Quantifying Educational Expenditure Disparities in The Gambia: A Heavy- Tailed Extreme Value Approach

Dr. Ebrima Sisawo, Permanent Secretary Ministry of Local Government and Lands.

By Dr. Ebrima Sisawo

Ministry of Lands, Regional Government & Religious Affairs, The Gambia
October 2026

Abstract — Standard metrics of economic inequality, such as sample means, medians, or classical non-parametric Gini indices, fail when applied to heavy-tailed distributions where population variance is infinite (1/2 < γ < 1). In educational economics, household expenditures and institutional grant allocations frequently exhibit long upper tails driven by elite private tuition fees and capital spending spikes. This paper applies the score-moment t-Hill estimator and semi-parametric Pareto extrapolation developed by Sisawo (2024) to evaluate the Quintile Share Ratio (QSR) index across the seven administrative regions of The Gambia (N = 1,200). Using 300 non-parametric bootstrap resamples, we derive robust point estimates and 95% confidence intervals. The empirical results reveal significant regional variation, ranging from extreme inequality in Brikama/West Coast (22.39x QSR; 95% CI: 13.65x–63.45x) and Banjul (18.60x QSR; 95% CI: 9.58x–37.96x) to more compact structures in Kanifing (12.38x QSR; 95% CI: 10.07x–23.02x). We present detailed policy implications for resource allocation, scholarship targeting, and statistical integration into national education monitoring systems. Keywords: Extreme Value Theory (EVT), Quintile Share Ratio (QSR), Heavy-Tailed Distributions, Educational Inequality, t-Hill Estimator, Weissman Extrapolation, Sub-Saharan Africa Economics, The Gambia.

1. Introduction

Measuring economic inequality in developing education systems presents severe statistical challenges. In Sub-Saharan Africa, household spending on education—encompassing tuition fees, uniforms, textbooks, transportation, and private tutoring—is characterized by extreme skewness. While the majority of rural households allocate modest sums toward basic public schooling, a wealthy urban minority spends orders of magnitude more on elite private and international institutions.

When the underlying distribution F possesses a heavy right tail with tail index γ ∈ (1/2, 1), the second moment is infinite (E[X²] = ∞). Under these conditions, standard sample averages and classical non-parametric estimators exhibit unbounded influence functions and infinite asymptotic variance. Consequently, standard confidence intervals fail to provide reliable inference.

To overcome this structural limitation, this study operationalizes the theoretical Extreme Value Theory (EVT) framework established by Sisawo (2024). We focus on the Quintile Share Ratio (QSR)—a core social inclusion indicator comparing the total expenditure of the top 20% to the bottom 20%. By replacing raw empirical upper tail sums with t-Hill score-moment estimators and Weissman semi-parametric Pareto extrapolation, we restore asymptotic normality and finite sample variance.

2. Mathematical Framework and Estimator Derivations

2.1 The Quintile Share Ratio (QSR) Index

Let X₁, X₂, …, Xₙ be independent and identically distributed positive random variables with heavy-tailed distribution function F. The Quintile Share Ratio η(α, β) compares the upper conditional expectation above percentile β = 0.8 to the lower conditional expectation below percentile α = 0.2:

where Q(s) = F⁻¹(s) denotes the quantile function.

2.2 Robust t-Hill Tail Index Estimator

In the presence of heavy tails, classical maximum-likelihood estimators such as the Hill estimator γ̂_H are hyper-sensitive to upper-tail reporting anomalies or extreme outliers. To address this, Sisawo (2024) utilizes the score-moment t-Hill estimator γ̂^{tH}:

where X₁,ₙ ≤ X₂,ₙ ≤ … ≤ Xₙ,ₙ are the order statistics, and k represents the number of upper-tail observations. Because the reciprocal ratio X_{n-k,n}/X_{n-i+1,n} is bounded within (0, 1], an arbitrarily large outlier cannot exert unbounded leverage on the estimator.

2.3 Semi-Parametric QSR Integration

The upper expectation U_F(β) is decomposed at intermediate threshold X_{n-k,n}:

Combining Ũ_{n,k}^{tH} with the empirical lower expectation L_F(α) yields the semi-parametric t-Hill QSR estimator:

2.4 Semi-Parametric Bias-Reduced Least-Squares Estimator

To eliminate first-order asymptotic extrapolation bias, Sisawo (2024) specifies an exponential regression model on the log-spacings of upper-tail order statistics:

where ρ < 0 is the second-order parameter governing the speed of convergence. Fitting this structure via least-squares yields the bias-corrected estimator:

2.5 Automatic Threshold Selection (k*)

Optimal threshold selection k* balances asymptotic bias against variance using the Thomas & Reiss sequential minimization algorithm:

3. Data Description and Empirical Data Processing

3.1 Survey and Empirical Micro-Data (N = 1,200)

The empirical framework is operationalized using a representative household education expenditure survey dataset across The Gambia (N = 1,200 households). The dataset spans all seven administrative regions (Banjul, Kanifing, Brikama/West Coast, Mansakonko/LRR, Kerewan/NBR, Janjanbureh/CRR, and Basse/URR) and captures institutional spending dynamics across Public Lower Basic, Public Senior Secondary, Grant-Aided, and Private International schools.

Table 1: Household Education Expenditure Summary Statistics by Region (N = 1,200)

Region	Sample (N)	Mean Spending (GMD)	Median Spending (GMD)	Max Spending (GMD)
Brikama (West Coast)	360	25,648.10	8,243.60	1,224,030.99
Banjul	119	25,381.30	11,242.80	485,120.50
Janjanbureh (CRR)	126	12,916.70	5,549.85	185,400.00
Basse (URR)	110	21,217.80	8,127.26	312,050.00
Kanifing	276	20,723.80	8,988.78	295,400.00
Mansakonko (LRR)	89	10,619.60	6,165.31	98,500.00
Kerewan (NBR)	120	16,606.80	6,088.83	210,300.00
National Sample	1,200	20,727.41	8,117.75	1,224,030.99

3.2 Heavy-Tail Screening Diagnostics (γ > 0)

Diagnostic Pareto log-quantile plots confirm strong heavy-tail characteristics across all regions, with estimated tail indices γ lying strictly in the heavy-tail domain (0.40 < γ < 0.86). This validates the theoretical necessity of applying EVT semi-parametric extrapolation.

4. Empirical Results and Sub-National Inequality Analysis

Executing Sisawo’s robust t-Hill algorithm with B = 300 non-parametric bootstrap resamples yields the sub-national inequality baseline detailed in Table 2.

Table 2: Sub-National QSR Point Estimates and 95% Bootstrap Confidence Intervals (B = 300)

Region	Sample (N)	QSR Point Est.	95% CI Lower	95% CI Upper	Std Error
Brikama (West Coast)	360	22.39x	13.65x	63.45x	15.08
Banjul	119	18.60x	9.58x	37.96x	6.45
Janjanbureh (CRR)	126	16.05x	8.04x	29.10x	6.03
Basse (URR)	110	15.82x	8.41x	36.62x	6.50
Kanifing	276	12.38x	10.07x	23.02x	5.82
Mansakonko (LRR)	89	11.48x	4.90x	17.76x	3.67
Kerewan (NBR)	120	10.28x	7.15x	35.30x	9.18

Figure 1: Regional Quintile Share Ratio (QSR) Estimates with 95% Bootstrap Confidence Intervals across The Gambia.

5. Policy Implications and System Strategic Applications

The empirical findings demonstrate that educational expenditure in The Gambia is highly fragmented along regional and socio-economic lines. Applying these statistical results leads to four core strategic recommendations for national education planning and resource allocation:

1. Targeted Equity Bursaries and Means-Testing: National policy interventions must move beyond flat capitation grants. Because regional averages fail to account for severe internal disparities, national bursary programs must be strictly means-tested. In regions such as Brikama (West Coast) and Banjul, where the top 20% spend over 18 to 22 times more than the bottom 20%, direct cash transfers and fee waivers should target households in the bottom two quintiles to prevent low-income students from dropping out due to out-of-pocket costs.

2. Formula-Based Equity Operating Grants for Public Schools: To counteract the heavy upper-tail spending driven by private education, public school capitation grants should be adjusted using a weighted equity index. Schools operating in regions with elevated QSR indices (e.g., CRR, URR, and West Coast) should receive supplementary operational subsidies to ensure equal access to essential learning materials, science laboratories, and digital infrastructure.

3. Integration into National EMIS Monitoring Systems: Educational planning bodies should incorporate t-Hill heavy-tailed statistical algorithms directly into national Education Management Information Systems (EMIS). Automated QSR calculation scripts can process annual household survey data dynamically, providing decision-makers with early warning indicators of expanding educational inequality gaps.

4. National Survey Enhancement with Statistics Agencies: National statistical offices should refine future household survey frameworks—such as the Gambia Integrated Household Survey (IHS)—to ensure detailed disaggregation of out-of-pocket educational costs (e.g., transport, exam fees, private tutoring). Capturing granular tail data will improve threshold tuning (k*) and enhance the precision of sub-national equity metrics.

6. Conclusion

This study demonstrates that classical non-parametric metrics understate or destabilize inequality measurement in heavy-tailed educational systems. By applying Dr. Ebrima Sisawo’s t-Hill score-moment estimator and semi-parametric Pareto extrapolation, we establish a robust inequality benchmark for The Gambia. The empirical results demonstrate that household educational spending disparities are most severe in urban and suburban hubs, providing a clear quantitative foundation for targeted bursaries, equity-based school grants, and enhanced national policy monitoring.

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