Estimated t Test and F Test with Interval-Censored Normal Data

In this paper we derive the asymptotic sampling distributions of maximum likelihood estimators based on an interval-censored data sample from a normal population and propose two test statistics analogous to the Student\u27s t-ratio test and the Snedecor\u27s F-ratio test in the analysis of variance. And the results of simulation studies of these tests for means are exhibited

A Bivariate Permutation Test for Analysis of Three Interval Data Samples

Let X, Y, and Z be three random variables with unknown continuous cumulative distribution functions F, G, and H. D.R. Whitney1) proposed the (U,V) statistics as a bivariate extension of Wilcoxon\u27s U statistic in 1951. It is well known that Whitney\u27s (U,V) tests based on these statistics are particularly powerful against the following two types of alternatives, respectively: (1) F(t)>G(t) and F(t)>H(t) (for all t), (2) G(t)>F(t)>H(t) (for all t). A bivariate permutation (U,V) test that is an extension of Whitney\u27s (U,V) test is proposed for samples in which observations are specified only by intervals with known probability density functions. Here, the two statistics, U and V, are based on generalized signs instead of ranks. The proposed bivariate permutation (U,V) test is not rough even in small samples, because the value of the generalized sign can take on real value densely. In the same way, we can construct a multivariate permutation test for many samples in which observations are specified only by intervals. Computer programs were developed for determining the critical region of (U,V) in a given sample of nx x\u27s, ny y\u27s and nz z\u27s

A Generalized Wilcoxon Test for Comparing Interval Data Samples

A distribution-free two-sample test is proposed that is an extension of the Wilcoxon test to interval-censored data or more generally to interval data samples. Interval-censored data are often observed in medical or biological studies and the idea of interval data is extension of interval-censored data. We define a generalized sign of difference between two observations based on their interval data under the estimated distribution of each observation. The generalized sign may be interpreted as the probability that the one is larger than the other. The test statistic is defined as the sum of the generalized signs based on all combinations of the two samples. The test is conditional on the pattern of observations. The null hypothesis is H0 : F (t) = G (t) against either H1 : F (t) G(t), where F, G are cumulative distribution functions of the observations. The test is shown to be asymptotically normal. Working examples are presented and the tests are performed by a BASIC program which was developed for the proposed test

<原著>環境汚染の程度を比較するパックデータ・アナリシス

環境汚染の問題は,きわめて広範多岐にわたり,汚染度の観測・観測値の解析に問題を限定しても複雑多様に過ぎる。そこでこの小論では,騒音公害・悪臭公害など,それらの観測値の平均値や合計値よりも,有害とみなされる高レベルの観測値の頻度や強度が問題となる場合を取りあげ,各地域間の汚染度の高低を検定する問題に限定することにする。さて,騒音や悪臭の場合,汚染状況は概ね日・週・季節を周期として変動する。そしてその基本となる最短周期の一日においても,時刻・晴雨・気温・風向きなどによりレ観測値は大きく変動する。従って,地域間の汚染度の比較においては,日々の観測値の生起する順番はあまり重要ではない。かくして,このような公害データ解析の手始めとしては,各観測地点ごとに一日間の観測値をパックし,一日ごとに汚染の程度を比較をするのが妥当であろう。そのとき,各観測値のもつ情報を出来るだけ有効に利用する方法として,Wilcoxonの検定法を一般化したWilcoxonの精密検定法を提案する。The problem of environmental pollution covers a wide range of subjects, and is too complicated even if we limit the problem to study the methods of observation and statistical analyses of pollution. So in this article, we will take up the noise pollution and the bad smell pollution. In these cases, the frequency and the strength of observations which are considered harmful to the health are more important than their means or totals. To make the points clear, we limit ourselves to test which of the two areas is higher in the degree of pollution. And furthermore, we assume that the observation system is decided in advance. Now in the cases of noise pollution and bad smell pollution, the state of pollution fluctuates as a period of a day, week and season approximately. And even in a day which is the shortest period, the observations change by the tine, weather, temperature, the direction of the wind and so on. Therefore, the time order of observations in one day does not play a vital role, in order to compare the pollution level between areas. Under these conditions, as the first step of these pollution data analysis, it is appropriate to pack the observations, day by day, every areas, and compare the pollution levels among areas. Then we propose the generalized Wilcoxon exact test, as a method which utilizes the information of every observations effectively