Kernel Characterisations of Stochastic Orders Within Parametric Density Families

Abstract: We develop kernel criteria for the likelihood-ratio, hazard-rate, usual stochastic, and relative log-concavity orders in parametric families of univariate probability laws with densities. The score is the derivative of the log density with respect to the parameter, and a kernel equals the score up to an additive term depending only on the parameter. Kernel monotonicity gives likelihood-ratio order, kernel concavity gives relative log-concavity, and two tail-conditional mean inequalities give the hazard-rate and usual stochastic orders. The same construction applies along joint-parameter paths and to comparisons between two laws whose densities admit parameter-dependent factors, where the log-factor ratio is used as the kernel. For compound sums with a random number of i.i.d. terms, the induced kernel is the posterior mean of the kernel of the summand count. The applications recover standard one-parameter orderings, give likelihood-ratio comparisons for compound laws, and handle nonmonotone examples through the tail-conditional criteria.

• The paper establishes if-and-only-if kernel criteria for likelihood-ratio, relative log-concavity, hazard-rate, and usual stochastic orders, reducing comparisons to monotonicity, concavity, or tail-conditional inequalities.
• The paper extends the framework to nonmonotone kernels, parameter paths, direct pairwise comparisons, and compound laws, recovering explicit results for beta-binomial, Poisson, negative-binomial, and related families.
• The paper shows that stochastic and hazard-rate ordering can hold even when likelihood-ratio ordering fails, as demonstrated by half-Student and zero-inflated Poisson or exponential examples.

Overview and main idea

This paper, by Zakaria Derbazi (Queen Mary University of London), develops a unified set of criteria for establishing four stochastic orders — likelihood-ratio ($\lr$), relative log-concavity ($\lc$), hazard-rate ($\hr$), and usual stochastic ($\st$) — between members of parametric families of univariate probability laws. The central object is the kernel: for a density family $(f_\nu)$ with score $s_\nu(x)=\partial_\nu\log f_\nu(x)$, a kernel is any measurable function $K_\nu$ on the support whose centring under $P_\nu$ recovers the score,

$$s_\nu(x) = K_\nu(x) - \int_J K_\nu(y)\,\mathrm{d}P_\nu(y).$$

Since the centring term is constant in $x$, the kernel inherits every shape property of the score that is invariant under additive constants (monotonicity, concavity). The practical advantage is that kernels are often computed at the level of parameter-dependent factors $\lc$0 via $\lc$1, thereby eliminating normalising terms before any analysis begins.

The four-order characterisation

The technical core rests on two identities. First, Proposition 3.2 shows that for any nonnegative test function $\lc$2 satisfying an integrable-majorant regularity condition,

$\lc$3

where $\lc$4 is the $\lc$5-tilted law. Second, Lemma 3.3 instantiates this at three levels: the density derivative equals $\lc$6, the survival-function derivative equals $\lc$7, and the hazard-rate derivative equals $\lc$8. Integrating these local identities over a parameter interval yields the pairwise comparison functions for the log-density ratio, log-survival ratio, and log-hazard ratio.

Theorem 3.4 then gives an if-and-only-if characterisation of all four orders in terms of the kernel alone:

Order	Kernel condition
$\lc$9	$\hr$0 nondecreasing on $\hr$1
$\hr$2	$\hr$3 concave on $\hr$4
$\hr$5	$\hr$6 for all $\hr$7
$\hr$8	$\hr$9 for all $\st$0

The necessity directions are proved by contradiction using continuity of the score in $\st$1 under Assumption 2.1 (differentiation under the integral sign with a compact-interval majorant): a local violation of the kernel condition persists over a parameter interval, producing a strict violation of the corresponding order after integration. This uniform four-order statement is stronger than the classical likelihood-ratio-based criteria: monotonicity of the kernel simultaneously delivers $\st$2, hence $\st$3 and $\st$4 through the standard implication chain.

Beyond monotone and concave kernels

When global shape properties fail, the paper supplies a superlevel-set criterion (Proposition 3.7): if the superlevel set $\st$5 is a nonempty initial interval of the support and $\st$6 has finite left endpoint, then the family is $\st$7-ordered; adding right-tail monotonicity of $\st$8 yields $\st$9. Two corollaries specialise this to concave kernels and to unimodal kernels with a common mode, reducing verification to a boundary check $(f_\nu)$0. These are kernel-level analogues of the pairwise likelihood-ratio conditions of Derbazi's earlier work.

Two applications illustrate the reach of these weaker hypotheses:

• Half-Student family: the score satisfies $(f_\nu)$1, so it is unimodal with mode $(f_\nu)$2 independent of $(f_\nu)$3, and $(f_\nu)$4 follows from concavity of the digamma function. Hence increasing degrees of freedom raises the law in both $(f_\nu)$5 and $(f_\nu)$6 — even though the Student-$(f_\nu)$7 kernel on $(f_\nu)$8 is neither monotone nor concave.
• Zero-inflated Poisson and zero-inflated exponential: atom inflation breaks kernel monotonicity at the origin (for large $(f_\nu)$9, $s_\nu(x)=\partial_\nu\log f_\nu(x)$0 while $s_\nu(x)=\partial_\nu\log f_\nu(x)$1), ruling out $s_\nu(x)=\partial_\nu\log f_\nu(x)$2 and $s_\nu(x)=\partial_\nu\log f_\nu(x)$3; nevertheless the tail-conditional inequalities hold explicitly because the inflated atom is excluded from upper tails, so $s_\nu(x)=\partial_\nu\log f_\nu(x)$4 and $s_\nu(x)=\partial_\nu\log f_\nu(x)$5 survive. The zero-inflated exponential example makes the failure of $s_\nu(x)=\partial_\nu\log f_\nu(x)$6 concrete: the likelihood ratio jumps upward at the atom.

These examples demonstrate a genuine separation between the orders attainable by the method and those that are not.

Extensions: paths, pairwise comparisons, and compound laws

Three extensions broaden the scope beyond single-parameter families. For joint-parameter paths, the chain rule expresses the path kernel as a weighted sum of single-parameter kernels, recovering e.g. $s_\nu(x)=\partial_\nu\log f_\nu(x)$7 along simultaneous shape–success-probability paths, and gamma comparisons under jointly increasing shape and scale. An interpolation construction connects laws of distinct factor form; notably, if $s_\nu(x)=\partial_\nu\log f_\nu(x)$8 then $s_\nu(x)=\partial_\nu\log f_\nu(x)$9, proved by a path whose kernel forward difference has an explicit sign determined exactly by this threshold condition.

For pairwise comparisons between laws admitting compatible factorisations, the geometric-interpolation path has constant path score equal to the log-factor ratio $K_\nu$0, so Theorem 3.4 applies directly without choosing a path. This yields a catalogue of Katz-class comparisons (binomial–Poisson, binomial–negative-binomial, Poisson–negative-binomial) with explicit endpoint tests such as $K_\nu$1, CMP dispersion orderings, and a beta-binomial versus hypergeometric criterion governed by the single inequality $K_\nu$2.

For compound sums $K_\nu$3 with PF$K_\nu$4 summand law $K_\nu$5, Proposition 3.11 shows the compound score is the posterior expectation $K_\nu$6 of the counting-law score, and Corollary 3.12 transports factor-level kernels similarly. Since PF$K_\nu$7 convolution powers form TP$K_\nu$8 kernels, posterior laws of $K_\nu$9 given $P_\nu$0 are stochastically increasing in $P_\nu$1, so monotone $P_\nu$2 implies the compound law is $P_\nu$3-monotone in $P_\nu$4 (Proposition 3.13). A single table covers five classical counting laws, including compound Poisson ordering in the rate — relevant to actuarial compound-geometric models — and the harmonic-kernel case of compound negative-binomial in its shape parameter. Notably, the compound geometric case extends Xia and Lv's relative-log-concavity framework, which is unavailable there, to full likelihood-ratio comparisons. The Poisson-binomial application recovers coordinatewise likelihood-ratio monotonicity $P_\nu$5 under coordinatewise domination of success probabilities.

Limitations and open questions

The criteria are inherently one-dimensional: multi-parameter comparisons require either fixing all but one parameter or exhibiting a suitable path, and the choice of path can affect which comparisons are certifiable. All results assume a common support interval and differentiability under the integral (Assumption 2.1); laws with parameter-dependent supports fall outside the framework. The compound-law extension requires PF$P_\nu$6 summands, and the posterior-averaging argument delivers $P_\nu$7 but not $P_\nu$8 in cases such as the compound negative-binomial in its shape parameter. The tail-conditional inequalities in Theorem 3.4(iii)–(iv) must be verified pointwise when kernel shape conditions fail, and no general algorithm is offered for doing so. Two directions flagged in the conclusion — multivariate extensions and connections to other partial orders — remain open.

Conclusion

The paper establishes that the likelihood-ratio, relative log-concavity, hazard-rate, and usual stochastic orders within parametric density families admit complete characterisations through a single object, the kernel, equivalently the centred score. Because kernels live at the factor level before normalisation, the same criteria transport cleanly to joint-parameter paths, direct pairwise factor comparisons, and compound constructions via posterior averaging. The framework recovers standard orderings for exponential, power-series, Pochhammer-block, and location families, handles nonmonotone cases through superlevel-set and tail-conditional tests, and extends compound-geometric comparisons beyond the existing relative-log-concavity literature.