Finite mixture models (FMMs) arise naturally in statistical analysis of biological and sociological data [11, 13]. The model of mixture with varying concentrations (MVC) is a modification of the FMM where the mixing probabilities may be different for different observations. Namely, we consider a sample of subjects $O_{1},\dots ,O_{N}$ where each subject belongs to one of subpopulations (mixture components) $\mathcal{P}_{1},\dots ,\mathcal{P}_{M}$. The true subpopulation to which the subject $O_{j}$ belongs is unknown, but we know the probabilities ${p_{j;N}^{m}}=\operatorname{\mathsf{P}}[O_{j}\in \mathcal{P}_{m}]$ (mixing probabilities, concentrations of $\mathcal{P}_{m}$ in the mixture at the jth observation, $j=1,\dots ,N$, $m=1,\dots ,M$). For each subject O, a variable $\xi (O)$ is observed, which is considered as a random element in a measurable space $\mathfrak{X}$ equipped by a σ-algebra $\mathfrak{F}$. Let be the distribution of $\xi (O)$ for subjects O that belong to the mth component. Then the unconditional distribution of $\xi _{j;N}=\xi (O_{j})$ is The observations $\xi _{j;N}$ are assumed to be independent for $j=1,\dots ,N$.

We consider the nonparametric MVC model where the concentrations ${p_{j;N}^{m}}$ are known but the component distributions $F_{m}$ are completely unknown. Such models were applied to analyze gene expression level data [8] and data on sensitive questions in sociology [12]. An example of sociological data analysis based on MVC is presented in [9]. In this paper, we consider adherents of different political parties in Ukraine as subpopulations $\mathcal{P}_{i}$. Their concentrations are deduced from 2006 parliament election results in different regions of Ukraine. Individual voters are considered as subjects; their observed characteristics are taken from the Four-Wave Values Survey held in Ukraine in 2006. (Note that the political choices of the surveyed individuals were unknown. So, each subject must be considered as selected from mixture of different $\mathcal{P}_{i}$.) For example, one of the observed characteristics is the satisfaction of personal income (in points from 1 to 10).

A natural question in the analysis of such data is homogeneity testing for different components. For example, if $\mathfrak{X}=\mathbb{R}$, then we may ask if the means or variances (or both) of the distributions $F_{i}$ and $F_{k}$ are the same for some fixed i and k or if the variances of all the components are the same.

In [8], a test is proposed for the hypothesis of two-means homogeneity. In this paper, we generalize the approach from [8] to a much richer class of hypotheses, including different statements on means, variances, and other generalized functional moments of component distributions.

Hypotheses of equality of MVC component distributions, that is, $F_{i}\equiv F_{k}$, were considered in [6] (a Kolmogorov–Smirnov-type test is proposed) and [1] (tests based on wavelet density estimation). The technique of our paper also allows testing such hypotheses using the “grouped ${\chi }^{2}$”-approach.

Parametric tests for different hypotheses on mixture components were also considered in [4, 5, 13].

The rest of the paper is organized as follows. We describe the considered hypotheses formally and discuss the test construction in Section 2. Section 3 contains auxiliary information on the functional moments estimation in MVC models. In Section 4, the test is described formally. Section 5 contains results of the test performance analysis by a simulation study and an example of real-life data analysis. Technical proofs are given in Appendix A.

In the rest of the paper, we use the following notation.

The zero vector from ${\mathbb{R}}^{k}$ is denoted by $\mathbb{O}_{k}$. The unit $k\times k$-matrix is denoted by $\mathbb{I}_{k\times k}$, and the $k\times m$-zero matrix by $\mathbb{O}_{k\times m}$. Convergences in probability and in distribution are denoted $\stackrel{P}{\longrightarrow }$ and $\stackrel{d}{\longrightarrow }$, respectively.

We consider the set of concentrations $p=({p_{j;N}^{m}},j=1,\dots ,N;\hspace{2.5pt}m=1,\dots ,M;N=1,\dots \hspace{0.1667em})$ as an infinite array, ${p_{\operatorname{\mathbf{\cdot }};N}^{\operatorname{\mathbf{\cdot }}}}=({p_{j;N}^{m}},j=1,\dots ,N;\hspace{2.5pt}m=1,\dots ,M)$ as an $(N\times m)$-matrix, and ${p_{\operatorname{\mathbf{\cdot }};N}^{m}}=({p_{j;N}^{m}},j=1,\dots ,N)\in {\mathbb{R}}^{d}$ and ${p_{j,N}^{\operatorname{\mathbf{\cdot }}}}=({p_{j;N}^{m}},m=1,\dots ,M)$ as column vectors. The same notation is used for arrays of similar structure, such as the array a introduced further.

Angle brackets with subscript N denote averaging of an array over all the observations, for example, Multiplication, summation, and other similar operations inside the angle brackets are applied to the arrays componentwise, so that and so on.

Angle brackets without subscript mean the limit of the corresponding averages as $N\to \infty $ (assuming that this limit exists): We introduce formally random elements $\eta _{m}\in \mathfrak{X}$ with distributions $F_{m}$, $m=1,\dots ,M$.

Consider a set of $K\le M$ measurable functions $g_{k}:\mathfrak{X}\to {\mathbb{R}}^{d_{k}}$, $k=1,\dots ,K$. Let ${\bar{g}_{k}^{m}}$ be the (vector-valued) functional moment of the mth component with moment function $g_{k}$, that is,

Fix a measurable function $T:{\mathbb{R}}^{d_{1}}\times {\mathbb{R}}^{d_{2}}\times \cdots \times {\mathbb{R}}^{d_{K}}\to {\mathbb{R}}^{L}$. For data described by the MVC model (1) we consider testing a null-hypothesis of the form against the general alternative $T({\bar{g}_{1}^{1}},\dots ,{\bar{g}_{K}^{K}})\ne \mathbb{O}_{L}$.

To test $H_{0}$ defined by (3), we adopt the following approach. Let there be some consistent estimators ${\hat{g}_{k;N}^{m}}$ for ${\bar{g}_{k}^{m}}$. Assume that T is continuous. Consider the statistic $\hat{T}_{N}=T({\hat{g}_{1;N}^{1}},\dots ,{\hat{g}_{K;N}^{K}})$. Then, under $H_{0}$, $\hat{T}_{N}\approx \mathbb{O}_{L}$, and a far departure of $\hat{T}_{N}$ from zero will evidence in favor of the alternative.

To measure this departure, we use a Mahalanobis-type distance. If $\sqrt{N}\hat{T}_{N}$ is asymptotically normal with a nonsingular asymptotic covariance matrix D, then, under $H_{0}$, $N{\hat{T}_{N}^{T}}{D}^{-1}\hat{T}_{N}$ is asymptotically ${\chi }^{2}$-distributed. In fact, D depends on unknown component distributions $F_{i}$, so we replace it by its consistent estimator $\hat{D}_{N}$. The resulting statistic $\hat{s}_{N}=N{\hat{T}_{N}^{T}}{\hat{D}_{N}^{-1}}\hat{T}_{N}$ is a test statistic. The test rejects $H_{0}$ if $\hat{s}_{N}>{Q}^{{\chi _{L}^{2}}}(1-\alpha )$, where α is the significance level, and ${Q}^{G}(\alpha )$ denotes the quantile of level α for distribution G.

Possible candidates for the role of estimators ${\hat{g}_{k;N}^{m}}$ and $\hat{D}_{N}$ are considered in the next section.

Let us start with the nonparametric estimation of $F_{m}$ by the weighted empirical distribution of the form where ${a_{j;N}^{m}}$ are some nonrandom weights to be selected “in the best way.” Denote $e_{m}={(\mathbb{1}\{k=m\},k=1,\dots ,M)}^{T}$ and Assume that $\varGamma _{N}$ is nonsingular. It is shown in [8] that, in this case, the weight array yields the unbiased estimator with minimal assured quadratic risk.

The simple estimator ${\hat{g}_{i;N}^{m}}$ for ${\bar{g}_{i}^{m}}$ is defined as We denote $\varGamma =\lim _{N\to \infty }\varGamma _{N}={(\langle {p}^{i}{p}^{m}\rangle )_{i,m=1}^{M}}$. Let $h:\mathfrak{X}\to {\mathbb{R}}^{d}$ be any measurable function.

To formulate the asymptotic normality result for the simple moment estimators, we need some additional notation.

We consider the set of all moments ${\bar{g}_{k}^{k}}$, $k=1,\dots ,K$, as one long vector belonging to ${\mathbb{R}}^{d}$, $d:=d_{1}+\cdots +d_{K}$: The corresponding estimators also form a long vector We denote the matrices of mixed second moments of $g_{k}(x)$, $k=1,\dots ,K$, and the corresponding estimators as

We consider the function T as a function of d-dimensional argument, that is, $T(y):=T({y}^{1},\dots ,{y}^{K})$. Then $\hat{T}_{N}:=T(\hat{g}_{N})=T({\hat{g}_{1;N}^{1}},\dots ,{\hat{g}_{K;N}^{K}})$.

Let us define the following matrices (assuming that the limits exist):

Then the asymptotic covariance matrix of the normalized estimate $\sqrt{N}(\hat{g}_{N}-\bar{g})$ is Σ, where Σ consists of the blocks ${\varSigma }^{(k,l)}$:

Thus, to construct a test for $H_{0}$, we need a consistent estimator for Σ. The matrices $\alpha _{r,s;N}$ and $\beta _{m;N}$ are natural estimators for $\alpha _{r,s}$ and $\beta _{m}$. It is also natural to estimate ${\bar{g}_{k,l}^{m}}$ by ${\hat{g}_{k,l;N}^{m}}$ defined in (5b). In view of Theorem 1, these estimators are consistent under the assumptions of Theorem 2. But they can possess undesirable properties for moderate sample size. Indeed, note that $\hat{F}_{m;N}$ is not a probability distribution itself since the weights ${a_{j;N}^{m}}$ are negative for some j. Therefore, for example, the simple estimator of the second moment of some component can be negative, estimator (5b) for the positive semidefinite matrix ${\bar{g}_{k,k}^{m}}$ can be not positive semidefinite matrix, and so on. Due to the asymptotic normality result, this is not too troublesome for estimation of $\bar{g}$. But it causes serious difficulties when one uses an estimator of the asymptotic covariance matrix D based on ${\hat{g}_{k,l;N}^{m}}$ in order to calculate $\hat{s}_{N}$.

In [10], a technique is developed of $\hat{F}_{m;N}$ and ${\hat{h}_{N}^{m}}$ improvement that allows one to derive estimators with more adequate finite sample properties if $\mathfrak{X}=\mathbb{R}$.

So, assume that $\xi (O)\in \mathbb{R}$ and consider the weighted empirical CDF It is not a nondecreasing function, and it can attain values outside $[0,1]$ since some ${a_{j;N}^{m}}$ are negative. The transform yields a monotone function $\tilde{F}_{m;N}(x)$, but it still can be greater than 1 at some x. So, define as the improved estimator for $F_{m}(x)$. Note that this is an “improvement upward,” since ${\tilde{F}_{m;N}^{+}}(x)\ge \hat{F}_{m;N}(x)$. Similarly, a downward improved estimator can be defined as Any CDF that lies between ${\hat{F}_{m;N}^{-}}(x)$ and ${\hat{F}_{m;N}^{+}}(x)$ can be considered as an improved version of $\hat{F}_{m;N}(x)$. We will use only one such improvement, which combines ${\hat{F}_{m;N}^{-}}(x)$ and ${\hat{F}_{m;N}^{+}}(x)$:

Note that all the three considered estimators ${\hat{F}_{m;N}^{\ast }}$ (∗ means any symbol from +, −, or ±) are piecewise constants on intervals between successive order statistics of the data. Thus, they can be represented as where ${b_{j;N}^{m\ast }}$ are some random weights that depend on the data.

The corresponding improved estimator for ${\bar{g}_{i}^{m}}$ is Let $h:\mathbb{R}\to \mathbb{R}$ be a measurable function.

We first state an asymptotic normality result for $\hat{T}_{N}$. Denote

For the proof, see Appendix. Note that (iii) implies the nonsingularity of Σ.

Now, to estimate D, we can use where $\tilde{g}_{N}$ is any consistent estimator for $\bar{g}$, where ${\tilde{g}_{k,l;N}^{m}}$ is any consistent estimator for ${\bar{g}_{k,l;N}^{m}}$. For example, we can use if $\mathfrak{X}=\mathbb{R}$ and the assumptions of Theorem 3 hold for all $h(x)={g_{l}^{i}}(x){g_{k}^{n}}(x)$, $i,k=1,\dots ,K$, $i=1,\dots ,d_{l}$, $n=1,\dots ,d_{k}$, $g_{l}(x)={({g_{l}^{1}}(x),\dots ,{g_{l}^{d_{l}}}(x))}^{T}$.

Now let the test statistic be $\hat{s}_{N}=N{(\hat{T}_{N})}^{T}{\hat{D}_{N}^{-1}}\hat{T}_{N}$. For a given significance level α, the test $\pi _{N,\alpha }$ accepts $H_{0}$ if $\hat{s}_{N}\le {Q}^{\xi _{L}}(1-\alpha )$ and rejects $H_{0}$ otherwise.

The p-level of the test (i.e., the attained significance level) can be calculated as $p=1-G(\hat{s}_{N})$, where G means the CDF of ${\chi _{L}^{2}}$-distribution.

We developed a technique that allows one to construct testing procedures for different hypotheses on functional moments of mixtures with varying concentrations. This technique can be applied to test the homogeneity of means or variances (or both) of some components of the mixture. Performance of different modifications of the test procedure is compared in a small simulation study. The (ss) modification showed the worst first-type error and the highest power. The (ii) test has the best first-type error and the worst power. It seems that the (si) modification can be recommended as a golden mean.