Search Results (5)

Multivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building blocks of the Edgeworth expansions for the distribution of parameter estimates. Isserlis (1918) gave the bivariate normal moments and two special cases of trivariate moments. Beyond that, convenient expressions for multivariate variate normal moments are still not available. We compare three methods for obtaining them, the most powerful being the differential method. We give simpler formulas for the bivariate moment than that of Isserlis, and explicit expressions for the general moments of dimensions 3 and 4. Full article

Latent variables play a crucial role in psychometric research, yet traditional models often struggle to address context-dependent effects, ambivalent states, and non-commutative measurement processes. This study proposes a quantum-inspired framework for latent variable modeling that employs Hilbert space representations, allowing questionnaire items to be treated as pure or mixed quantum states. By integrating concepts such as superposition, interference, and non-commutative probabilities, the framework captures cognitive and behavioral phenomena that extend beyond the capabilities of classical methods. To illustrate its potential, we introduce quantum-specific metrics—fidelity, overlap, and von Neumann entropy—as complements to correlation-based measures. We also outline a machine-learning pipeline using complex and real-valued neural networks to handle amplitude and phase information. Results highlight the capacity of quantum-inspired models to reveal order effects, ambivalent responses, and multimodal distributions that remain elusive in standard psychometric approaches. This framework broadens the multivariate analysis theoretical and methodological toolkit, offering a dynamic and context-sensitive perspective on latent constructs while inviting further empirical validation in diverse research settings. Full article

Diverse multivariate statistics are powerful tools for musical analysis. A recent study identified relationships among different versions of the composition Sadhukarn from Thailand, Laos, and Cambodia using non-metric multidimensional scaling (NMDS) and cluster analysis. However, the datasets used for NMDS and cluster analysis require musical knowledge and complicated manual conversion of notations. This work aims to (i) evaluate a novel approach based on multivariate statistics of potential note degree of rhyme structure and pillar tone (Look Tok) for musical analysis of the 26 versions of the composition Sadhukarn from Thailand, Laos, and Cambodia; (ii) compare the multivariate results obtained by this novel approach and with the datasets from the published method using manual conversion; and (iii) investigate the impact of normalization on the results obtained by this new method. The result shows that the novel approach established in this study successfully identifies the 26 Sadhukarn versions according to their countries of origin. The results obtained by the novel approach of the full version were comparable to those obtained by the manual conversion approach. The normalization process causes the loss of identity and uniqueness. In conclusion, the novel approach based on the full version can be considered as a useful alternative approach for musical analysis based on multivariate statistics. In addition, it can be applied for other music genres, forms, and styles, as well as other musical instruments. Full article

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4. Full article

The comparison of group means in latent variable models plays a vital role in empirical research in the social sciences. The present article discusses an extension of invariance alignment and Haberman linking by choosing the robust power loss function $\rho \left(x\right)={\left|x\right|}^p$$(p>0)$. This power loss function with power values p smaller than one is particularly suited for item responses that are generated under partial invariance. For a general class of linking functions, asymptotic normality of estimates is shown. Moreover, the theory of M-estimation is applied for obtaining linking errors (i.e., inference with respect to a population of items) for this class of linking functions. In a simulation study, it is shown that invariance alignment and Haberman linking have comparable performance, and in some conditions, the newly proposed robust Haberman linking outperforms invariance alignment. In three examples, the influence of the choice of a particular linking function on the estimation of group means is demonstrated. It is concluded that the choice of the loss function in linking is related to structural assumptions about the pattern of noninvariance in item parameters. Full article