The Chow test is a test of equality of two sets of regression coefficients in two regression models under the assumption of homoscedasticity. Toyoda (1974) studied the actual size of the Chow test under heteroscedasticity. In this paper, we reexamine the same problem using the method of Takemura and Honda (1994), and pay particular attention to cases of small heteroscedasticity.

View full abstract

This paper discusses a testing problem of the linear hypothesis for the expectation matrix in a balanced growth curve model with random parameters. We consider this testing problem from an invariance point of view, and obtain a maximal invariant under a group of transformations which leaves the problem invariant. We also derive the likelihood ratio statistic and show that its least favorable distribution is given by Λ-distribution.

View full abstract

Recently, a new higher order approximation to a percentage point for a non-central t-distribution was proposed by Akahira [1] using the Cornish-Fisher expansion and was shown to be better than prior approximations. In this paper approximation formulae for percentage points of the non-central x² and F distribution are proposed in a similar way to the above. An approximation formula of a percentage point of the non-central x² distribution is also given by a direct application of the Cornish-Fisher expansion for the chi-square statistic. Further, the numerical comparison of these formulae with former formulae shows that the new approximation formulae behave better than the others.

View full abstract

This paper deals with the density for a class of estimators _??__{c1 c2} (c₁, c₂≥0) in Gaussian AR (1) process. Here _??__{c1 c2} includes various estimators if the constants c₁ and c₂ are specified appropriately. Applying the saddlepoint method to the general formula by Geary [5], the density of _??__{c1 c2} is approximated. Although Phillips [14] pointed out that the saddlepoint density is undefined in a substantial part of the tails, we elucidate that the resulting approximation is always defined if c₁ and c₂ are appropriately chosen. Some numerical comparisons are made among the Edgeworth approximation, the saddlepoint approximation, and the exact distribution for _??__{1/2, 1/2}. We also approximate the density for the mean corrected estmator _??__{c1 c2}.

View full abstract

The Holt-Winters method has been widely used to forecast a seasonal time series in application fields as a nonparametric forecasting technique. In this paper, we investigate the asymptotic forecast errors of the Holt-Winters method. For that purpose, we show that the nonlinear least squares estimates of the smoothing parameters included in the smoothing algorithm hold strong convergence properties under suitable conditions. Then we show the mean squared errors and the limiting distributions of the forecast errors for some stochastic processes. Finally, numerical studies are performed to evaluate the forecasting performance of the Holt-Winters method.

View full abstract

In this paper we obtain an approximation of a jackknife estimator of the variance of a linear combination of U-statistics and obtain an approximation of a studentized linear combination statistic substituting the jackknife estimator of the variance. Additionally, an Edgeworth expansion with a remainder term of o(n^-1) is established for the studentized linear statistic. It is shown that both the studentized U-statistic and the von Mises V-statistic have same Edgeworth expansion.

View full abstract

This paper describes a large deviation principle for the sample covariance function ⁿΣ_i=1X_iX_i-1/n of a first order non-explosive Gaussian autoregressive process with unknown autoregressive parameter θ, the rate function is also provided. The asymptotic rate of convergence of the tail probability of the sample covariance function is also studied.

View full abstract

Using a generalization of the divergence of degree λ on the location model, a distance between two populations is proposed for mixed data of categorical and continuous variables. Three propositions about the proposed distance are derived from properties of the divergence. Two real examples and some numerical examples are given in order to study the propositions in detail and have a suitable value of λ. The information of degree λ is considered as the distance between an observation and a population, and minimizing this distance establishes an allocation rule on the location model. The allocation rule is shown to be equivalent to the optimal rule based on the maximum-likelihood method.

View full abstract

Some methods of construction for nested group divisible designs are give. Cyclic nested group divisible designs are further discussed. Some individual plans are also tabulated with 4 new E-optimal designs.

View full abstract

Scale invariant tests based on M-statistics are proposed in order to test homogeneity in a multivariate k sample model. Asymptotic noncentral x²-distributions are drawn under a contiguous sequence of location-alternatives without assuming Fisher consistency, and asymptotic robustness is derived. Permutation tests based on the proposed M-test statistics are considered. Using a Monte Carlo simulation, the power of these tests is compared with permutation tests based on parametric test statistics. Next, robust estimators for location parameters are proposed, based on scale-invariant M-statistics, and the asymptotic normality of these estimators is drawn. After a simple algorithm is studied, the risks of the M-estimators and the least squares estimators are compared in a simulation. For the univariate case, it is found that (i) the asymptotic relative efficiency (ARE) of the proposed M-procedures relative to parametric procedures agrees with the ARE of one-sample M-estimator proposed by Huber (1964) relative to the sample mean, and that (ii) for small sample sizes, the M-procedures are more efficient than parametric procedures except for the case where the underlying distribution is normal.

View full abstract