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.