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

This function performs the kernel-based quadratic distance goodness-of-fit tests. It includes tests for multivariate normality, two-sample tests and \(k\)-sample tests.

Usage

kb.test(
  x,
  y = NULL,
  h = NULL,
  method = "subsampling",
  B = 150,
  b = NULL,
  Quantile = 0.95,
  mu_hat = NULL,
  Sigma_hat = NULL,
  centeringType = "Nonparam",
  K_threshold = 10,
  alternative = "skewness"
)

# S4 method for ANY
kb.test(
  x,
  y = NULL,
  h = NULL,
  method = "subsampling",
  B = 150,
  b = 0.9,
  Quantile = 0.95,
  mu_hat = NULL,
  Sigma_hat = NULL,
  centeringType = "Nonparam",
  K_threshold = 10,
  alternative = "skewness"
)

# S4 method for kb.test
show(object)

Arguments

x

Numeric matrix or vector of data values.

y

Numeric matrix or vector of data values. Depending on the input y, the corresponding test is performed.

• if y = NULL, the function performs the tests for normality on x

• if y is a data matrix, with same dimensions of x, the function performs the two-sample test between x and y.

• if y is a numeric or factor vector, indicating the group memberships for each observation, the function performs the k-sample test.

h

Bandwidth for the kernel function. If a value is not provided, the algorithm for the selection of an optimal h is performed automatically. See the function select_h for more details.

method

The method used for critical value estimation ("subsampling", "bootstrap", or "permutation")(default: "subsampling").

B

The number of iterations to use for critical value estimation (default: 150).

b

The size of the subsamples used in the subsampling algorithm (default: 0.8).

Quantile

The quantile to use for critical value estimation, 0.95 is the default value.

mu_hat

Mean vector for the reference distribution.

Sigma_hat

Covariance matrix of the reference distribution.

centeringType

String indicating the method used for centering the normal kernel ('Param' or 'Nonparam').

K_threshold

maximum number of groups allowed. Default is 10. It is a control parameter. Change in case of more than 10 samples.

alternative

Family of alternative chosen for selecting h, between "location", "scale" and "skewness" (only if h is not provided).

object

Object of class kb.test

Value

An S4 object of class kb.test containing the results of the kernel-based quadratic distance tests, based on the normal kernel. The object contains the following slots:

• method: Description of the kernel-based quadratic distance test performed.

• x Data list of samples X (and Y).

• Un The value of the U-statistic.

• H0_Un A logical value indicating whether or not the null hypothesis is rejected according to Un.

• CV_Un The critical value computed for the test Un.

• Vn The value of the V-statistic (if available).

• H0_Vn A logical value indicating whether or not the null hypothesis is rejected according to Vn (if available).

• CV_Vn The critical value computed for the test Vn (if available).

• h List with the value of bandwidth parameter used for the normal kernel function. If select_h is used, the matrix of computed power values and the corresponding power plot are also provided.

• B Number of bootstrap/permutation/subsampling replications.

• var_Un exact variance of the kernel-based U-statistic.

• cv_method The method used to estimate the critical value (one of "subsampling", "permutation" or "bootstrap").

Details

The function kb.test 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, the non-parametric two-sample tests or the 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 \(x_1, x_2, ..., x_n\) 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 statistics 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 $$\hat{D}_{ij} = \frac{1}{n_i n_j} \sum_{\ell=1}^{n_i} \sum_{r=1}^{n_j}K_{\bar{F}}(\mathbf{x}^{(i)}_\ell,\mathbf{x}^{(j)}_r), \qquad \mbox{ for }i \not= j$$ and in the diagonal $$\hat{D}_{ii} = \frac{1}{n_i (n_i -1)} \sum_{\ell=1}^{n_i} \sum_{r\not= \ell}^{n_i} K_{\bar{F}}(\mathbf{x}^{(i)}_\ell, \mathbf{x}^{(i)}_r), \qquad \mbox{ for }i = j,$$ where \(K_{\bar{F}}\) denotes the Normal kernel \(K_h\) centered non-parametrically with respect to $$\bar{F} = \frac{n_1 \hat{F}_1 + \ldots + n_k \hat{F}_k}{n}, \quad \mbox{ with } n=\sum_{i=1}^k n_i.$$

  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 by employing numerical techniques such as the 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 \(95^{th}\) quantile of the obtained values.

• Two-sample test:
  Let \(x_1, x_2, ..., x_{n_1} \sim F\) and \(y_1, y_2, ..., y_{n_2} \sim G\) 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 mu_hat and Sigma_hat indicate the normal model considered for the normality test, that is \(H_0: F = N(\)mu_hat, Sigma_hat). For the two-sample and \(k\)-sample tests, mu_hat and Sigma_hat can be used for the parametric centering of the kernel, in the case we want to specify the reference distribution, with centeringType = "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 the 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 centeringType = "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 $$K_{cen(\bar{F})}(\mathbf{s},\mathbf{t}) = K(\mathbf{s},\mathbf{t}) - \frac{1}{n}\sum_{i=1}^{n} K(\mathbf{s},\mathbf{z}_i) - \frac{1}{n}\sum_{i=1}^{n} K(\mathbf{z}_i,\mathbf{t}) + \frac{1}{n(n-1)}\sum_{i=1}^{n} \sum_{j \not=i}^{n} K(\mathbf{z}_i,\mathbf{z}_j).$$

Note

For the two- and \(k\)-sample tests, the slots Vn, H0_Vn and CV_Vn are empty, while the computed statistics are both reported in slots Un, H0_Un and CV_Un.

A U-statistic is a type of statistic that is used to estimate a population parameter. It is based on the idea of averaging over all possible distinct combinations of a fixed size from a sample. A V-statistic considers all possible tuples of a certain size, not just distinct combinations and can be used in contexts where unbiasedness is not required.

References

Markatou, M. and Saraceno, G. (2024). “A Unified Framework for Multivariate Two- and k-Sample Kernel-based Quadratic Distance Goodness-of-Fit Tests.”
https://doi.org/10.48550/arXiv.2407.16374

Lindsay, B.G., Markatou, M. and Ray, S. (2014) "Kernels, Degrees of Freedom, and Power Properties of Quadratic Distance Goodness-of-Fit Tests", Journal of the American Statistical Association, 109:505, 395-410, DOI: 10.1080/01621459.2013.836972

See also

kb.test for the class definition.

Examples

# create a kb.test object
x <- matrix(rnorm(100),ncol=2)
y <- matrix(rnorm(100),ncol=2)

# Normality test
my_test <- kb.test(x, h=0.5)
my_test
#> 
#>  Kernel-based quadratic distance Normality test 
#> 		U-statistic	V-statistic
#> ------------------------------------------------
#> Test Statistic:	 0.3027069 	 0.6189598 
#> Critical Value:	 1.262023 	 6.071062 
#> H0 is rejected:	 FALSE 		 FALSE 
#> Selected tuning parameter h:  0.5 
#> 

# Two-sample test
my_test <- kb.test(x,y,h=0.5, method="subsampling",b=0.9,
                     centeringType = "Nonparam")
my_test
#> 
#>  Kernel-based quadratic distance two-sample test 
#> U-statistic	 Dn 		 Trace 
#> ------------------------------------------------
#> Test Statistic:	 0.239788 	 0.2819997 
#> Critical Value:	 0.9875162 	 1.162661 
#> H0 is rejected:	 FALSE 		 FALSE 
#> CV method:  subsampling 
#> Selected tuning parameter h:  0.5 
#> 

# k-sample test
z <- matrix(rnorm(100,2),ncol=2)
dat <- rbind(x,y,z)
group <- rep(c(1,2,3),each=50)
my_test <- kb.test(x=dat,y=group,h=0.5, method="subsampling",b=0.9)
my_test
#> 
#>  Kernel-based quadratic distance k-sample test 
#> U-statistic	 Dn 		 Trace 
#> ------------------------------------------------
#> Test Statistic:	 7.325505 	 11.45482 
#> Critical Value:	 0.7402039 	 1.158313 
#> H0 is rejected:	 TRUE 		 TRUE 
#> CV method:  subsampling 
#> Selected tuning parameter h:  0.5 
#>