Calibrating the Excess Mass and Dip Tests of Modality

Abstract

Nonparametric tests of modality are a distribution-free way of assessing evidence about inhomogeneity in a population, provided that the potential subpopulations are suf®ciently well separated. They include the excess mass and dip tests, which are equivalent in univariate settings and are alternatives to the bandwidth test. Only very conservative forms of the excess mass and dip tests are available at present, however, and for that reason they are generally not competitive with the bandwidth test. In the present paper we develop a practical approach to calibrating the excess mass and dip tests to improve their level accuracy and power substantially. Our method exploits the fact that the limiting distribution of the excess mass statistic under the null hypothesis depends on unknowns only through a constant, which may be estimated. Our calibrated test exploits this fact and is shown to have greater power and level accuracy than the bandwidth test has. The latter tends to be quite conservative, even in an asymptotic sense. Moreover, the calibrated test avoids dif®culties that the bandwidth test has with spurious modes in the tails, which often must be discounted through subjective intervention of the experimenter.

Why Do Online Product Reviews Have a J-Shaped Distribution? Overcoming Biases in Online Word-of-Mouth Communication

“…For example, Everitt and Hand (1981) approximated the sampling distribution by a parametric mixture of unimodal distributions to assess goodness of fit (Aitkin andRubin 1985, Roeder 1994). A disadvantage of this approach is that such assessment may be influenced by the validity of the particular parametric model and the hypothesis of homogeneity (Cheng and Hall 1998). This is because testing the goodness of fit along with using the goodness of fit can be very awkward (Cheng and Hall 1998 5 Bimodality is not due to a truncated distribution since consumers cannot write reviews higher than 5 or lower than 1 star.…”

“…A disadvantage of this approach is that such assessment may be influenced by the validity of the particular parametric model and the hypothesis of homogeneity (Cheng and Hall 1998). This is because testing the goodness of fit along with using the goodness of fit can be very awkward (Cheng and Hall 1998 5 Bimodality is not due to a truncated distribution since consumers cannot write reviews higher than 5 or lower than 1 star. Graphic plots of the mean of product reviews other than 3.0 or 2.5 stars reveal that there are fewer consumers writing a review of 5-stars than those writing a 4-star review.…”

Historical Ethnic Homelands and Income Convergence in Africa

“…In other words, the reference distribution for estimating the Dip statistic is the uniform unimodal distribution. Following Cheng and Hall (1998), P −values are calculated by comparing the Dip statistic obtained with those for repeated samples of the same size from a uniform distribution.…”

Avoiding arbitrary exclusion restrictions using ratios of reduced-form estimates

“…A formal way to detect modality is employing a multimodality test. Several tests for modality have appeared in the literature as can be seen, for instance, in Cheng and Hall (1998). Among them, the Silverman (1981) test seems to be the easiest to implement, which is described in the following and will be used later in our empirical section.…”