In a generalized linear model of binary data, we consider models based on a general link function including a logistic regression model and a probit model as special cases. For testing the null ...hypothesis H0 that the considered model is correct, we consider a family of ϕ-divergence goodness-of-fit test statistics Cϕ that includes a power divergence family of statistics Ra. We propose a transformed Cϕ statistics that improves the speed of convergence to a chi-square limiting distribution and show numerically that the transformed Ra statistic performs well. We also give a real data example of the transformed Ra statistic being more reliable than the original Ra statistic for testing H0.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In a generalized linear model with binary response, the role of a link function is important to find a model that fits data well. Aranda-Ordaz (1981) proposed a family of link functions that includes ...a logistic link function and a complementary log-log function. In this paper, we propose a new family of models on the basis of a family of link functions by extending the family proposed by Aranda-Ordaz (1981). We also consider tests to determine whether the new model fits data well. Examples of artificial and real data showing that our new model is more appropriate than the Aranda-Ordaz model are presented.
We consider φ-divergence statistics Cφ for the test of one factor independence from the other two in an r × s × t contingency table. Statistics Cφ include the statistics Ra based on the power ...divergence as a special case. Statistic R0 is the log likelihood ratio statistic and R1 is Pearson's X2 statistic. Statistic R2/3 corresponds to the statistic for the goodness-of-fit test recommended by Cressie and Read (1984). Statistics Cφ have the same chi-square limiting distribution under the hypothesis that one factor and the other two are independent. In this paper, when we assume that the distribution of Cφ is continuous, we show the derivation of an expression of approximation based on a multivariate Edgeworth expansion for the distribution of Cφ under the hypothesis that one factor and the other two are independent. Using the expression, we propose a new approximation of the distribution of Cφ. In addition, on the basis of the approximation, we obtain transformed statistics that improve the speed of convergence to a chi-square limiting distribution of Cφ. By numerical comparison in the case of Ra, we show that the transformed statistics perform well for a small sample.
Tests of the hypothesis of conditional independence in J×K×L contingency tables are considered. An expression to approximate the null distribution of the test statistics is derived. Using this ...expression, transformed statistics are obtained which converge to a chi-square limiting distribution faster than the original statistics do. Simulations are used to compare the transformed statistics with the original ones, and transformed statistics are proposed based on a Bartlett-type adjustment. Through this work and earlier ones, we cover testing hierarchical loglinear models in three-way tables except for a model having three interaction terms.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Statistics R
a
based on power divergence can be used for testing the homogeneity of a product multinomial model. All R
a
have the same chi-square limiting distribution under the null hypothesis of ...homogeneity. R
0
is the log likelihood ratio statistic and R
1
is Pearson's X
2
statistic. In this article, we consider improvement of approximation of the distribution of R
a
under the homogeneity hypothesis. The expression of the asymptotic expansion of distribution of R
a
under the homogeneity hypothesis is investigated. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, a new approximation of the distribution of R
a
is proposed. A moment-corrected type of chi-square approximation is also derived. By numerical comparison, we show that both of the approximations perform much better than that of usual chi-square approximation for the statistics R
a
when a ≤ 0, which include the log likelihood ratio statistic.
Full text
Available for:
BFBNIB, GIS, IJS, KISLJ, NUK, PNG, UL, UM, UPUK
Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics,
R
a
, as multinomial goodness-of-fit statistics. Each
R
a
has a limiting noncentral chi-square ...distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of
R
a
under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of
R
a
under nonlocal alternatives can be improved. We apply the expansion to the power approximation for
R
a
. The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In the multinomial goodness-of-fit test, it is well known that the log-likelihood ratio statistic T has a limiting chi-square distribution under a simple null hypothesis. Siotani and Fujikoshi (1984) ...derived an asymptotic expansion for the null distribution of the log-likelihood ratio statistic T under a simple null hypothesis. They proposed J1+Ĵ2 as an approximation of the lower probability of T, where J1 is a term of multivariate Edgeworth expansion and Ĵ2 is a first order approximation of a discontinuous term. In this paper, we introduce discrete Bartlett-type transformed statistics T* and T** of the log-likelihood ratio statistic T that are constructed from J1 and Ĵ2. By numerical comparison, we found that the accuracy of approximation of the log-likelihood ratio statistic T to a chi-square distribution is improved by T* or T**.
N. Cressie and T. R. C. Read (1984, J. Roy. Statist. Soc. B46, 440–464) introduced a class of multinomial goodness-of-fit statistics Ra based on power divergence. All Ra have the same chi-square ...limiting distribution under null hypothesis and have the same noncentral chi-square limiting distribution under local alternatives. In this paper, we investigate asymptotic approximations for the distributions of Ra under local alternatives. We obtain an expression of approximation for the distribution of Ra under local alternatives. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the power of Ra. We call the approximation AE approximation. By numerical investigation of the accuracy of the AE approximation, we present a range of sample size n that the omission of the discontinuous term exercises only slight influence on power approximation of Ra. We find that the AE approximation is effective for a much wider range of the value of a than the other power approximations, except for an approximation method which requires high computer performance.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider a class of statistics
C
φ
based on
φ
-divergence for the test of independence in
r
×
s
contingency tables. The class of statistics
C
φ
includes the statistics
R
a
based on the power ...divergence as a special case. Statistic
R
0
is the log likelihood ratio statistic and
R
1
is Pearson's
X
2
statistic. Statistic
R
2
/
3
corresponds to the statistic recommended by Cressie and Read Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. B 46 (1984) 440–464 for the goodness-of-fit test. All members of statistics
C
φ
have the same chi-square limiting distribution under the hypothesis of independence. In this paper, we show the derivation of an expression of approximation for the distribution of
C
φ
under the hypothesis of independence. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the distribution of
C
φ
. Furthermore, on the basis of the approximation, we obtain transformations that improve the speed of convergence to the chi-square limiting distribution of
C
φ
. As a competitor of the transformed statistic, we derive a moment-corrected-type statistic. By numerical comparison in the case of
R
a
, we show that the transformed
R
1
statistic performs very well.
Full text
Available for:
GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider multinomial goodness-of-fit tests for a specified simple hypothesis under the assumption of sparseness. It is shown that the asymptotic normality of the Pearson X2 statistic (X2k) and the ...log-likelihood ratio statistic (G2k) assuming sparseness. In this paper, we improve the asymptotic normality of X2k and G2k statistics based on two kinds of normalizing transformation. The performance of the transformed statistics is numerically investigated. PUBLICATION ABSTRACT
PDF
