{"title":"Estimating Regression Coefficients by Minimizing the Dispersion of the Residuals","authors":"Louis A. Jaeckel","doi":"10.1214/AOMS/1177692377","DOIUrl":"https://doi.org/10.1214/AOMS/1177692377","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"84 1","pages":"1449-1458"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74971056","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":"Sharp One-Sided Confidence Bounds Over Positive Regions","authors":"R. Bohrer, George K. Francis","doi":"10.1214/AOMS/1177692386","DOIUrl":"https://doi.org/10.1214/AOMS/1177692386","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"29 1","pages":"1541-1548"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81552992","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 ${p_n}_0^N$ be a discrete distribution on $0 leqq n leqq N$ and let $g(u) = sum^infty_0 p_n u^n$ be its $operatorname{pgf}$. Then for $0 leqq t < infty g_t(u) = g(u + t)/g(1 + t) = sum^N_0 p_n(t)u^n$ is a family of $operatorname{pgf}$'s indexed by $t$. It is shown that there is a unique value $t^ast$ such that ${p_n(t)}_0^N$ is $log$-concave $(PF_2)$ for all $t geqq t^ast$ and is not $log$-concave for $0 < t < t^ast$. As a consequence one finds the infinite set of moment inequalities ${mu_{lbrack rrbrack}/mathbf{r}!}^{1/r} geqq {mu_{lbrack r+1rbrack}/(r + 1)!}^{1/r+1} mathbf{r} = 1,2,3,cdots$ etc. where $mu_{lbrack rrbrack}$ is the $mathbf{r}$th factorial moment of ${p_n}_0^N$ when the lattice distribution is $log$-concave. The known set of inequalities for the continuous analogue is shown to follow from the discrete inequalities.
{"title":"A Threshold for Log-Concavity for Probability Generating Functions and Associated Moment Inequalities","authors":"J. Keilson","doi":"10.1214/AOMS/1177692406","DOIUrl":"https://doi.org/10.1214/AOMS/1177692406","url":null,"abstract":"Let ${p_n}_0^N$ be a discrete distribution on $0 leqq n leqq N$ and let $g(u) = sum^infty_0 p_n u^n$ be its $operatorname{pgf}$. Then for $0 leqq t < infty g_t(u) = g(u + t)/g(1 + t) = sum^N_0 p_n(t)u^n$ is a family of $operatorname{pgf}$'s indexed by $t$. It is shown that there is a unique value $t^ast$ such that ${p_n(t)}_0^N$ is $log$-concave $(PF_2)$ for all $t geqq t^ast$ and is not $log$-concave for $0 < t < t^ast$. As a consequence one finds the infinite set of moment inequalities ${mu_{lbrack rrbrack}/mathbf{r}!}^{1/r} geqq {mu_{lbrack r+1rbrack}/(r + 1)!}^{1/r+1} mathbf{r} = 1,2,3,cdots$ etc. where $mu_{lbrack rrbrack}$ is the $mathbf{r}$th factorial moment of ${p_n}_0^N$ when the lattice distribution is $log$-concave. The known set of inequalities for the continuous analogue is shown to follow from the discrete inequalities.","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"49 1","pages":"1702-1708"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77608323","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 Iterative Scaling for Log-Linear Models","authors":"J. Darroch, D. Ratcliff","doi":"10.1214/AOMS/1177692379","DOIUrl":"https://doi.org/10.1214/AOMS/1177692379","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"1 1","pages":"1470-1480"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81962820","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":"Rates of Convergence for Weighted Sums of Random Variables","authors":"F. T. Wright","doi":"10.1214/AOMS/1177692403","DOIUrl":"https://doi.org/10.1214/AOMS/1177692403","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"59 1","pages":"1687-1691"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86694457","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 independent sampling model, Rao-Blackwell distribution function estimators $tilde{F}_n(x)$ obtained by conditioning on sufficient statistics $T_n(X_1, cdots, X_n)$ are considered. If for each $n geqq 1, T_n$ is symmetric in $X_1,cdots, X_n$ and $T_{n+1}$ is $mathscr{B}(T_n, X_{n+1})$ measurable, it is shown that $tilde{F}_n(x)$ converges strongly to the corresponding $F(x)$ and uniformly in $x$. This is a direct generalization of the Glivenko-Cantelli theorem.
{"title":"Uniform Strong Consistency of Rao-Blackwell Distribution Function Estimators","authors":"F. O'Reilly, C. Quesenberry","doi":"10.1214/AOMS/1177692401","DOIUrl":"https://doi.org/10.1214/AOMS/1177692401","url":null,"abstract":"In the independent sampling model, Rao-Blackwell distribution function estimators $tilde{F}_n(x)$ obtained by conditioning on sufficient statistics $T_n(X_1, cdots, X_n)$ are considered. If for each $n geqq 1, T_n$ is symmetric in $X_1,cdots, X_n$ and $T_{n+1}$ is $mathscr{B}(T_n, X_{n+1})$ measurable, it is shown that $tilde{F}_n(x)$ converges strongly to the corresponding $F(x)$ and uniformly in $x$. This is a direct generalization of the Glivenko-Cantelli theorem.","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"18 1","pages":"1678-1679"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75717961","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":"Conditions for Metric Transitivity for Stationary Gaussian Processes on Groups","authors":"J. Blum, B. Eisenberg","doi":"10.1214/AOMS/1177692412","DOIUrl":"https://doi.org/10.1214/AOMS/1177692412","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"55 1","pages":"1737-1741"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79650737","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 the Reduction of Associate Classes for Certain PBIB Designs","authors":"S. Kageyama","doi":"10.1214/AOMS/1177692385","DOIUrl":"https://doi.org/10.1214/AOMS/1177692385","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"6 1","pages":"1528-1540"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73033913","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 Bonferroni-Type Inequalities of the Same Degree for the Probability of Unions and Intersections","authors":"M. Sobel, V. Uppuluri","doi":"10.1214/AOMS/1177692387","DOIUrl":"https://doi.org/10.1214/AOMS/1177692387","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"71 1","pages":"1549-1558"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89884195","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":"A Note on the Classical Occupancy Problem","authors":"C. J. Park","doi":"10.1214/AOMS/1177692405","DOIUrl":"https://doi.org/10.1214/AOMS/1177692405","url":null,"abstract":"","PeriodicalId":50764,"journal":{"name":"Annals of Mathematical Statistics","volume":"3 1","pages":"1698-1701"},"PeriodicalIF":0.0,"publicationDate":"1972-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80274301","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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