Theil Index¶

An additively decomposable generalized-entropy inequality measure comparing population shares with shares of a positive resource, with Theil T weighting log relative shares by the resource and Theil L weighting log shortfalls by population.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7780
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Distributional Analysis → Economics & Finance

Core Idea¶

The Theil Index is a family of inequality measures derived from generalized entropy and information theory.^[1] It compares each unit's share of a positive resource such as income with its share of the population.^[2] Equality occurs when the two distributions coincide; divergence increases as resource shares become more concentrated.

For N units with positive values x_i and mean μ, Theil T is T = (1/N) Σ (x_i/μ) ln(x_i/μ).^[3] Each relative value is weighted by its resource share. Theil L, also called the mean log deviation, is L = (1/N) Σ ln(μ/x_i), weighting each unit equally by population.^[4]

The two are different members of the generalized entropy family.^[5] Theil T corresponds to parameter α=1, while Theil L corresponds to the limiting case α=0. Calling both “the Theil index” without a formula is ambiguous. They respond differently to changes in different parts of a distribution.

At perfect equality, every x_i=μ, the logarithmic ratios equal zero, and both indexes are zero. Larger values indicate greater inequality under their assumptions. The scale is not a percentage and has no universal intuitive upper bound independent of population size and zero conventions.

For a finite population with nonnegative resources, Theil T reaches ln N when one unit holds the entire positive total and the others hold zero, using the limiting convention that 0 ln 0 = 0. Dividing by ln N creates a zero-to-one normalization but changes comparability and replication properties. The normalization must be disclosed.

Theil L requires strictly positive x_i because ln(μ/0) diverges. Zero incomes or resource values therefore make L infinite or undefined under the literal formula. Analysts may restrict populations, add an offset, top/bottom code, or use another measure, but each alters the estimand. Negative incomes are incompatible with ordinary log-ratio formulas.

The information-theoretic interpretation is precise. Let each person have prior population share 1/N and posterior resource share x_i/(Nμ). Theil T is the Kullback–Leibler divergence from resource shares to population shares. It is often described as redundancy or expected information in moving from equal population weights to observed resource weights.

This does not mean income is literally a probability-generating message. Normalized shares form probability distributions, enabling information geometry. The interpretation supplies mathematical structure, not a claim that inequality and communication are socially equivalent.

Additive decomposability is the measure's signature advantage.^[6] Partition a population into groups. Total inequality can be written as a weighted sum of within-group inequality plus between-group inequality computed from group means. For Theil T, weights follow group resource shares for the within component; for Theil L, population-share weights appear. Exact formulas should accompany a decomposition.

The between-group component measures inequality that would remain if everyone within each group received the group mean. The within component measures residual inequality inside groups. It does not prove that group membership caused inequality. Changing group definitions changes the decomposition.

Nested decomposition can allocate inequality across regions, industries, demographic categories, or organizational levels. Overlapping groups and intersecting identities do not fit a simple mutually exclusive partition without additional methods. Summing decompositions from incompatible partitions double-counts.

The transfer principle holds: a small progressive transfer from a richer to a poorer unit, without reversing their rank, lowers the index under standard conditions. Sensitivity differs. Theil T is relatively responsive to changes at the upper end, while Theil L is more sensitive near the bottom, where log shortfalls grow strongly.

Scale invariance means multiplying every value by the same positive constant leaves relative ratios unchanged. Replication invariance means duplicating the entire population distribution leaves the unnormalized index unchanged. Population-size-normalized variants can violate replication invariance because their denominator changes.

The measure is anonymous: permuting unit labels does not change it. That supports distributional comparison but omits identity and need. Two populations can have the same Theil value while inequality falls on very different people or intersects with discrimination differently.

The Theil index can measure segregation when shares represent group distributions across locations or institutions. The carrier and reference distribution must be specified. Income inequality and racial segregation use analogous divergences but different units, weights, normative interpretations, and possible zero patterns.

Comparing indexes across years requires consistent resource definition, price adjustment, population unit, equivalence scale, survey coverage, and treatment of taxes and transfers. Household income, individual earnings, consumption, wealth, and opportunity are not interchangeable. A mathematically exact index can answer the wrong substantive question.

Survey weights generalize population shares. Each observed unit represents a number of population units, and resource totals must use the same weights. Replicate weights or design-based methods propagate sampling uncertainty. Applying the unweighted formula to a complex survey can bias the estimate.

Top coding and heavy tails matter because resource weighting makes T sensitive to large observations. Administrative and survey data often miss or censor high incomes. Pareto interpolation or linked tax records can change estimates. Bottom coding and zeros especially affect L.

The index is not a welfare function by itself. Choosing T, L, or another α embodies sensitivity choices, but no single number states how society values transfers, poverty, mobility, or procedural fairness. Report distributions and complementary measures such as Gini, percentile ratios, and poverty rates.

Structural Signature¶

Sig role-phrases:

Weighted population units — declare the persons, households, regions, or other observations and their population representation.
Additive resource — assigns the positive income, wealth component, exposure, or other quantity whose distribution is being compared.
Aligned share distributions — normalize population weights and resource totals over the same units, with equality represented by coincident shares.
Logarithmic share contrast — compares each unit's resource share with its population share through a log ratio.
T-or-L branch — chooses resource-share weighting for Theil T or population weighting of log shortfalls for Theil L.
Aggregate divergence — sums unit contributions into a nonnegative inequality value that is zero at equality under the declared convention.
Additive partition — decomposes an exhaustive, mutually exclusive grouping into variant-correct within-group and between-group terms.^[7]
Boundary policy — states how zeros, negative values, censoring, missingness, survey weights, log base, and optional normalization alter the estimand.
Interpretive limit — preserves the declared inequality or segregation carrier and prevents decomposition or index magnitude from being read as causal attribution, welfare judgment, or a percentage.

What It Is Not¶

Not one unique formula unless the variant is named. Theil T weights log share contrasts by resource share, while Theil L or mean log deviation weights log shortfalls by population share and responds differently across the distribution.
Not entropy itself. The measures use generalized-entropy and Kullback–Leibler structure on normalized shares, but the reported value is inequality in a declared resource distribution rather than a thermodynamic state.^[8]
Not a percentage. Zero marks equality, while the upper behavior depends on population size, variant, data boundary, log base, and any disclosed normalization.
Not the Gini coefficient, Atkinson index, or poverty rate. Those measures apply different weighting, sensitivity, thresholds, and welfare assumptions even when computed on the same observations.
Not compatible with zeros and negatives in one uniform way. Theil T can use a limiting convention for zero resource shares, whereas Theil L diverges at zero and ordinary log-ratio forms do not admit negative values.^[9]
Not comparable across normalized and unnormalized implementations without conversion. Dividing by ln N, changing log base, censoring, survey weighting, or adding an offset changes the scale or estimand.
Not proof of causal discrimination or a welfare verdict. Decomposition attributes measured inequality arithmetically to within- and between-group components; it does not identify why the distribution arose or whether it is just.

Scope of Application¶

The Theil Index is a precondition-bounded distributional measure: it applies where aligned population shares and shares of a declared positive additive resource are compared through the exact Theil T or Theil L logarithmic formula.^[10] Every habitat must state units, population, resource, survey or exposure weights, T/L variant, log base, normalization, zero and negative policy, partition, data source, adjustments, and uncertainty; an entropy metaphor or unaligned concentration score is outside scope.

Individual income inequality. Persons' population shares are compared with their shares of disposable, market, gross, or another explicitly defined income concept under a common period and price basis.
Household income inequality. Households enter with declared equivalence scales, member weights, taxes and transfers, survey coverage, and treatment of zero or negative income.
Earnings and wage dispersion. Workers, jobs, hours, or full-time equivalents form the population while pay period, labor-force selection, top coding, and nonemployment boundaries remain explicit.
Wealth-component inequality. Net worth or a positive asset component can be measured only under a policy for debts, negative net wealth, missing high-wealth observations, valuation dates, and household units.
Regional inequality. Regions' population shares and resource shares yield total and between-region divergence, with within-region contributions preserved under the chosen T or L weights.
Urban–rural decomposition. Exhaustive, mutually exclusive residence categories separate inequality within each area from inequality between their mean resources without implying a causal effect of location.
Industry and sector decomposition. Workers, establishments, or output units are partitioned by a declared classification to compare within-sector dispersion with differences among sector means.
Productivity dispersion. Firms, plants, or workers carry a positive productivity or output measure under consistent deflation, size weighting, sampling, and zero-value treatment.
Multilevel geographic decomposition. Country, state, county, district, or other nested partitions allocate measured inequality across levels only when memberships are exhaustive and nonoverlapping at each step.
Demographic group decomposition. Age, sex, race, education, or other declared groups support within/between accounting while overlapping identities require methods beyond a simple additive partition.
Longitudinal inequality comparison. Years or cohorts can be compared after harmonizing units, resource definitions, prices, equivalence scales, coverage, censoring, and data vintages.
Complex-survey estimation. Design weights align sample units with population shares, and replicate or design-based methods quantify uncertainty rather than applying an unweighted sample formula.
Spatial or institutional segregation. Members of a group and their reference population are distributed across neighborhoods, schools, workplaces, or other units under an explicit share construction and substantive interpretation.
Irrigation-system performance. Water, service, or another additive delivery resource is compared across declared fields or users only when the same aligned-share and T/L formula is retained.
Software-metric distribution. Code or project units may be compared by an additive positive metric under the literal formula, with the result interpreted as metric concentration rather than economic welfare.
Ecological abundance distribution. Species or sites can supply aligned shares for the same instrument when zero policy and sampling are explicit, but the result describes abundance concentration or isolation rather than economic inequality.
Robustness and estimator comparison. Analysts vary T versus L, tail reconstruction, zero policy, normalization, log base, weights, or group partition and compare companion distributional measures to expose what the scalar omits.

Clarity¶

Naming the Theil Index makes inequality legible as a logarithmic divergence between population shares and shares of a positive resource. It prevents “Theil” from hiding two different measures: Theil T weights relative shares by the resource and emphasizes the upper tail, whereas Theil L weights units by population and is especially sensitive near zero. The formula, weights, logarithm base, normalization, and zero-value policy therefore belong to the result's identity.

This recognition also sharpens decomposition: within- and between-group components are an accounting partition for exhaustive, mutually exclusive groups, not evidence that group membership caused inequality. A value such as 0.2 is not “20% inequality,” and normalized and unnormalized results are not silently interchangeable. The better practitioner question is: which T or L estimand, population and resource weights, boundary treatment, and group partition produced this number?

Manages Complexity¶

The Theil Index compresses a full distribution into the summed logarithmic divergence between population shares and shares of a declared positive resource. The analyst tracks the T or L variant, unit and survey weights, log base, normalization, treatment of zeros and negatives, and any exhaustive partition. Those coordinates yield a total inequality value and unit or group contributions; additive decomposition then makes readable how much of that measured divergence lies within groups and how much remains between their means, including nested regional or organizational levels.

The main branches follow from weighting and boundary choices. Theil T emphasizes resource-rich observations and tolerates zeros by a limiting convention, whereas Theil L is especially sensitive near the bottom and is undefined at zero under its literal formula; normalizing by ln N changes replication behavior. The compression stops at what the scalar and partition discard: rank identity, distribution shape, overlapping memberships, causal mechanisms, welfare judgments, and missing or censored tails. Quantile displays, uncertainty estimates, alternative measures, and the exact resource and population definitions must therefore accompany substantive interpretation.

Abstract Reasoning¶

The analyst begins with a declared population, a positive additive resource, and aligned population and resource weights, then normalizes both over the same units. Equality predicts coincident shares and a value of zero; departures are accumulated as logarithmic share ratios. Choosing Theil T or Theil L determines the direction of weighting and therefore the inference: T gives greater leverage to resource-rich units, whereas L gives equal population leverage and becomes increasingly sensitive near zero. A result is uninterpretable until the variant, log base, normalization, and treatment of zero or negative observations are known.

For a mutually exclusive, exhaustive partition, the total can be recomputed as within-group inequality plus inequality among group means under the variant's proper weights. This supports a diagnostic from totals to levels: if the between-group term is small, most measured divergence remains inside the chosen groups, not that group membership is causally irrelevant. Changing the group partition, applying survey weights, restoring a censored upper tail, or replacing zeros with an offset tests how much the conclusion depends on design choices. The scalar cannot determine rank identity, overlapping-group contributions, welfare significance, or causal mechanism; those claims require information that the index deliberately discards.

Knowledge Transfer¶

Within distributional economics, the Theil Index transfers literally across individual or household income, wages, wealth components, regional disparity, productivity dispersion, survey-weighted estimates, and nested group decompositions when the population, positive resource, weights, and T or L variant are fixed. The carried method normalizes population and resource shares, accumulates the appropriate logarithmic divergence, and partitions total inequality into within- and between-group terms under variant-specific weights. Diagnostics compare T with L, restore censored tails, change partitions, apply survey weights, and test zero, negative, normalization, or log-base policies; interventions alter one estimator choice while preserving the underlying distribution. Theil T, Theil L, mean log deviation, resource share, population share, generalized entropy, decomposition, and replication remain literal measurement vocabulary.

Across segregation, concentration, fairness, and some ecological distribution studies, the reach is primarily (C) instrument/measure, with (B) shared abstract mechanism through Generalized Entropy Index and an (A) analogy boundary around informal uses of “entropy.” The Theil formula remains literal when units, a nonnegative or positive additive resource, reference shares, variant, boundary policy, and valid partition are supplied; the information-theoretic divergence structure then carries unchanged. What remains home-bound to inequality interpretation is the economic meaning of income or wealth, population representation, transfer and welfare assumptions, survey coverage, and the normative significance of within- and between-group components. A diverse ecosystem or compressed file is not economically unequal merely because its entropy expression resembles Theil. The stopping boundary is absence of aligned shares and the declared T or L formula; beyond it the comparison is an analogy, not a Theil Index.

Examples¶

Canonical¶

Four equally weighted people receive positive incomes [1, 1, 2, 4], whose mean is 2. Theil T is (1/4)Σ(x_i/2)ln(x_i/2) ≈ 0.1733; Theil L is (1/4)Σln(2/x_i) ≈ 0.1733 for this particular distribution. Both would be zero at [2, 2, 2, 2], but their numerical equality here is incidental: T weights relative incomes by resource share, whereas L gives equal population weight to log shortfalls. Neither 0.1733 is “17.33 percent inequality.”

Mapped back: The four people supply Weighted population units and their incomes the Additive resource. Dividing population and total income over the same people creates Aligned share distributions; the logarithms perform the Logarithmic share contrast. Choosing each formula exposes the T-or-L branch, and summing gives the Aggregate divergence. The percentage warning enforces the Interpretive limit.

Applied / In Practice¶

An analyst partitions the same four people into region A with incomes [1, 1] and region B with [2, 4]. For Theil T, region A's within value is zero, region B's is about 0.05663, and resource-share weighting gives a total within component of (2/8)×0 + (6/8)×0.05663 ≈ 0.04247. Replacing each income by its region mean yields the between component ≈ 0.13081; together they recover 0.17329 up to rounding. The split says how measured inequality is allocated under this partition, not that region caused it.

Mapped back: The mutually exclusive regions instantiate the Additive partition, while the same people and income total preserve Weighted population units, Additive resource, and Aligned share distributions. Resource-share weighting selects the T side of the T-or-L branch and reconstructs the Aggregate divergence. Fixing the partition and retaining the causal warning satisfy the Boundary policy and Interpretive limit.

Structural Tensions¶

T1: Upper-tail sensitivity versus lower-tail sensitivity. Theil T weights logarithmic contrasts by resource shares and is comparatively responsive to large values, while Theil L weights units by population and reacts strongly near zero. Calling either one simply “the Theil index” suppresses a consequential choice about which part of the distribution has leverage.

Diagnostic: Does the stated question and data boundary justify T, L, or an explicitly reported comparison of both?

T2: Logarithmic sensitivity versus zero and negative observations. Log ratios give the family its divergence structure, but Theil L diverges at zero and the ordinary formulas do not admit negative resources; T handles zero shares only by a limiting convention. Offsets, restrictions, and coding rules make computation possible by changing the estimand.

Diagnostic: Which observations are zero or negative, and what declared treatment preserves or alters the intended quantity?

T3: Additive decomposition versus causal attribution. Within- and between-group terms reveal where measured divergence sits under an exhaustive partition, yet they do not explain why the distribution arose. A clean accounting identity can invite an unwarranted claim that group membership caused the between-group component.

Diagnostic: Is the conclusion limited to arithmetic allocation under the chosen partition, or does separate evidence identify a causal mechanism?

T4: Scalar compression versus distributional shape. One number supports comparison and decomposition, while distinct distributions can share that number despite different ranks, gaps, and concentration patterns. The compression is useful precisely because it discards detail that may remain substantively important.

Diagnostic: Which distributional features relevant to the decision are invisible in the reported Theil value?

T5: Bounded normalization versus replication invariance. Dividing Theil T by ln N supplies an appealing zero-to-one scale for a fixed population size, but duplicating the same distribution changes that denominator. Ease of presentation is purchased by losing a property of the unnormalized measure.

Diagnostic: Is a bounded display more important here than comparability under population replication, and is the normalization disclosed?

T6: Scale invariance versus substantive comparability. Multiplying every observation by one positive constant leaves relative shares unchanged, but changing the resource definition, unit of observation, equivalence scale, or population boundary can alter what is being measured. Mathematical invariance cannot repair an inconsistent substantive design.

Diagnostic: Are the compared estimates based on the same population, additive resource, weights, adjustments, and period conventions?

T7: Tail responsiveness versus incomplete data. Theil T can expose concentration among resource-rich units and Theil L can expose severe shortfalls near the bottom, while top coding, nonresponse, censoring, or bottom coding often weaken evidence exactly where each variant is most sensitive. Greater diagnostic leverage therefore brings greater dependence on boundary data.

Diagnostic: How do restored tails, alternative coding, or uncertainty estimates change the result in the region emphasized by the chosen variant?

T8: Anonymous distribution versus situated inequality. Permuting unit labels leaves the index unchanged, enabling distributional comparison without privileging identities. The same anonymity prevents the scalar from showing who bears a shortfall, how disadvantages intersect, or whether a measured transfer is socially just.

Diagnostic: Which identity, need, mobility, discrimination, or welfare claims require evidence beyond the anonymous resource distribution?

T9: Theil Index autonomy versus reduction to Generalized entropy index (Generalized Entropy Index). The immediate domain-specific parent abstraction carries the wider parameterized family. Every Theil Index is a strict kind of Generalized Entropy Index—its T and L branches are the named α=1 and limiting α=0 cases—but the child fixes their resource- and population-weighting interpretations, information-theoretic reading, and characteristic decomposition practice. Reduction loses those identities; total autonomy hides the broader family.

Diagnostic: Does the account preserve the T/L formulas and their distinct weights as differentia of this Generalized Entropy Index?

Structural–Framed Character¶

Theil Index is mixed-structural. Its vocab_travels is moderate because logarithmic shares, weighted sums, and decomposition are general, while population weights, positive resources, inequality, and segregation fix the application. Its evaluative_weight is low: the name identifies a descriptive formula, and choosing a population, resource, T/L branch, zero policy, or grouping changes the estimand without itself issuing a welfare verdict. Its institutional_origin lies in economic and statistical measurement practice. Its human_practice_bound is moderate: exact distributions support the computation, but empirical estimands depend on population and resource definitions. On import_vs_recognize, distributional shares are observed or stipulated while the T/L branch and boundary policy are imposed.

The exact immediate parent, Generalized entropy index, is the in-domain umbrella: Theil T and L are fixed members of that parameterized inequality family. The smallest reviewed portable skeleton is Aggregation: weighted logarithmic share contrasts are combined into one scalar while declared information is discarded. Portable and cross-domain reach belongs to that Prime. The Theil identity additionally requires aligned population and resource shares, T-or-L weighting, scale invariance, optional valid group decomposition, and explicit zero and negative-value policy.

Its character: mixed-structural because the weighted aggregate and decomposition laws are exact, while the measured distribution, population, resource, weighting, and interpretation remain framed.

Structural Core vs. Domain Accent¶

This decomposition shows why the Theil Index is a domain-specific abstraction rather than a Prime.

What is skeletal (could lift toward a cross-domain prime). The abstract carrier is a set of aligned units bearing two normalized share distributions. A declared logarithmic weighting rule maps unit-level share contrasts into one nonnegative scalar; equality remains the zero invariant, and recognition requires that carrier, weights, formula branch, and boundary conventions remain fixed. This many-to-one information-losing operation instantiates Aggregation. Within inequality measurement, the candidate strictly specializes Generalized entropy index: remove the fixed Theil branches and the parameterized generalized-entropy family remains, whereas removing mean-relative transformation and decomposable family structure destroys the Theil identity.

What is domain-bound. The units represent a population and the second distribution represents shares of a declared positive additive resource. Theil T fixes the generalized-entropy parameter at α=1 and resource-share weights; Theil L fixes the limiting α=0 form and population-weighted log shortfalls. Variant-correct within/between decomposition, zero and negative-value policy, survey weights, log base, and any normalization determine what the number means. Replace the aligned population/resource shares with an informal notion of entropy, interchange T and L, or omit the resource and boundary policy, and the resulting statistic is not the Theil Index.

Why this does not clear the prime bar. The complete population-share, positive-resource, T-or-L, and inequality-interpretation signature does not recur literally across three unrelated domains; portable reach belongs to Aggregation, while the immediate family relation remains Generalized entropy index. Stripping the domain accent leaves a weighted scalar aggregation or a broader entropy-family member, not the named index. Conversely, retaining Theil or inequality vocabulary while removing the aligned-share logarithmic operation and its fixed branch leaves a label without the candidate-level structure.

This entry is a kind of Generalized entropy index.

Immediate domain parent — Generalized entropy index (Generalized entropy index). Theil T and Theil L are the named α = 1 and limiting α = 0 members of the parameterized generalized-entropy family. They preserve its mean-relative carrier, power-or-log transformation, scale invariance, and additive decomposability while fixing two formulas, two weighting interpretations, and their distinct upper- and lower-tail sensitivities. The child therefore specializes rather than merely resembles the family.

Related to — Measurement (Measurement). In empirical use, inequality is the target attribute, a declared distribution of a positive additive resource supplies the carrier, and the dimensionless Theil T or L value supplies a scale. Resource definition, population and resource weighting, log base, normalization, zero policy, and survey treatment determine the estimand; sampling and censoring can also require uncertainty analysis. This is not class-wide subsumption: an exact computation on a mathematical distribution need not include an instrument-target interaction, calibration chain, observer frame, or uncertainty envelope, and the formula alone is a transformation rule rather than an instrument. Consistent changes of physical units do not alter the index because it uses shares and is scale invariant.

Instantiates — Aggregation (Aggregation). Individual logarithmic share contrasts are the many inputs, and their variant-specific weighted sum is the one scalar summary. The operation preserves selected information about distributional divergence while deliberately discarding unit identity, ranks, distribution shape, and causal history. The aggregation remains valid under permutation and common positive rescaling, but ceases to be this index when the T/L weighting rule, aligned-share basis, or boundary conventions are lost.

Related to — Decomposition (Decomposition). When an exhaustive, mutually exclusive partition is supplied, the total Theil value can be separated into variant-correct within-group and between-group contributions whose weighted recombination exactly restores the whole. The partition is the decomposition axis; group contributions are the parts; exact additivity is the recomposition invariant. This is a characteristic conditional capability rather than the identity of every Theil computation: without an actual partitioning operation, the statistic remains a Theil index but no decomposition instance has been performed. The capability fails when groups overlap, omit units, mix population and resource weights, or when components from incompatible partitions are summed, and arithmetic decomposition never by itself warrants a causal account.

Relationships to Other Abstractions¶

Current abstraction Theil Index Domain-specific

Parents (1) — more general patterns this builds on

Theil Index is a kind of Generalized entropy index Domain-specific

Theil T and Theil L are the named α = 1 and limiting α = 0 members of the parameterized generalized-entropy family.

Hierarchy path (1) — routes to 1 parentless root

Theil Index → Generalized entropy index → Measurement

Neighborhood in Abstraction Space¶

Theil Index sits in a sparse region of the domain-specific corpus (65^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Gini Coefficient — 0.87
Foster–Greer–Thorbecke Poverty Measures — 0.85
Genetic Load — 0.84
Class stratification — 0.84
Divisia Index — 0.83

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Generalized entropy index. The generalized entropy index is the broader parameterized inequality-measure family; Theil T and Theil L are its α = 1 and limiting α = 0 members, respectively. Tell: Is α free to select another sensitivity profile, or is the formula fixed to the T or L case conventionally grouped under the Theil name?
Theil T. Theil T is the resource-share-weighted α = 1 member conventionally grouped under the umbrella Theil Index family, computed from (x_i/μ) ln(x_i/μ) contributions; it is one variant, not an alias for the whole family or an interchangeable name for Theil L. Tell: Are log relative shares weighted by each unit's resource share, with zero shares handled by the 0 ln 0 limit?
Theil L or mean log deviation. Theil L, also called mean log deviation, is the population-weighted limiting α = 0 member conventionally grouped under the umbrella Theil Index family; it is computed from ln(μ/x_i) and is undefined or divergent at a literal zero value. Tell: Does every population unit receive its population weight in a log-shortfall formula, making a zero resource value inadmissible under the unmodified estimator?
Shannon entropy. Shannon entropy measures uncertainty in a probability distribution; it supplies information-theoretic structure for the Theil interpretation, whereas a Theil result measures divergence between aligned population and resource shares under a declared inequality carrier. Tell: Is the reported scalar the uncertainty of one distribution, or the inequality obtained by comparing observed resource shares with population shares?
Kullback–Leibler divergence. Kullback–Leibler divergence is a directional discrepancy between two probability distributions; Theil T is its specific distributional application from resource shares to population shares, with economic or segregation units and boundary policies supplied. Tell: Are arbitrary probability distributions being compared, or are the two arguments specifically aligned resource and population shares over the same units?
Gini coefficient. The Gini coefficient is a distinct inequality measure derived from pairwise or Lorenz-curve disparity and does not inherit the Theil family's same additive within/between partition. Tell: Does the analysis use logarithmic share ratios with variant-specific decomposition weights, or rank and cumulative-share geometry?
Atkinson index. The Atkinson index is a welfare-based inequality family parameterized by inequality aversion; it can be mathematically related to generalized-entropy measures but answers a differently parameterized evaluative question. Tell: Is sensitivity chosen through a welfare or aversion parameter, or through the T/L logarithmic share formula and its stated weighting branch?

References¶

[1] Theil, Inequality Indices and Decomposition registry ↩ Show verification details

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