Inference on Scale-Inflated Gamma Distribution Applied to Malaysian Household Income

Mallak Ahmad AlhouraniShamsul Rijal Muhammad Sabri^*

• School of Mathematical Sciences, University of Science Malaysia, Penang, Malaysia

support. A key challenge, however, lies in evaluating the extent to which household income inflates over time. While income is inherently random, it exhibits a persistent upward trend, and fitting income distributions using conventional models often leads to inconsistent parameter estimates (Majid and Ibrahim, 2021; Majid et al., 2023). This highlights the necessity of explicitly incorporating inflation-adjusted scaling to preserve proper statistical properties. To address this gap, we introduce the Scale-Inflated Gamma (SIG) distribution, which extends the standard Gamma distribution by including an inflation-adjusted scale parameter (delta), thereby providing greater flexibility in capturing heterogeneous income dynamics. Standard models such as the Lognormal, Pareto, or Generalized Beta of the Second Kind (GB2) systematically underestimate upper-tail incomes and fail to capture inflation-adjusted heterogeneity across subgroups (B40, M40, T20). The SIG model, in contrast, strikes a balance between parsimony and flexibility by directly adjusting for inflationary scale shifts. For instance, while the Gamma distribution underestimates the 95th percentile by 10–12 percent in 2019, the SIG model reduces this bias to approximately 3 percent, accurately reflecting income dynamics across the B40, M40, and T20 groups (Jenkins, 2009; Majid et al., 2023). We develop the theoretical foundations of the SIG distribution by deriving its probability density function (PDF), cumulative distribution function (CDF), and moments. Parameters are initially estimated using the method of moments and then refined through maximum likelihood estimation (MLE). To assess estimator precision, we derive the Fisher information matrix, using the inverse Hessian to approximate the variance–covariance matrix, thus ensuring reliable inference. A Monte Carlo simulation study is conducted to evaluate the consistency and efficiency of the estimators under various sample sizes. The SIG model is subsequently applied to Malaysian Household Income Survey (HIS) data spanning the period from 2007 to 2022. Results demonstrate that the SIG distribution offers a superior fit for modeling income inequality and upper-tail behavior compared to conventional models. Overall, the study establishes the SIG distribution as a theoretically robust and policy-relevant framework for analyzing income patterns in inflation-sensitive and structurally diverse economies.