Asian Journal of Probability and Statistics

Comparative Machine Learning Modeling of Self-help Groups’ Impact on Livelihoods in Murang’a East Sub-county

Jane Wangui Runo — 2026-09-09

Self-help groups are widely used within communities to mobilise savings, access credit and support income-generating activities. This study compared three machine learning techniques: Logistic Regression, Naïve Bayes and Support Vector Machine, to model wealth status among self-help group members in Murang’a East Sub-County, Kenya. Primary data were collected through structured questionnaires from 969 self-help group members included in the final analysis, drawn from a target population of 2,250 members. Principal Component Analysis identified the factors most strongly associated with self-help groups’ performance. The PCA results ranked frequency of meetings and quality of discussions as the most important factors, followed by access to financial resources and credit facilities, collaboration with other organisations and stakeholders, member participation and engagement, community support, supportive policies, training and capacity building, and leadership and management. Logistic Regression, Naïve Bayes and Support Vector Machine were then fitted to classify members according to whether their wealth status had improved since joining a self-help group. Logistic Regression achieved an accuracy of 88.04%, Naïve Bayes 92.34% and Support Vector Machine 84.62%. Naïve Bayes also recorded the highest precision (94.92%), recall (96.89%) and F1-score (95.89%). The results indicated that the choice of classification method affected predictive performance, with Naïve Bayes providing the strongest overall performance. The study contributes to knowledge of self-help groups’ role in improving rural livelihoods and reducing poverty in Kenya and makes a methodological contribution by comparing classification models for predicting members’ wealth status and demonstrating the applicability of Naïve Bayes for analysing self-help group livelihood outcomes.

Comparing Mathematics Students' Interest in Learning Algebra Using the Friedman Test: A Nonparametric Approach

Bright Asare — 2026-09-10

Purpose: This study explored how students’ interest in learning algebra changes over time and whether teaching methods influence that interest.

Design/Methodology/Approach: The research involved ten first-year Mathematics Education students, whose interest in algebra was measured at three stages: before instruction, midway through the semester, and at the end of the course. A Likert-scale questionnaire was used to capture students’ interest, and because the same students were measured repeatedly using ordinal data, the Friedman test was applied for analysis.

Findings: The results showed a clear and significant change in interest levels, with students becoming more interested after the initial lessons and maintaining a relatively high level of interest by the end of the semester. These findings suggest that effective and engaging instruction can positively shape students’ attitudes toward algebra. The study also demonstrates that the Friedman test is a suitable and reliable method for analysing repeated interest data in mathematics education research. Finally, the result implies that students' interest in mathematics is not significantly different between the two linked groups.

Originality/Novelty: Despite its suitability, the Friedman test remains underutilized in mathematics education research, with many studies continuing to rely on parametric techniques even when data characteristics suggest otherwise.

Maximum Likelihood Estimation and Forecasting Performance of the GARCH Model Based on the Geometric Measure of Variation

Nkatet Siololo — 2026-09-12

Volatility models in the GARCH family typically build the conditional scale from an arithmetic average of squared or absolute deviations, an averaging rule that is disproportionately sensitive to large shocks and, through the triangular inequality, tends to systematically overstate the true average displacement of returns. This paper studies G-GARCH(1,1), a GARCH-type model in which the conditional scale Gt is instead driven by the geometric measure of variation through the log-linear recursion ln Gt = \(\omega\) + a ln |\(\varepsilon\)t-1 | + b ln Gt-1, so that the model’s response to a shock is logarithmic rather than quadratic in its magnitude. Building on established theoretical properties of this specification, namely its stationarity, ergodicity, positivity, long-run behaviour, identifiability, and a formal bound showing that the geometric scale can never exceed the classical conditional mean absolute deviation, this paper derives the model’s maximum likelihood estimator under both Gaussian and Student’s t innovations, obtaining the score function analytically through a recursive sensitivity argument, proving that no closed-form estimator exists, establishing consistency and asymptotic normality under regularity conditions analogous to those used for the classical GARCH estimator, and supplying a constrained-to-unconstrained reparameterisation together with a Broyden–Fletcher–Goldfarb–Shanno (BFGS) estimation algorithm that enforces the stationarity restrictions automatically. We then evaluate the model empirically against four competing specifications, standard GARCH(1,1) and exponential GARCH(1,1), each under Gaussian and Student’s t innovations, across three daily return series chosen to span a range of market conditions: Safaricom PLC on the Nairobi Securities Exchange, a frontier-market single stock; Google (Alphabet Inc.) on the Nasdaq, a developed-market single stock; and the S&P 500 index, a developed-market diversified benchmark. Applied to more than four thousand daily observations per series (January 2010 to June 2026) with a held-out test window drawn from the first half of 2026, every G-GARCH parameter is estimated with high precision (p < 0.001) in every market. The standard GARCH(1,1) model with Student’s t innovations attains the best in-sample fit by both the Akaike and Bayesian information criteria in all three markets, yet ranks last or second-to-last on every out-of-sample accuracy metric examined, a pattern traced to its squared-residual score letting a handful of extreme training returns pull its estimated tail thickness toward the boundary of finite kurtosis ( \(\hat{v}\) between 3.09 and 5.32 across the three series). The G-GARCH(1,1) model with Student’s t innovations is, by contrast, the best or joint-best out-of-sample forecaster of the five specifications compared for the two single-stock series, while the exponential GARCH(1,1) model forecasts more accurately for the S&P 500, whose crash-dominated, high-kurtosis return profile differs markedly from the two individual equities. Taken together, these results show that the theoretical conservatism established for the geometric conditional scale translates into a genuine, if market-dependent, forecasting advantage in practice.

Mathematical and Statistical Determination of Seismicity Index, Recurrence Interval and Seismic Risk of Shallow Earthquake Occurrence in Anatolian Fault in Turkey

J. U. Atsu — 2026-09-12

This study investigates the seismicity index, recurrence interval, and seismic risk of shallow earthquakes along the Anatolian Fault in Turkey using the least squares regression technique. Earthquake data were obtained from the International Seismological Centre (ISC) catalogue, considering events with Mb ≥ 4.0 and focal depths of 0–40 km. A total of 1,008 earthquake events were analysed. The least squares method produced seismicity parameters of a = 6.787 and b = 0.98666, representing the level of seismic activity and the relative occurrence of small versus large earthquakes. These parameters were used to estimate annual occurrence rates, recurrence intervals, and occurrence probabilities for different earthquake magnitudes. Results indicate that seismicity along the Anatolian Fault is dominated by low- to moderate-magnitude earthquakes, while larger events occur less frequently. The seismicity index decreases with increasing magnitude, whereas recurrence intervals become longer, demonstrating an inverse relationship between earthquake size and frequency. Seismic risk increases with longer exposure periods but decreases for higher-magnitude events because of their lower recurrence rates. These findings enhance understanding of the seismic behaviour of the Anatolian Fault and provide valuable information for seismic hazard assessment, earthquake preparedness, resilient infrastructure design, and strategies to reduce the impacts of future earthquakes.

Gerber–Shiu Penalty Function for a Dependent Risk Model with Stochastic Premium Income

Chunyu Xue — 2026-09-14

This study considers a dependent insurance risk model in which premium income is represented by a stochastic compound Poisson process and the claim mechanism incorporates dependence between claim amounts and subsequent inter-claim times through a stochastic threshold structure. The Gerber–Shiu expected discounted penalty function is investigated for two states determined by the relationship between the original loss amount and the corresponding stochastic threshold. Under the assumption that individual premium amounts follow an exponential distribution, integral equations for the state-dependent Gerber–Shiu functions are established and transformed using Laplace-transform techniques. The resulting expressions are analysed through a characteristic equation, from which the required transform quantities are determined. A defective renewal-equation representation of the Gerber–Shiu functions is subsequently derived using the transformed expressions and appropriate operator identities. Two numerical examples are considered to illustrate the analytical results. In the first example, both the premium and original loss amounts follow exponential distributions. In the second example, a mixture of exponential distributions is introduced for one component of the loss-threshold structure while the premium amount remains exponentially distributed. The resulting ruin probabilities decrease as the initial surplus increases, while differences between the two dependence states are also observed. The analysis provides a mathematical framework for examining ruin-related quantities in a dependent risk model where premium income is stochastic rather than deterministic.

Feature Selection-Assisted Support Vector Machine Classification of Breast Cancer Mammographic Outcomes

Beryl Kanali — 2026-09-14

Aims: This study assessed the effect of feature selection on Support Vector Machine classification of breast cancer mammographic outcomes using image-derived mammographic predictors.

Study Design: The study adopted a secondary-data comparative statistical learning design.

Place and Duration of Study: The study used the publicly available Curated Breast Imaging Subset of the Digital Database for Screening Mammography. The analysis was conducted at the Department of Mathematics, Multimedia University of Kenya.

Methodology: After image matching and preprocessing, 2,863 training observations and 704 testing observations were retained. Thirty-four image-derived predictors were extracted from cropped mammogram images and region of interest masks. The predictors comprised texture, shape and intensity features. Correlation-Based Feature Selection and Elastic Net-regularised multinomial logistic regression were applied before radial basis function Support Vector Machine classification. Model performance was assessed using accuracy, precision, recall, macro F1-score, weighted F1-score and McNemar’s test.

Results: Correlation-Based Feature Selection retained five predictors, reducing the feature space by 85.3%. Elastic Net retained sixteen predictors after coefficient ranking and cross-validated subset selection, reducing the feature space by 52.9%. Elastic Net-SVM achieved the highest accuracy of 0.5369, while the baseline SVM trained on all extracted predictors achieved the highest macro F1-score of 0.5200. McNemar’s tests showed no statistically significant differences in paired classification error rates among the three SVM models at the 5% level.

Conclusion: Feature selection reduced the input feature space and improved model interpretability but did not produce a statistically significant improvement in overall SVM classification performance. The findings highlight the need to report class-level performance, particularly malignant recall, when evaluating feature selection-assisted mammographic classification models.

Integrating Factor Analysis with the Cox Proportional Hazards Model for Neonatal Mortality Analysis Using KDHS 2022 Data

Berit Heddy Atieno — 2026-09-24

Aims/Objectives: To integrate Exploratory Factor Analysis (EFA) with Cox proportional hazards regression to reduce multicollinearity among correlated covariates and identify neonatal mortality risk factors in Kenya.

Study Design: Secondary analysis of a nationally representative cross-sectional demographic health survey, using a retrospective cohort (survival analysis) design.

Place and Duration of Study: The Catholic University of Eastern Africa, using data from the 2022 Kenya Demographic and Health Survey (KDHS); analysis conducted between April 2026 and August 2026.

Methodology: Data from 77,089 live births recorded in the 2022 KDHS were analyzed. Latent dimensions from correlated variables relating to socioeconomic status and fertility history were extracted. Resulting factor scores, together with selected observed covariates, were incorporated into a stratified Cox proportional hazards model. Five alternative model specifications were compared using Akaike and Bayesian Information Criteria, pseudo-R2, concordance index (C- index), events per variable, and generalized variance inflation factors.

Results: Child sex, birth order, preceding birth interval, maternal age at childbirth, pregnancy losses, multiple births, socioeconomic status, and fertility context were significant predictors of neonatal mortality. Female neonates had 30.5% lower hazard of death than males (HR = 0.695; 95% CI: 0.610–0.793), while multiple births had nearly six times the hazard of singletons (HR = 5.795; 95% CI: 4.849–6.925). The EFA-Cox framework required only 13 parameters versus 48 in the conventional model, while improving explained variation (28.7% vs. 14.3%) and discrimination (C-index: 0.787 vs. 0.712) and reducing multicollinearity. Embedding prior child mortality within a latent fertility-context factor, rather than including it directly, retained prognostic value while reducing predictive bias.

Conclusion: The EFA-Cox framework offers a concise, stable, and interpretable approach for modeling neonatal mortality and identifying its determinants in complex demographic survey data.

Probabilistic Forecasting of Mortality in Kenya Using a Zero-truncated Conway-Maxwell-Poisson Bayesian Generalized Additive Model

Grace Mukami Mwangi — 2026-09-30

Aims: To develop the Zero-Truncated Conway-Maxwell-Poisson Bayesian Generalized Additive Model (ZTCMP-BGAM) for probabilistic mortality forecasting in Kenya, estimate its parameters, evaluate its performance against three models, and generate probabilistic forecasts of mortality and life expectancy for 2024 to 2053.

Study Design: Quantitative secondary data analysis and mathematical modeling.

Place and Duration of Study: Kenya, 1950 to 2023 (observation); 2024 to 2053 (forecast horizon).

Methodology: UN WPP 2024 graduated death counts and population exposures for Kenya across 101 ages (1950-2023) were used. The ZTCMP-BGAM embeds a zero-truncated Conway-Maxwell-Poisson distribution with an age-specific dispersion parameter within a Bayesian generalized additive model. Parameters were estimated via penalized log-likelihood with a Laplace approximation, implemented in Template Model Builder. The ZTCMP-BGAM was compared against three models in a two-by-two factorial design crossing zero-truncation and flexible dispersion. Performance was evaluated using BIC, in-sample accuracy, an independent dispersion diagnostic, and backtesting over two holdout periods. Forecasts were generated by extrapolating the additive predictor.

Results: The ZTCMP-BGAM achieved the lowest BIC (66,003.42). Flexible dispersion accounted for 99,290 log-likelihood units of improvement; zero-truncation contributed 1.5 and 0.36 units within the equidispersion and flexible-dispersion classes, respectively. Departures from equidispersion were confirmed at 96 of 101 ages. Primary backtesting (2014-2023) yielded errors of 19.5% and 29.2% for equidispersion and flexible-dispersion models, respectively; sensitivity backtesting (1994-2023) reversed the ranking (31.8% vs. 24.5%). Life expectancy was projected to rise from 65.94 (95% CI: 65.75-66.15) to 81.23 years (95% CI: 78.87-83.59) by 2053.

Conclusion: Flexible age-specific dispersion improves distributional fit; equidispersion is inappropriate across most ages. Out-of-sample forecast performance depends on the mortality dynamics in the holdout period. The framework is transferable to any country relying on UN WPP estimates.

Modelling Volatility Dynamics of Monthly Changes in Nigeria’s Savings Deposit Rate: A GARCH Approach

Ogike Theresa U. — 2026-09-30

This study modelled the volatility dynamics of monthly changes in Nigeria's Savings Deposit Rate using the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) framework. Monthly Savings Deposit Rate data covering January 2013 to April 2026 were obtained from the Central Bank of Nigeria Statistical Bulletin and analysed using EViews 10. Descriptive statistics, the Augmented Dickey-Fuller (ADF) test and the ARCH-LM test were employed as preliminary procedures, while ARCH and GARCH models were estimated using Maximum Likelihood. The ADF test showed that the Savings Deposit Rate was non-stationary at level but became stationary after first differencing. The ARCH-LM test provided significant evidence of conditional heteroskedasticity, with an F-statistic of 21.47946 and a probability value of 0.0000, thereby justifying the use of conditional volatility models. The estimated model produced a positive and statistically significant ARCH coefficient of 0.241450 (p = 0.0210), indicating that recent shocks significantly affect current volatility. The GARCH coefficient was positive at 0.284001 but was statistically insignificant at the 5% level (p = 0.0943). The estimated volatility persistence measure, α+β=0.525451, was below unity, indicating a stationary conditional variance process. The estimated half-life of a volatility shock was approximately 1.08 months, suggesting relatively rapid dissipation of volatility shocks. Post-estimation diagnostics showed no significant remaining ARCH effects or residual serial correlation, while the standardised residuals did not significantly depart from normality. The study concludes that monthly changes in Nigeria's Savings Deposit Rate exhibit time-varying conditional volatility and that the model provides an adequate and parsimonious representation of its conditional variance dynamics.