A panel-trained neural integer-valued autoregressive network for multi-country COVID-19 mortality forecasting, with an analysis of Poisson–Weighted Sujatha innovations

Integer-valued autoregressive (INAR) models are a standard framework for count time series, but their innovation law is usually fixed in advance and their thinning probability is usually constant. We study a three-parameter Poisson–Weighted Sujatha (PWS) count distribution, obtained by Poisson-mixing a weighted Sujatha density with a free rate λ. With λ = 1 it coincides with the Poisson-weighted Sujatha law of Shanker and Shukla; our additions are the rate parameter, proofs, and an analysis of the likelihood. The pmf is a three-component Negative Binomial mixture; we derive moments, limiting relations to the Poisson, Negative Binomial and Poisson–Sujatha laws, and, for the continuous mixing density, exact conditions for a monotone hazard rate and mean residual life: they hold for c ≥1/27 but not for all θ>0 when c is small. The model is strictly identifiable but (θ, λ) are practically unidentified along a likelihood ridge (Fisher information for logθ decays like θ −2 ), whereas the mean, variance and p = θ/(θ+λ) are estimated with decreasing mean squared error. On three COVID-19 series the Negative Binomial has the best AIC and BIC: every PWS component (for c ≥0) is a Negative Binomial with shape at least one, while the series require shape below one, so ĉ lies on the boundary c = 0. We embed PWS in a GRU-driven INAR network with time-varying thinning and innovation parameters (NS-INAR), trained on a panel by an exact convolution likelihood, and evaluate it on weekly COVID-19 mortality in 83 countries under a protocol in which country selection, tuning and training use only pre-test information. The network reduces the aggregate negative log score of a static per-country INAR(1)-NB by 23.2% (95% bootstrap CI [14.1 to 32.2]), but the held-out window is one of collapsed counts, and a persistence forecast with a Negative Binomial law does at least as well; INGARCH-NB and DeepAR-style networks are comparable; seasonal inputs do not help; and PWS innovations beat Negative Binomial innovations in one network design but not under per-series scaling. We therefore attribute the improvement to panel-trained, time-varying parameterisation, not to the PWS law, and restrict our conclusions to one disease, one data source and one forecast window.