Abstract
In this paper we are interested in checking whether the conditional variances are equal in k ≥ 2 location-scale models. Our procedure is fully nonparametric and is based on the comparison of the error distributions under the null hypothesis of equality of variances and without making use of this null hypothesis. We propose four test statistic based on empirical distribution functions (Kolmogorov-Smirnovand and Cramer-von Mises type test statistics) and two test statistics based on empirical characteristic functions. The limiting distributions of these six test statistics are established under the null hypothesis and under a local alternative. We show how to approximate the critical values under null using either an estimated version of the asymptotic null distribution or a bootstrap procedure. Simulation studies are conducted to assess the finite sample performance of the proposed tests. We also apply our tests to data on monthly expenditure.
Highlights
When comparing k (k ≥ 2) populations it is interesting comparing the means, and other characteristics, like the variances
This section studies some asymptotic properties of the empirical CDFs (ECDFs)-based test statistics TK1 S, TC1M, TK2 S and TC2M
The results in Corollaries 5 and 8 reveal that the asymptotic null distributions of the proposed test statistics are in all cases unknown because they depend on unknown quantities
Summary
When comparing k (k ≥ 2) populations it is interesting comparing the means, and other characteristics, like the variances. In this paper we will consider both cases, that is, to test H0 we will compare consistent estimators of the CDFs and CFs of the random variables εj and ε0j, 1 ≤ j ≤ k With this aim, the paper is organized as follows. Xk), diag(x) is the k × k diagonal matrix whose (i, i) entry is xi, 1 ≤ i ≤ k; for any complex number z = a + ib, Re(z) = a is its real part, Im(z) = b is its imaginary part, z = a − ib is its conjugate and |z| is its modulus; Nk(μ, Σ) denotes the k-variate normal distribution with mean vector μ and variance-covariance matrix Σ; an unspecified integral denotes integration over the whole real line R; supt stands for supt∈R; I(S) denotes the indicator function of a set S
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