Goodness-of-fit Test Using Sample Quantiles

Abstract

Summary A goodness‐of‐fit test is constructed using sample quantiles for a p.d.f. F(x, θ) (where θ is to be estimated) which is continuous and strictly increasing at the quantiles of interest. It is shown to be asymptotically equivalent, under alternatives “close” to the null, to the usual goodness‐of‐fit test based on grouped observations, the advantage being that we may avoid the difficulties of choosing cell boundaries for grouping.

Goodness-of-Fit Statistics for Discrete Multivariate Data

“…(a) The order statistics or sample quantiles i = 0, ... , n + 1 (7.3) (e.g., Bofinger, 1973;Miyamoto, 1976). (b) The transformed spacings (or generalizations thereof) i= 1, ... ,n+ 1 (7.4) (e.g., Greenwood, 1946;Pyke, 1965;Cressie, 1976;Hall, 1986).…”

“…First we consider a test statistic based on a subset of the order statistics 1(1) ::;; 1(2) ::;; ... ::;; 1(n)· If m i is the integer part of (n + I)Yi; i = 1, ... , k + 1 for some fixed k and Yi (0;: Y1 < ... < Yk+1 ;: 1), Bofinger (1973) The similarity of(7.6) and Pearson's X2 statistic for grouped data (i.e., (7.2) with A = 1) can be observed when b i from (7.2) is defined to satisfy Fo(bi; 9*) = Yi; i = 1, ... , k + 1. Provided Fo(Y; 9*) is continuous and strictly increasing in Y in the neighborhood of bi' these b i will be unique, although not computable in practice since 9*, the true value of 9, is unknown.…”

“…Provided Fo(Y; 9*) is continuous and strictly increasing in Y in the neighborhood of bi' these b i will be unique, although not computable in practice since 9*, the true value of 9, is unknown. If, in addition, certain standard regularity conditions are required of 0 and F o , then Bofinger (1973) shows that Pearson's X 2 and the sample quantile statistic X~ of (7.6) are asymptotically equivalent both under the null hypothesis and under certain local alternatives (for further details see Miyamoto, 1976;Durbin, 1978). A consequence of this asymptotic equivalence is that a preference for using either a sample quantile statistic or power-divergence statistic (7.2) will have to be based on criteria that address more "distant" alternatives and smallsample considerations.…”

Goodness of Fit

“…Although (4) and (8) are the standard tests recommended in textbooks and have received the most attention by practitioners and theoreticians, there are others. For example, a general class of goodness-of-fit tests can be based on quadratic form* theory for multinormally distributed random variables.…”

“…. , k. Under suitable regularity conditions on f X (x), (4). This bypasses the problem of constructing intervals I j and uses part of the natural ordering of the sample.…”