In this paper the limiting distribution of Carmér-von Mises-Smirnov goodness of fit test statistic discussed when the parameters are estimated from the samle. Using the general theory of Durbin (1973 a and b) a method of deriving the limiting moment generating function for the test statistic is proposed under null and under a sequence of local alternatives. Applying this method, the limiting moment generating functions are evaluated for logistic and Cauchy distributions in the case of efficient estimation. Some results of numerical computation are also given.

View full abstract

The multivariate Studentized range R_max is defined as the positive square root of the maximum of generalized Studentized distances between any two points in a random sample drawn from a multivariate distribution. A short note on the background of R_max is given. Then, for obtaining good approximation to the exact upper percentiles of R_max the modified second approximation method is explained. Under the assumption of normality, the asymptotic expansion formulas necesary to evaluate those percentiles are drived for the large number of drgrees of freedom of the sample covariance matrix. The accuracy of the estimated percentiles is investigated extensively based on Monte Carlo studies. In the studies the simulated distributions of R_max are calculated for selected values of parameters, some of which are presented in this paper. The results obtained by numerical examinations are summarized, in which the valid ranges of parameters for the formulas of the modified second approximation are discussed. Finally, simultaneous pairwise comparesions of mean vectors of m mulitinormal populations are discussed as an example of the use of R_max.

View full abstract

Using Malliavin calculus we derived asymptotic expansions of mean values of statistics related to small diffusions. Applications to statistics and economics are discussed.

View full abstract

This paper analyzes the small sample properties of GMM estimator under the condition that errors are serially correlated. Some of the interesting results are as follows: 1) The standard deviation of parameter estimates tends to decrease as the number of lags used for instruments increases. 2) Tauchen (1986) insisted that there is a variance/bias trade-off regarding the number of lags used for instruments. However, we found that this result does not generally hold true. It depends on the law of motion of exogenous variables. 3) The test of overidentifying restrictions is biased toward accepting the specification of the model. The magnitude of this bias increases as the number of lags used for forming instruments increases. The overidentifying restriction test is not affected by the difference of the law of motion. Thus, it can be said that, since the desirable choice of instruments depends on the specification of exogenous variables, it is generally recommended to check the law of motion of the exogenous variables in advance in applied works.

View full abstract

This paper is concerned with a multiple decision problem having location slippage hypotheses in a multivariate regression model. The underlying distribution is assumed to be any member of a class of elliptically contoured distributions. We propose a decision rule based on a generalized version of the sample Mahalanobis squared distance, and show that the decision rule is admissible and minimax for the multiple decision problem. Our multiple decision problem includes the problem of classifying a future observation into one of two multivariate elliptically contoured populations with unknown locations and unknown but equal scale matrices. This is an extension of Kudô (1959) and Das Gupta (1965) where the underlying distribution was normal. Our results indicate that their classification rule is robust in the sense that the optimal properties of their classification rule are carried over to non-normal distributions. Kariya and Sinha (1985, 1989) call this optimality robustness.

View full abstract

This paper examines parsimonious models for both the British and Danish Social Occupational Mobility data. The models examined vary from the uniform association model (Goodman, 1981) to the variable distance models (Haberman, 1974). Both the Akaike (AIC) and the Bayesian (BIC) information criteria are employed for model selection. For the British data, the parsimonious UV (uniform-variable distance) association model is preferred, while for the Danish data, the UF (Uniform-fixed distance) association model is considered to be a good fit and it is also thus preferred. We also study the local odds pertaining to adjacent categories for the chosen models. Our models are preferable (based on the BIC criterion) to the parsimonious QU models introduced by Tomizawa (1990a, b).

View full abstract

To investigate bivariate linear structural relationships from unpaired and unequally replicated observations, we consider five models with certain types of the error variances. We discuss the maximum likelihood estimation of the parameters for each model and give the asymptotic variance-covariance matrices of the estimators when the number of replication tends to infinity and that of incidental parameters remain fixed.

View full abstract

For testing problems of the coefficient vectors of p-variate linear functional and structural relationships, the statistic which is based on the multiple correlation coefficient between the null variate and the uncorrelated (p-1) variates is considered. The asymptotic distributions of the test statistic under null hypothesis, where the estimators are substituted for other unknown parameters, are obtained for the functional and the structural relationship respectively. The testing procedures considered in this article are shown to be the same either for the functional or the structural relationship. The computations are illustrated with an example.

View full abstract

When a household chooses a residential land, it takes account of its neighborhood setting. Since the rent of a land is supposed to be a function of its environmental attributes as well as its site characteristics, the valuation of the environmental attributes of a household can be derived from the relation.
This paper explores some possibilities for imputing an index of the environment among districts in an urban area by using rent data. The values of environmental attributes are estimated, multiplied by the corresponding attributes and summed to obtain a score for each district.
Various methods are proposed to construct an all-inclusive index of the environment. The choice of weights and units is crucial to synthesize individual indices. This method seems superior to those in the sense that the index is derived from households' own preferences and expressed in terms of money.

View full abstract

A lot of studies have been done by many researchers on Hayashi's third method of quantification, which together with Guttman's scale analysis and Benzécri's correspondence analysis are useful in multivariate data analysis. But some important questions remain unanswered so far. One of the most conspicuous problems is acomparison between the two types of the data, free-choice type and item-categories type, which seem to be equivalent with each other with respect to information contained. It is required to find some criterion to decide which of them should be used in analyzing the data. A newt question is concerned with Hayashi's objective to diagonalize the response pattern of individuals to categories as well as possible by assigning suitable scores to the individuals and categories. Whether the objective is fulfilled for the item-categories type is questioned, assuming a success for the free-choice type. Other questions are concerned with the behavior of an individual which responds either to no categories or all categories. This paper challenges these questions. The author's answer to the first question is the criterion of choosing the type that makes the largest correlation coefficient greater.

View full abstract