Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests

Contents

Kernel-based quadratic distance (KBQD) Goodness-of-Fit tests#

The class KernelTest performs the kernel-based quadratic distance tests using the Gaussian kernel with bandwidth parameter h. Depending on the shape of the input y the function performs the tests of multivariate normality, non-parametric two-sample tests, or k-sample tests.

The quadratic distance between two probability distributions \(F\) and \(G\) is defined as

\[d_{K}(F,G)=\iint K(x,y)d(F-G)(x)d(F-G)(y),\]

where \(G\) is a distribution whose goodness of fit we wish to assess and \(K\) denotes the Normal kernel defined as

\[K_{{h}}(\mathbf{s}, \mathbf{t}) = (2 \pi)^{-d/2} \left(\det{\mathbf{\Sigma}_h}\right)^{-\frac{1}{2}} \exp\left\{-\frac{1}{2}(\mathbf{s} - \mathbf{t})^\top \mathbf{\Sigma}_h^{-1}(\mathbf{s} - \mathbf{t})\right\},\]

for every \(\mathbf{s}, \mathbf{t} \in \mathbb{R}^d \times \mathbb{R}^d\), with covariance matrix \(\mathbf{\Sigma}_h=h^2 I\) and tuning parameter \(h\).

  • Test for Normality:

    Let \(x_1, x_2, \ldots, x_n\) be a random sample with empirical distribution function \(\hat F\). We test the null hypothesis of normality, i.e. \(H_0:F=G=\mathcal{N}_d(\mu, \Sigma)\).

    We consider the U-statistic estimate of the sample KBQD

    \[U_{n}=\frac{1}{n(n-1)}\sum_{i=2}^{n}\sum_{j=1}^{i-1} K_{cen}(\mathbf{x}_{i}, \mathbf{x}_{j}),\]

    then the first test statistic is

    \[T_{n}=\frac{U_{n}}{\sqrt{Var(U_{n})}},\]

    with \(Var(U_n)\) computed exactly following Lindsay et al. (2014), and the V-statistic estimate

    \[V_{n} = \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}K_{cen}(\mathbf{x}_{i}, \mathbf{x}_{j}),\]

    where \(K_{cen}\) denotes the Normal kernel \(K_h\) with parametric centering with respect to the considered normal distribution \(G = \mathcal{N}_d(\mu, \Sigma)\).

    The asymptotic distribution of the V-statistic is an infinite combination of weighted independent chi-squared random variables with one degree of freedom. The cutoff value is obtained using the Satterthwaite approximation \(c \cdot \chi_{DOF}^2\), where \(c\) and \(DOF\) are computed exactly following the formulas in Lindsay et al. (2014).

    For the \(U\)-statistic, the cutoff is determined empirically:

    • Generate data from the considered normal distribution;

    • Compute the test statistics for B Monte Carlo (MC) replications;

    • Compute the 95th quantile of the empirical distribution of the test statistic.

  • k-sample test:

    Consider \(k\) random samples of i.i.d. observations \(\mathbf{x}^{(i)}_1, \mathbf{x}^{(i)}_{2}, \ldots, \mathbf{x}^{(i)}_{n_i} \sim F_i\), \(i = 1, \ldots, k\). We test if the samples are generated from the same unknown distribution, that is \(H_0: F_1 = F_2 = \ldots = F_k\) versus \(H_1: F_i \not= F_j\), for some \(1 \le i \not= j \le k\).

    We construct a matrix distance \(\hat{\mathbf{D}}\), with off-diagonal elements

    \[\hat{D}_{ij} = \frac{1}{n_i n_j} \sum_{\ell=1}^{n_i} \sum_{r=1}^{n_j} K_{\bar{F}}(\mathbf{x}^{(i)}_\ell,\mathbf{x}^{(j)}_r), \quad \text{for } i \not= j\]

    and in the diagonal

    \[\hat{D}_{ii} = \frac{1}{n_i (n_i -1)} \sum_{\ell=1}^{n_i} \sum_{r\not= \ell}^{n_i} K_{\bar{F}}(\mathbf{x}^{(i)}_\ell,\mathbf{x}^{(i)}_r), \quad \text{for } i = j,\]

    where \(K_{\bar{F}}\) denotes the Normal kernel \(K_h\) centered non-parametrically with respect to

    \[\bar{F} = \frac{n_1 \hat{F}_1 + \ldots + n_k \hat{F}_k}{n}, \quad \text{with } n=\sum_{i=1}^k n_i.\]

    We compute the trace statistic

    \[\mathrm{trace}(\hat{\mathbf{D}}_n) = \sum_{i=1}^{k}\hat{D}_{ii}\]

    and \(D_n\), derived considering all the possible pairwise comparisons in the k-sample null hypothesis, given as

    \[D_n = (k-1) \mathrm{trace}(\hat{\mathbf{D}}_n) - 2 \sum_{i=1}^{k}\sum_{j > i}^{k}\hat{D}_{ij}.\]

    We compute the empirical critical value using numerical techniques such as bootstrap, permutation, and subsampling algorithms:

    • Generate k-tuples, of total size \(n_B\), from the pooled sample following one of the sampling methods;

    • Compute the k-sample test statistic;

    • Repeat B times;

    • Select the 95th quantile of the obtained values.

  • Two-sample test:

    Let \(x_1, x_2, \ldots, x_{n_1} \sim F\) and \(y_1, y_2, \ldots, y_{n_2} \sim G\) be random samples from the distributions \(F\) and \(G\), respectively. We test the null hypothesis that the two samples are generated from the same unknown distribution, that is \(H_0: F=G\) vs \(H_1:F\not=G\). The test statistics coincide with the k-sample test statistics when \(k=2\).

Kernel Centering#

The arguments \(\hat{\mu}\) (mu_hat) and \(\hat{\Sigma}\) (sigma_hat) indicate the normal model considered for the normality test, that is \(H_0: F = N(\hat{\mu},\hat{\Sigma})\). For the two-sample and k-sample tests, mu_hat and sigma_hat can be used for the parametric centering of the kernel, in case we wish to specify the reference distribution, with centering_type = "param". This is the default method when the test for normality is performed. The normal kernel centered with respect to \(G \sim N_d(\mathbf{\mu}, \mathbf{V})\) can be computed as

\[K_{cen(G)}(\mathbf{s}, \mathbf{t}) = K_{\mathbf{\Sigma_h}}(\mathbf{s}, \mathbf{t}) - K_{\mathbf{\Sigma_h} + \mathbf{V}}(\mathbf{\mu}, \mathbf{t}) - K_{\mathbf{\Sigma_h} + \mathbf{V}}(\mathbf{s}, \mathbf{\mu}) + K_{\mathbf{\Sigma_h} + 2\mathbf{V}}(\mathbf{\mu}, \mathbf{\mu}).\]

We consider non-parametric centering of the kernel with respect to \(\bar{F}=(n_1 F_1 + \ldots + n_k F_k)/n\) where \(n=\sum_{i=1}^k n_i\), with centering_type = "nonparam", for the two- and k-sample tests. Let \(\mathbf{z}_1,\ldots, \mathbf{z}_n\) denote the pooled sample. For any \(s,t \in \{\mathbf{z}_1,\ldots, \mathbf{z}_n\}\), it is given by

\[K_{cen(\bar{F})}(\mathbf{s},\mathbf{t}) = K(\mathbf{s},\mathbf{t}) - \frac{1}{n}\sum_{i=1}^{n} K(\mathbf{s},\mathbf{z}_i) - \frac{1}{n}\sum_{i=1}^{n} K(\mathbf{z}_i,\mathbf{t}) + \frac{1}{n(n-1)}\sum_{i=1}^{n} \sum_{j \not=i}^{n} K(\mathbf{z}_i,\mathbf{z}_j).\]
Contents