### Citations

427 |
Statistical Inference Under Order Restrictions
- Barlow, Bartholomew, et al.
- 1972
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Citation Context ...values. Then we(P ) with P defined in (2.4) is equal to { P (e +1,a; p1·,...,pa·), 1 ≤ e ≤ a − 1, we(P )= 0, e = a. (2.16) The expressions of P (l, k; n1,...,nk) for k ≤ 4 are given in Barlow, et al. =-=[2]-=-, Section 3.3. For general k, Miwa, et al. [24] pointed out that P (l, k; n1,...,nk) can be evaluated numerically using a successive integration technique. From a practical point of view there is no d... |

328 |
Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions.
- Self, Liang
- 1987
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Citation Context ...of Chernoff [5], who discussed the asymptotic distribution of the likelihood ratio test statistic when the true parameter is on the boundary of the hypothesis parameter space (also see Self and Liang =-=[31]-=-). A proof is given in Section 4. Theorem 2.2 Assume that an a×b table {nij} is a sample from the multinomial ((a×b)nominal) distribution under the RC model (1.3) with the order restrictions (1.4) in ... |

310 |
Categorical data analysis. 2nd ed.
- Agresti
- 2002
(Show Context)
Citation Context ...g order-restricted models In this section a contingency table is analyzed as an example. This is a cross-classified table of job satisfaction by income given in Table 3.1, which is cited from Agresti =-=[1]-=-, Table 142.8. Everitt [11] reanalyzed the data using the (unrestricted) RC association model. This is a typical example of categorical data with both row and column variables ordinal. We reanalyze t... |

269 |
Curvature measures
- Federer
- 1959
(Show Context)
Citation Context ... (Takemura and Kuriki [34]). The following theorem states the continuity of the volume of the tube with respect to the distance δ(·, ·). This is a spherical version of Theorems 5.6 and 5.9 of Federer =-=[12]-=-. The proof is parallel to that of Federer [12] and is omitted. 25Theorem 4.2 Suppose that ɛ>0. LetAi, i =1, 2,..., and B be closed subsets of S n−1 with the critical radii θc(Ai),θc(B) ≥ ɛ such that... |

144 |
On the volume of tubes
- Weyl
- 1939
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Citation Context ...per probabilities of the χ2 distributions with coefficients characterized by some geometric quantities of the index set M. This theory is called the tube method, originating from Hotelling [19], Weyl =-=[37]-=-, and developed by Sun [33], Takemura and Kuriki [34], [35], and Kuriki and Takemura [22], [23]. In the following we give a brief summary of the tube method. Define a geodesic distance between two poi... |

97 |
On the Distribution of the Likelihood Ratio,”
- CHERNOFF
- 1954
(Show Context)
Citation Context ...ution of the likelihood ratio criterion for testing H : φ = 0 in the RC association model with the order restrictions is proved to be the same as that of n max{ ˆ φ, 0} 2 using the theory of Chernoff =-=[5]-=-. A formula for approximating their tail probabilities is then given by the tube method. In Section 3 we give an example of data analysis. Some techniques for fitting order-restricted models are propo... |

95 |
Simple models for the analysis of association in cross-classifications having ordered categories.
- Goodman
- 1979
(Show Context)
Citation Context ...evels of the ith row and the jth column, respectively. The multiplicative model for the same purpose is the RC association model (the rowcolumn-effect association model, RC model) proposed by Goodman =-=[14]-=-, [15], [16]: log pij = αi + βj + φµiνj. (1.3) To ensure identifiability, the same side conditions (1.2) are imposed. The RC association model can be regarded as a natural extension of the model by Jo... |

72 |
Tail probabilities of the maxima of Gaussian Random Fields.
- Sun
- 1993
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Citation Context ...aximum of a certain Gaussian random field. Recently an integral-geometric approach called the tube method has been developed for deriving the distribution of the maxima of Gaussian random fields (Sun =-=[33]-=-, Kuriki and Takemura [23], Takemura and Kuriki [35]). With the help of the tube method, we derive an expression approximating the upper tail probabilities of “inequality-restricted canonical correlat... |

60 |
The analysis of cross-classified data having ordered and/or unordered 20 Biometrics,
- Goodman
- 1985
(Show Context)
Citation Context ... ith row and the jth column, respectively. The multiplicative model for the same purpose is the RC association model (the rowcolumn-effect association model, RC model) proposed by Goodman [14], [15], =-=[16]-=-: log pij = αi + βj + φµiνj. (1.3) To ensure identifiability, the same side conditions (1.2) are imposed. The RC association model can be regarded as a natural extension of the model by Johnson and Gr... |

53 |
Tubes and spheres in n-spaces, and a class of statistical problems,
- Hotelling
- 1939
(Show Context)
Citation Context ...ation of upper probabilities of the χ2 distributions with coefficients characterized by some geometric quantities of the index set M. This theory is called the tube method, originating from Hotelling =-=[19]-=-, Weyl [37], and developed by Sun [33], Takemura and Kuriki [34], [35], and Kuriki and Takemura [22], [23]. In the following we give a brief summary of the tube method. Define a geodesic distance betw... |

30 |
Hotelling’s theorem on the volumes of tubes: some illustrations in simultaneous inference and data analysis,
- Johansen, Johnstone
- 1990
(Show Context)
Citation Context ...ness and local convexity of M (Assumptions 2.1 and 2.2) ensure the positiveness of θc. θc can be evaluated by the following theorem, which is an extension of Proposition 4.3 of Johansen and Johnstone =-=[20]-=-. Theorem 2.3 (Takemura and Kuriki [35], Lemma 2.1.) Let Nv(M) =Sv(M) ∗ ∩ span{v} ⊥ , (2.9) where Sv(M) ∗ = {x ∈ R n | x ′ y ≤ 0, ∀y ∈ Sv(M)} 9is the dual cone of Sv(M) in R n , span{v} ⊥ is the orth... |

27 | On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains - Takemura, Kuriki |

24 |
An analysis of a two-way model with interaction and no
- Johnson, Graybill
- 1972
(Show Context)
Citation Context ...ij = αi + βj + φµiνj. (1.3) To ensure identifiability, the same side conditions (1.2) are imposed. The RC association model can be regarded as a natural extension of the model by Johnson and Graybill =-=[21]-=- for two-way ANOVA in two-way categorical data analysis. In the RC association model the maximum likelihood method is usually used to estimate parameters. Numerical algorithms for maximizing likelihoo... |

23 |
The asymptotic distribution of singular values with applications to canonical correlations and correspondance analysis,”
- L, TYLER
- 1994
(Show Context)
Citation Context ...0 converges in distribution to the square root of the largest eigenvalue of an (a − 1) × (a − 1) Wishart random matrix with (b − 1) degrees of freedom, Wa−1(b − 1,Ia−1) (O’Neill [27], Eaton and Tyler =-=[9]-=-). Haberman [17] proved that in the RC association model (1.3) the likelihood ratio criterion for testing 4H : φ = 0 has the same asymptotic distribution as the largest eigenvalue of the Wishart matr... |

22 | Tail probabilities of the maxima of multilinear forms and their applications, Ann
- Kuriki, Takemura
- 2001
(Show Context)
Citation Context ...an random field. Recently an integral-geometric approach called the tube method has been developed for deriving the distribution of the maxima of Gaussian random fields (Sun [33], Kuriki and Takemura =-=[23]-=-, Takemura and Kuriki [35]). With the help of the tube method, we derive an expression approximating the upper tail probabilities of “inequality-restricted canonical correlations”, which includes ˆ φ ... |

20 |
Association models and canonical correlation in the analysis of cross classifications having ordered categories
- Goodman
- 1981
(Show Context)
Citation Context ...of the ith row and the jth column, respectively. The multiplicative model for the same purpose is the RC association model (the rowcolumn-effect association model, RC model) proposed by Goodman [14], =-=[15]-=-, [16]: log pij = αi + βj + φµiνj. (1.3) To ensure identifiability, the same side conditions (1.2) are imposed. The RC association model can be regarded as a natural extension of the model by Johnson ... |

16 |
Weights of χ 2 distribution for smooth or piecewise smooth cone alternatives
- Takemura, Kuriki
- 1997
(Show Context)
Citation Context ...cients characterized by some geometric quantities of the index set M. This theory is called the tube method, originating from Hotelling [19], Weyl [37], and developed by Sun [33], Takemura and Kuriki =-=[34]-=-, [35], and Kuriki and Takemura [22], [23]. In the following we give a brief summary of the tube method. Define a geodesic distance between two points on the unit sphere Sn−1 by the length of the grea... |

14 |
Analysis of contingency tables by correspondence models subject to order constraints
- Ritov, Gilula
- 1993
(Show Context)
Citation Context ... above reasons, models with the order restrictions (1.4) or (1.5) have been proposed by many authors. See, for example, Nishisato and Arri [26], Tanaka [36], Saito and Otsu [30], and Ritov and Gilula =-=[29]-=- in the context of the correspondence analysis, and Goodman [16] and Ritov and Gilula [28] in the context of the RC association model. Douglas and Fienberg [8], and Etzioni, et al. [10] give excellent... |

13 |
Consistency and asymptotic normality of MLE’s for exponential models
- Berk
- 1972
(Show Context)
Citation Context ...oof. Under the assumptions that ωjs are locally compact and contain the true value, the MLEs ˆ θn,j ∈ ωj under Hj exist for n sufficiently large, and converge to θ o with probability one as n →∞(Berk =-=[4]-=-, Example 4). From this fact and the assumption of the existence of approximating cones, we can see that all of the assumptions of Theorem 1 of Chernoff [5] are fulfilled, and −2 log Λn is proved to c... |

9 | Restricted canonical correlations
- Das, Sen
- 1994
(Show Context)
Citation Context ...aper the maximum ˆρ = max v∈K, w∈L √ v ′ ˆ Σxxv v ′ Σxyw ˆ √ w ′ ˆ Σyyw (2.1) is called the (sample) inequality-restricted canonical correlation. Our definition is an extension of that of Das and Sen =-=[6]-=-, [7]. Note that the maximum ˆρ exists unless v ′ Σxxv ˆ = 0, ∀v ∈ K or w ′ Σyyw ˆ =0, ∀w ∈ L. Obviously when K = Rp and L = Rq , (2.1) is reduced to the largest canonical correlation in the usual def... |

7 |
Asymptotic distribution of restricted canonical correlations and relevant resampling methods
- Das, Sen
- 1996
(Show Context)
Citation Context ...arity in the models (1.1) or (1.3), such that the scores µi and νj are not identifiable under the independence model φ = 0. For this reason our problem is crucially different from that of Das and Sen =-=[7]-=-, who proved asymptotic normality of the inequality-restricted canonical correlation when the true canonical correlation is nonzero and maximizing scores (µi, νj, in our case) are identifiable. The co... |

7 |
Test for Independence in Two-way Contingency Tables Based on Canonical Correlation and on Linear-By-Linear Interaction. The Annals of Statistics
- Haberman
(Show Context)
Citation Context ...distribution to the square root of the largest eigenvalue of an (a − 1) × (a − 1) Wishart random matrix with (b − 1) degrees of freedom, Wa−1(b − 1,Ia−1) (O’Neill [27], Eaton and Tyler [9]). Haberman =-=[17]-=- proved that in the RC association model (1.3) the likelihood ratio criterion for testing 4H : φ = 0 has the same asymptotic distribution as the largest eigenvalue of the Wishart matrix Wa−1(b − 1,Ia... |

7 |
Distribution of the maximum of Gaussian random tube method and Euler characteristic method
- Kuriki, Takemura
(Show Context)
Citation Context ...ic quantities of the index set M. This theory is called the tube method, originating from Hotelling [19], Weyl [37], and developed by Sun [33], Takemura and Kuriki [34], [35], and Kuriki and Takemura =-=[22]-=-, [23]. In the following we give a brief summary of the tube method. Define a geodesic distance between two points on the unit sphere Sn−1 by the length of the great circle joining the two points: dis... |

7 |
The order-restricted RC model for ordered contingency tables: Estimation and testing for t
- Ritov, Gilula
- 1991
(Show Context)
Citation Context ...ny authors. See, for example, Nishisato and Arri [26], Tanaka [36], Saito and Otsu [30], and Ritov and Gilula [29] in the context of the correspondence analysis, and Goodman [16] and Ritov and Gilula =-=[28]-=- in the context of the RC association model. Douglas and Fienberg [8], and Etzioni, et al. [10] give excellent surveys of the relevant area. However, almost all of these studies have treated fitting t... |

6 |
Nonlinear programming approach to optimal scaling of partially ordered categories
- Nishisato, Arri
- 1975
(Show Context)
Citation Context ...(1.7), respectively (also see Gilula and Ritov [13]). For the above reasons, models with the order restrictions (1.4) or (1.5) have been proposed by many authors. See, for example, Nishisato and Arri =-=[26]-=-, Tanaka [36], Saito and Otsu [30], and Ritov and Gilula [29] in the context of the correspondence analysis, and Goodman [16] and Ritov and Gilula [28] in the context of the RC association model. Doug... |

6 |
On differentiability of metric projections in R n
- Shapiro
- 1987
(Show Context)
Citation Context ...R k | ∑ k i=1 ˜ θi =0} be the tangent space of Ω at θ o . Assume that, for j =0, 1, ωj has the approximating cone (tangent cone in the sense of Definition 2 of Chernoff [5] or Definition 1 of Shapiro =-=[32]-=-) S(ωj) ⊂ T (Ω) at θ o . That is, for each ω = ωj, when θ o is an accumulating point of ω, a closed cone S(ω) ⊂ T (Ω) exists, satisfying: (i) for any sequence yl ∈ ω \{θ o } such that liml→∞ yl = θ o ... |

5 |
Inferential ordinal correspondence analysis : motivation, derivation and limitations
- Gilula, Ritov
- 1990
(Show Context)
Citation Context ...elihood ratio. In other words, by imposing the order restrictions the models (1.1) and (1.3) give simple models that embody the partial orders (1.6) and (1.7), respectively (also see Gilula and Ritov =-=[13]-=-). For the above reasons, models with the order restrictions (1.4) or (1.5) have been proposed by many authors. See, for example, Nishisato and Arri [26], Tanaka [36], Saito and Otsu [30], and Ritov a... |

5 |
Asymptotic Distributions of the Canonical Correlations from Contingency Tables
- O'NEILL
- 1978
(Show Context)
Citation Context ...endence model H : φ = 0 converges in distribution to the square root of the largest eigenvalue of an (a − 1) × (a − 1) Wishart random matrix with (b − 1) degrees of freedom, Wa−1(b − 1,Ia−1) (O’Neill =-=[27]-=-, Eaton and Tyler [9]). Haberman [17] proved that in the RC association model (1.3) the likelihood ratio criterion for testing 4H : φ = 0 has the same asymptotic distribution as the largest eigenvalu... |

4 |
Maximum likelihood estimation of the RC(M) association model
- Becker
- 1990
(Show Context)
Citation Context ...alysis. In the RC association model the maximum likelihood method is usually used to estimate parameters. Numerical algorithms for maximizing likelihood are well developed (e.g., Goodman [16], Becker =-=[3]-=-). In this paper we focus on the analysis of two-way contingency tables where the row and/or the column variables are ordinal. To analyze such ordered categorical data, we use the correspondence analy... |

3 | Calculations of Level Probabilities for Normal Random Variables with Unequal Variances with Applications to Bartholomew’s Test in Unbalanced OneWay Models - Miwa, AJ, et al. - 2000 |

2 |
An overview of dependency models for cross-classi ed categorical data involving ordinal variables
- Douglas, Fienberg
- 1990
(Show Context)
Citation Context ...ito and Otsu [30], and Ritov and Gilula [29] in the context of the correspondence analysis, and Goodman [16] and Ritov and Gilula [28] in the context of the RC association model. Douglas and Fienberg =-=[8]-=-, and Etzioni, et al. [10] give excellent surveys of the relevant area. However, almost all of these studies have treated fitting the model for descriptive purposes. From the viewpoint of statistical ... |

2 |
The Analysis of Contingency Tables,” 2nd ed
- Everitt
- 1992
(Show Context)
Citation Context ...n this section a contingency table is analyzed as an example. This is a cross-classified table of job satisfaction by income given in Table 3.1, which is cited from Agresti [1], Table 142.8. Everitt =-=[11]-=- reanalyzed the data using the (unrestricted) RC association model. This is a typical example of categorical data with both row and column variables ordinal. We reanalyze the data using the order-rest... |

2 |
Use of cumulative efficient scores for testing ordered alternatives in discrete models
- Hirotsu
- 1982
(Show Context)
Citation Context ...gest eigenvalue of the Wishart matrix Wa−1(b − 1,Ia−1) under the null hypothesis. In contrast to these cases, the null distribution of ˆ φ under the order restrictions was completely unknown. Hirotsu =-=[18]-=- suggested the use of ˆ φ as a test statistic for testing independence, but he pointed out difficulties in handling its distribution. In this paper we first show that the asymptotic distribution of ˆ ... |

2 |
A method of optimal scaling for multivariate ordinal data and its extensions
- Saito, Otsu
- 1988
(Show Context)
Citation Context ...la and Ritov [13]). For the above reasons, models with the order restrictions (1.4) or (1.5) have been proposed by many authors. See, for example, Nishisato and Arri [26], Tanaka [36], Saito and Otsu =-=[30]-=-, and Ritov and Gilula [29] in the context of the correspondence analysis, and Goodman [16] and Ritov and Gilula [28] in the context of the RC association model. Douglas and Fienberg [8], and Etzioni,... |

2 |
Optimal scaling for arbitrarily ordered categories
- Tanaka
- 1979
(Show Context)
Citation Context ...tively (also see Gilula and Ritov [13]). For the above reasons, models with the order restrictions (1.4) or (1.5) have been proposed by many authors. See, for example, Nishisato and Arri [26], Tanaka =-=[36]-=-, Saito and Otsu [30], and Ritov and Gilula [29] in the context of the correspondence analysis, and Goodman [16] and Ritov and Gilula [28] in the context of the RC association model. Douglas and Fienb... |