Abstract This article considers exponential families of truncated multivariate normal distributions with one-sided truncation for some or all coordinates. We observe that if all components are one-sided truncated, then this family is not full. The family of truncated multivariate normal distributions is extended to a full family, and the extended family is investigated in detail. We identify the canonical parameter space of the extended family and establish that the family is not regular and not even steep. We also consider maximum likelihood estimation for the location vector parameter and the positive definite (symmetric) matrix dispersion parameter of a truncated non-singular multivariate normal distribution. It is shown that if the sample size is sufficiently large then, almost surely, the maximizer of the likelihood function is unique, provided that it exists. It is also shown that each solution to the score equations for the location and dispersion parameters satisfies the method-of-moments equations. Finally, it is observed that similar results arise in the case of an arbitrary number of truncated components.

A useful approach to exploratory functional data analysis is to work in the lower-dimensional principal component space rather than in the original functional data space. I demonstrate this approach by finding anomalies in age-specific US mortality rates between 1933 and 2022. The same approach can be employed for many other standard data analysis tasks and has the advantage that it allows immediate use of the vast array of multivariate data analysis tools that already exist, rather than having to develop new tools for functional data.

We appreciate this opportunity to discuss the paper by Prothero et al. (2024) that devised an interesting method, coined as Data Integration Via Analysis of Subspaces (DIVAS), to capture shared latent structures of multiple forms from multi-view data, in the hope to characterize common features across multi-modality data so to enhance data interpretability. One of key contributions in this paper pertains to that DIVAS offers an inference by the means of between-subspaces principal angles, which adds a rigorous uncertainty quantification to identify important signals with high confidence and validate scientific insights in integrative data analyses. Consider K -modality data {X 1 , . . . , X K } that are denoised by an additive model: The authors proposed a feature learning analytic that enables to derive different types of fully or partially shared latent structures, denoted by V i , through the following decomposition for the denoised K -modality data {A 1 , . . . , A K }: where V i are the shared latent features matrices (LFM) and L i,k are loading coefficients to be estimated, and index sets i ∈ 2 {1,••• ,K } whose definition is not straightforward. To understand the above decomposition, consider an example of K = 3, where the collection of all possible subsets is denoted by 2 {1,2,3} = {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. Index i ⊂ 2 {1,2,3} is a collection of some subsets; for instance, i = {i 1 , i 2 , i 3 , i 4 } = {{1, 2, 3}, {1, 2}, {1, 3}, {3}}; moreover, given such i when k = 1, notation i| k = 1 ∈ i = {{1, 2, 3}, {1, 2}, {1, 3}} and so forth. Suppose that each (denoised) data matrix A k ∈ R d k ×n for k = 1, 2, 3, and shared LFMs V i q ∈ R n×r q , q = 1, 2, 3, 4, where r q min(d 1 , d 2 , d 3 ) for dimension reduction. Then, a decomposition may take the following form:

We propose a goodness-of-fit test for a class of count time series models with covariates which includes the Poisson autoregressive model with covariates (PARX) as a special case. The test criteria are derived from a specific characterization for the conditional probability generating function, and the test statistic is formulated as a L2 weighting norm of the corresponding sample counterpart. The asymptotic properties of the proposed test statistic are provided under the null hypothesis as well as under specific alternatives. A bootstrap version of the test is explored in a Monte–Carlo study and illustrated on a real data set on road safety.

We would like to start by saying that this is a very interesting paper that outlines a powerful tool that can be applied in a wide range of areas. In our discussion, we focus on how the novel approach potentially can improve classification results in hard tasks within e.g., medicine and geoscience. In such situations, where it is hard to make precise predictions, it is natural to acquire information from many different sources and formats. The underlying idea is of course that, by utilizing all available information, improvements in terms of e.g., accuracy will be obtained. Although this sounds like a very natural claim, it is not obvious how different sources of information can be incorporated in a useful way. The method described by the authors make headway for many such situations and we would therefore like to congratulate them with a very impressive paper. It would be really interesting to have feedback from the authors concerning natural testbeds for their approach. Would e.g., early detection of cancer utilizing different modalities be possible to use for an evaluation of the method's performance in practice? In addition, we would like to hear the author's opinion if their method could be extended to handle data that are not on the form that they include so far in their framework. In particular, it would be interesting to hear if the method could be extended to handle unstructured data like text, which are frequently used in medical applications.