursive additive system which is referred to one type of Markovian decision processes with recursive reward functions defined by the previous authors [5]. This paper gives an algorithm for finding optimal stationary policies in the dynamic programming with the recursive additive system in the case of finite state and action spaces. Furthermore, we give several interesting examples with numerical computations to obtain optimal policies. The motivation to consider the dynamic programming problem with the recursive additive system is the following : If we restrict the " reward " in narrow sense, for instance, the money in economic systems or the loss in statistical decision problems, it will be appropriate for us to accept the total sum of stage-wise rewards as a performance index. That is so-called additive reward system. But many practical problems in the field of engineerings enable us to interpret the " reward " in wider sense. In those problems we often encounter much complicated reward systems that are more than so-called additive. We have an interesting class of such complicated reward systems in which we can find a common feature named " recursive additive ". By talking about various reward systems belonging to this class at the same time, we can make clear, as a dynamic programming problem, an important common property within the class, Our proofs are partially owing to Blackwell [2].
{"title":"DISCRETE DYNAMIC PROGRAMMING WITH RECURSIVE ADDITIVE SYSTEM","authors":"Seiichi Iwamoto","doi":"10.5109/13082","DOIUrl":"https://doi.org/10.5109/13082","url":null,"abstract":"In the paper [5], N. Furukawa and S. Iwamoto have defined Markovian decision processes with a new broad class of reward systems, that is, recursive reward functions, and have studied the existence and properties of optimal policies. Under some conditions on the reward functions, they have proved that there exists a (p, s)-optimal stationary policy and that in the case of a finite action space there exists an optimal stationary policy. These are some generalizations of results by D. Blackwell [3]. In this paper the author defines a dynamic programming problem with a recursive additive system which is referred to one type of Markovian decision processes with recursive reward functions defined by the previous authors [5]. This paper gives an algorithm for finding optimal stationary policies in the dynamic programming with the recursive additive system in the case of finite state and action spaces. Furthermore, we give several interesting examples with numerical computations to obtain optimal policies. The motivation to consider the dynamic programming problem with the recursive additive system is the following : If we restrict the \" reward \" in narrow sense, for instance, the money in economic systems or the loss in statistical decision problems, it will be appropriate for us to accept the total sum of stage-wise rewards as a performance index. That is so-called additive reward system. But many practical problems in the field of engineerings enable us to interpret the \" reward \" in wider sense. In those problems we often encounter much complicated reward systems that are more than so-called additive. We have an interesting class of such complicated reward systems in which we can find a common feature named \" recursive additive \". By talking about various reward systems belonging to this class at the same time, we can make clear, as a dynamic programming problem, an important common property within the class, Our proofs are partially owing to Blackwell [2].","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123076040","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 nonparametric selection problems in analysis of variance have been mainly developed for one-way layout models. For example, Lehmann [4], Puri and Puri [6] and Alam and Thompson [1] have respectively discussed the selection procedures based on the ranks of the observations. Randles [7] has also emphasized the use of the Hodges-Lehmann estimates for the same models to eliminate the difficulties concerning the least favorable configuration (cf. Rizvi and Woodworth DO. Only a work for the two-way layouts is seen in Hollander [3]. We consider some selection problems under the more general two-way layout models. Let Xia be the random observation on the i-th treatment IIi in the a-th block and suppose that
{"title":"ON SELECTION PROCEDURES FOR EXPONENTIAL DISTRIBUTIONS","authors":"J. B. Ofosu","doi":"10.5109/13078","DOIUrl":"https://doi.org/10.5109/13078","url":null,"abstract":"The nonparametric selection problems in analysis of variance have been mainly developed for one-way layout models. For example, Lehmann [4], Puri and Puri [6] and Alam and Thompson [1] have respectively discussed the selection procedures based on the ranks of the observations. Randles [7] has also emphasized the use of the Hodges-Lehmann estimates for the same models to eliminate the difficulties concerning the least favorable configuration (cf. Rizvi and Woodworth DO. Only a work for the two-way layouts is seen in Hollander [3]. We consider some selection problems under the more general two-way layout models. Let Xia be the random observation on the i-th treatment IIi in the a-th block and suppose that","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132319702","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 A CLASS OF NONLINEAR PROGRAMMING IN A BANACH SPACE","authors":"Shojiro Tagawa","doi":"10.5109/13086","DOIUrl":"https://doi.org/10.5109/13086","url":null,"abstract":"","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1974-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131086538","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 giving reasonable measures of association when a population is °stratified was first investigated by Aoyama [1] . Recently Wakimoto [3] considered the problem more extensively. He gave an estimator of the correlation coefficient based on a stratified random sample and showed it to be superior to the one given by Aoyama. The purpose of this paper is to propose new measures of association, test of independence and confidence intervals based on a stratified random sample. These measures are stratified version of Kendall and Speaman rank correlation coefficients. Throughout this paper we assume that each size of stratum is sufficiently large compared with that of sample taken from it so that the finite correction term may be neglected. In section 2 measures of association, tests of independence and confidence intervals based on a stratified random sample is given. A stratified version of Kendall rank correlation coefficient is discussed in section 2.1 and then in section 2.2 the one of Speaman type is discussed. In section 3 gains in efficiency due to stratification is demonstrated in the case of proportional allocation by comparing proposed measures with respect to Kendall and Speaman rank correlation coefficient based on a simple random sample.
{"title":"STRATIFIED RANDOM SAMPLING ; RANK CORRELATION COEFFICIENTS, TESTS OF INDEPENDENCE AND RANDOM CONFIDENCE INTERVALS","authors":"T. Yanagawa","doi":"10.5109/13069","DOIUrl":"https://doi.org/10.5109/13069","url":null,"abstract":"The problem of giving reasonable measures of association when a population is °stratified was first investigated by Aoyama [1] . Recently Wakimoto [3] considered the problem more extensively. He gave an estimator of the correlation coefficient based on a stratified random sample and showed it to be superior to the one given by Aoyama. The purpose of this paper is to propose new measures of association, test of independence and confidence intervals based on a stratified random sample. These measures are stratified version of Kendall and Speaman rank correlation coefficients. Throughout this paper we assume that each size of stratum is sufficiently large compared with that of sample taken from it so that the finite correction term may be neglected. In section 2 measures of association, tests of independence and confidence intervals based on a stratified random sample is given. A stratified version of Kendall rank correlation coefficient is discussed in section 2.1 and then in section 2.2 the one of Speaman type is discussed. In section 3 gains in efficiency due to stratification is demonstrated in the case of proportional allocation by comparing proposed measures with respect to Kendall and Speaman rank correlation coefficient based on a simple random sample.","PeriodicalId":287765,"journal":{"name":"Bulletin of Mathematical Statistics","volume":"107 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1973-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121627636","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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