Sieve Maximum Pseudo-Likelihood Estimation

• SMPLE is a semiparametric estimation technique that maximizes a pseudo-likelihood over sieve approximations of infinite-dimensional nuisances.
• It is applied in single-index and recurrent event models using wavelet and spline bases to extract finite-dimensional parameters effectively.
• The method provides rigorous finite-sample bounds and asymptotic guarantees, ensuring practical efficiency, robustness, and semiparametric efficiency.

Searching arXiv for the cited SMPLE papers and closely related terminology to ground the article in current arXiv records. arxiv_search(query="^^^^^^^^3\3 sample analysis of profile M-estimation in the Single Index model^^^^^^^^3\3 OR ^^^^^^^^3\3 Event Analysis with Ordinary Differential Equations^^^^^^^^3\3 OR ^^^^^^^^3\3 Maximum Pseudo-Likelihood Estimation^^^^^^^^3\3 max_results=^^^^3 OR \3\3^^^^, sort_by="relevance") Sieve Maximum Pseudo-Likelihood Estimation (SMPLE) is a sieve-based M-estimation methodology in which a pseudo-likelihood or quasi-likelihood is optimized jointly over a finite-dimensional parameter of interest and a growing finite-dimensional approximation to an infinite-dimensional nuisance component. In the arXiv literature represented here, SMPLE appears in two technically distinct but structurally related settings: sieve profile quasi maximum likelihood estimation for the single-index model with a nonparametric link approximated by PRESERVED_PLACEHOLDER_^^^^3\3^^^^-Daubechies wavelets (&&&^^^^3\3^^^^&&&), and sieve maximum pseudo-likelihood for recurrent event data in which the conditional mean function is modeled through an ordinary differential equation and the nuisance functions are approximated by B-splines under an NHPP working model (&&&^^^^3 OR \3^^^^&&&). Across these formulations, the common principle is to replace an infinite-dimensional nuisance problem by a sequence of finite-dimensional sieve spaces, optimize an empirical pseudo-likelihood, and derive finite-sample or asymptotic guarantees for the target finite-dimensional parameter.

^^^^3 OR \3^^^^. Conceptual definition and scope

SMPLE denotes estimation by maximizing a pseudo-likelihood over a sieve parameterization of nuisance functions. In the recurrent-event formulation, the method is introduced explicitly as a “Sieve Maximum Pseudo-Likelihood Estimation (SMPLE) method, employing the NHPP as a working model” (&&&^^^^3 OR \3^^^^&&&). In the single-index formulation, the paper describes a “sieve profile quasi maximum likelihood estimator” based on least squares, and states that “the paper’s ‘sieve profile quasi-MLE’ is an SMPLE” because it maximizes a quasi-likelihood over a sieve for the nonparametric link and profiles out that link to estimate the index parameter (&&&^^^^3\3^^^^&&&).

The “pseudo-likelihood” terminology is essential. In the single-index model, the criterion is least squares and “This is quasi-MLE because PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^ need not be Gaussian; as M-estimation it plays the role of a pseudo-likelihood” (&&&^^^^3\3^^^^&&&). In the recurrent-event model, the pseudo-likelihood is built from the NHPP log-likelihood even when the true process may be non-Poisson, so the estimator targets the parameters that best fit the conditional mean ODE “in the pseudo-likelihood sense (M-estimation)” (&&&^^^^3 OR \3^^^^&&&).

This suggests that SMPLE is not tied to a single stochastic model class. Rather, it is a general estimation strategy for semiparametric problems in which the parameter of interest is finite-dimensional, the nuisance structure is infinite-dimensional, and a working likelihood is available but may be misspecified.

^^^^3 OR \3^^^^. SMPLE in the single-index model

In the single-index setting, the starting point is the regression model

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^

with $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$ and i.i.d., centered, sub-Gaussian errors independent of $X_i$ (&&&^^^^3\3^^^^&&&). The paper defines the link through conditional expectation, $f(t) := E[g(X)\mid X^\top \theta = t]$, and uses the “true index” $\theta^*$ to characterize the target (&&&^^^^3\3^^^^&&&).

The sieve approximation uses a $C^3$-Daubechies wavelet basis with compact support on $[ -s_X, s_X]$. Writing $\{\psi_k\}_{k\ge 1}$ for the induced orthonormal basis on PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^, the link is approximated by

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^

with sieve dimension PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^ and resolution level PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^ satisfying PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^4 (&&&^^^^3\3^^^^&&&). The empirical criterion is

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^5

The joint sieve maximizer is

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^6

where PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^7 and the Euclidean ball constraint on PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^8 is used to “avoid boundary issues” (&&&^^^^3\3^^^^&&&). The profiled objective is

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^9

with

PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^

The theoretical targets distinguish the infinite-dimensional oracle

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^

from the finite-sieve target

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^

This separation makes explicit the two sources of deviation: stochastic estimation error and sieve approximation bias (&&&^^^^3\3^^^^&&&).

3. SMPLE for recurrent event analysis via ODEs

In the recurrent-event formulation, SMPLE is built around the conditional mean function

PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^

modeled as the solution to the ODE

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^4

with focus on the class

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^5

or equivalently

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^6

where PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^7 and PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^8 (&&&^^^^3 OR \3^^^^&&&).

Under the NHPP working model, the conditional intensity is set equal to the mean rate,

PRESERVED_PLACEHOLDER_^^^^3 OR \3^^^^9

and the pseudo-log-likelihood becomes

$$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$^^^^3\3^^^^

Here $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$^^^^3 OR \3^^^^ solves the subject-specific ODE with $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$^^^^3 OR \3^^^^ (&&&^^^^3 OR \3^^^^&&&).

The sieve spaces are spline-based. For $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$3 on $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$4, the paper chooses a polynomial spline space $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$5 with B-spline basis $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$6 and representation

$$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$7

For $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$8 on $$\theta \in S_1^{p,+} = \{\theta \in \mathbb{R}^p: \|\theta\| = 1, \theta_1 > 0\}$$9, it uses $X_i$^^^^3\3^^^^ with basis $X_i$^^^^3 OR \3^^^^ and

$X_i$^^^^3 OR \3^^^^

(&&&^^^^3 OR \3^^^^&&&).

The sieve parameter is $X_i$3, subject to the identifiability constraints $X_i$4 and $X_i$5, equivalently

$X_i$6

With these expansions, the objective becomes

$X_i$7

No explicit penalization is required; instead, identifiability constraints are enforced directly (&&&^^^^3 OR \3^^^^&&&).

4. Sieve construction, profiling, and identifiability

The two arXiv formulations differ in basis choice and data-generating structure, but they share a common architecture: the nuisance function is approximated by a finite-dimensional sieve, and the target finite-dimensional component is extracted either by profiling or by joint optimization.

In the single-index paper, smoothness of the true wavelet coefficients is imposed through

$X_i$8

together with bounded derivatives $X_i$9 and $f(t) := E[g(X)\mid X^\top \theta = t]$^^^^3\3^^^^ (&&&^^^^3\3^^^^&&&). Identifiability is enforced through the unit half-sphere restriction on $f(t) := E[g(X)\mid X^\top \theta = t]$^^^^3 OR \3^^^^, the lower bound on the design density $f(t) := E[g(X)\mid X^\top \theta = t]$^^^^3 OR \3^^^^, the conditional variance lower bound for orthogonal projections, and a salience condition requiring a region on which $f(t) := E[g(X)\mid X^\top \theta = t]$3 (&&&^^^^3\3^^^^&&&). The paper also allows model bias through

$f(t) := E[g(X)\mid X^\top \theta = t]$4

supplemented by curvature conditions away from $f(t) := E[g(X)\mid X^\top \theta = t]$5 (&&&^^^^3\3^^^^&&&).

In the recurrent-event paper, identifiability arises because $f(t) := E[g(X)\mid X^\top \theta = t]$6 and $f(t) := E[g(X)\mid X^\top \theta = t]$7 are both unspecified. The imposed normalizations $f(t) := E[g(X)\mid X^\top \theta = t]$8 and $f(t) := E[g(X)\mid X^\top \theta = t]$9 anchor the scale of the parametric and nonparametric components (&&&^^^^3 OR \3^^^^&&&). Regularity assumptions include compact covariate support, covariate density bounded below, non-singularity of $\theta^*$^^^^3\3^^^^, independent censoring given $\theta^*$^^^^3 OR \3^^^^, Hölder smoothness for $\theta^*$^^^^3 OR \3^^^^ and $\theta^*$3, and the technical conditions (C^^^^3 OR \3^^^^)–(C9), including a lower bound on $\theta^*$4, absence of strong collinearity between functional derivatives with respect to $\theta^*$5 and $\theta^*$6, existence of least favorable directions, and non-singularity of the information-type matrix $\theta^*$7 (&&&^^^^3 OR \3^^^^&&&).

A useful contrast is that the single-index paper develops the SMPLE within a profile M-estimation framework using the conditions of Andresen A. and Spokoiny V., including AssId, $\theta^*$8, $\theta^*$9, $C^3$^^^^3\3^^^^, $C^3$^^^^3 OR \3^^^^, and $C^3$^^^^3 OR \3^^^^ (&&&^^^^3\3^^^^&&&). The recurrent-event paper instead formulates the nuisance structure through ODE-constrained splines and develops semiparametric efficiency through least favorable directions and orthogonalized scores (&&&^^^^3 OR \3^^^^&&&).

5. Finite-sample and asymptotic theory

For the single-index SMPLE, the paper derives nonasymptotic Wilks and Fisher expansions. Let $C^3$3 and define the profiled information matrix

$C^3$4

together with the profile score

$C^3$5

where $C^3$6 (&&&^^^^3\3^^^^&&&). Under the stated assumptions and sample-size constraints, the paper proves that, for $C^3$7 large enough and with probability at least

$C^3$8

$C^3$9

and

$[ -s_X, s_X]$^^^^3\3^^^^

with

$[ -s_X, s_X]$^^^^3 OR \3^^^^

(&&&^^^^3\3^^^^&&&). The corresponding sieve bias bound is

$[ -s_X, s_X]$^^^^3 OR \3^^^^

When $[ -s_X, s_X]$3 and $[ -s_X, s_X]$4, these finite-sample results imply the asymptotic limits

$[ -s_X, s_X]$5

and, in the correctly specified Gaussian regression case,

$[ -s_X, s_X]$6

which the paper states is efficient in the considered setting (&&&^^^^3\3^^^^&&&).

For the recurrent-event SMPLE, the asymptotic theory is organized differently. Under (C^^^^3 OR \3^^^^)–(C7),

$[ -s_X, s_X]$7

where $[ -s_X, s_X]$8 is defined through the Euclidean norm for $[ -s_X, s_X]$9, $\{\psi_k\}_{k\ge 1}$^^^^3\3^^^^-distance for $\{\psi_k\}_{k\ge 1}$^^^^3 OR \3^^^^, and an $\{\psi_k\}_{k\ge 1}$^^^^3 OR \3^^^^-distance involving $\{\psi_k\}_{k\ge 1}$3 (&&&^^^^3 OR \3^^^^&&&). Under the conditions of Theorem ^^^^3 OR \3^^^^ together with (C8)–(C9),

$\{\psi_k\}_{k\ge 1}$4

where

$\{\psi_k\}_{k\ge 1}$5

is the efficient influence function after orthogonalization with respect to the nuisance tangent directions (&&&^^^^3 OR \3^^^^&&&).

Under the NHPP working model, the paper states that $\{\psi_k\}_{k\ge 1}$6, so

$\{\psi_k\}_{k\ge 1}$7

and the estimator “achieves the semiparametric efficiency bound” (&&&^^^^3 OR \3^^^^&&&). For the sieve components, the overall convergence rate satisfies

$\{\psi_k\}_{k\ge 1}$8

and in the balanced case $\{\psi_k\}_{k\ge 1}$9, PRESERVED_PLACEHOLDER_^^^^3 OR \3\3\3^^^^,

PRESERVED_PLACEHOLDER_^^^^3 OR \3\3 OR \3^^^^

(&&&^^^^3 OR \3^^^^&&&).

6. Algorithms, computation, and covariance estimation

The single-index paper analyzes an alternating maximization procedure. Initialization is performed on a coarse grid PRESERVED_PLACEHOLDER_^^^^3 OR \3\3 OR \3^^^^, with closed-form update

PRESERVED_PLACEHOLDER_^^^^3 OR \3\33^^^^

followed by iterative PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^4-updates and PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^5-updates (&&&^^^^3\3^^^^&&&). The PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^6-updates are linear least squares in wavelet coefficients, whereas the PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^7-updates are nonconvex (&&&^^^^3\3^^^^&&&). Under the stated conditions, the paper proves high-probability deviation bounds for the alternating sequence and a convergence-to-global-maximizer result via Theorem ^^^^3 OR \3^^^^.4 of AASPalternating (&&&^^^^3\3^^^^&&&). It also reports that each PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^8-update solves an PRESERVED_PLACEHOLDER_^^^^3 OR \3\3^^^^9 linear system with cost “PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3\3^^^^” (&&&^^^^3\3^^^^&&&).

The recurrent-event paper uses block coordinate ascent together with ODE-based gradients. Given

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3 OR \3^^^^

the gradient requires the sensitivity ODE

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3 OR \3^^^^

where PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \33^^^^ (&&&^^^^3 OR \3^^^^&&&). A central computational device is the time transformation

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^4

which yields

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^5

“independent of PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^6, enabling a single ODE solve” (&&&^^^^3 OR \3^^^^&&&). The optimization alternates between updating PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^7 subject to PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^8 and updating PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^9 subject to PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3\3^^^^, using a gradient-based optimizer (&&&^^^^3 OR \3^^^^&&&).

Covariance estimation is also treated differently in the two settings. The recurrent-event paper develops a resampling procedure to estimate the asymptotic covariance matrix without closed-form least favorable directions. The core approximation is

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3 OR \3^^^^

with PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3 OR \3^^^^ generated from a mean-zero distribution, so that PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \33^^^^ can be estimated row-wise by regression of perturbed scores on the perturbations (&&&^^^^3 OR \3^^^^&&&). This yields PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^4 (&&&^^^^3 OR \3^^^^&&&).

7. Applications, interpretation, and limitations

The single-index paper situates SMPLE within projection pursuit. After estimating one direction and link, it updates residuals by

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^5

and derives the bound

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^6

under the stated conditions (&&&^^^^3\3^^^^&&&). The paper remarks that if

PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^7

decays suitably, then the total approximation error improves over the full multivariate nonparametric rate PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^8 (&&&^^^^3\3^^^^&&&).

The recurrent-event paper evaluates SMPLE through simulations and an ICU readmission application. It reports NHPP scenarios, gamma-frailty non-Poisson scenarios, and a real-data analysis of the MIMIC-III ICU readmission frequency dataset with “^^^^3 OR \3^^^^8,^^^^3 OR \3^^^^5^^^^3 OR \3^^^^ patients, ^^^^3 OR \3\3^^^^,36^^^^3\3^^^^ events” and “^^^^3 OR \3\3^^^^ covariates after preprocessing” (&&&^^^^3 OR \3^^^^&&&). In that application, the fitted general linear transformation model with quartic B-splines yields estimated PRESERVED_PLACEHOLDER_^^^^3 OR \3 OR \3^^^^9 and PRESERVED_PLACEHOLDER_^^^^3 OR \33\3^^^^ that “are not constant nor linear in PRESERVED_PLACEHOLDER_^^^^3 OR \33 OR \3^^^^, indicating that Cox- or AFT-type models are inadequate” (&&&^^^^3 OR \3^^^^&&&).

Several limitations are explicit. In the single-index setting, smoothness conditions require PRESERVED_PLACEHOLDER_^^^^3 OR \33 OR \3^^^^, and “PRESERVED_PLACEHOLDER_^^^^3 OR \333^^^^ if PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^4 is needed to force PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^5 sufficiently fast” (&&&^^^^3\3^^^^&&&). The design assumptions include Lipschitz PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^6, bounded support, and conditional variance lower bounds (&&&^^^^3\3^^^^&&&). In the recurrent-event setting, the pseudo-likelihood relies on the NHPP working model for construction, and under “heavy-tailed or highly dependent processes, variance estimation may require larger resampling PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^7” (&&&^^^^3 OR \3^^^^&&&). Joint identifiability of PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^8 and PRESERVED_PLACEHOLDER_^^^^3 OR \33^^^^9 requires constraints, and spline-dimension selection may require BIC or AIC tuning (&&&^^^^3 OR \3^^^^&&&).

A common misconception is that pseudo-likelihood necessarily implies inefficiency. The two papers jointly show a more nuanced picture. In the single-index model, the profile quasi-MLE admits Wilks and Fisher expansions and, in the correctly specified Gaussian regression case, achieves the stated efficient asymptotic variance (&&&^^^^3\3^^^^&&&). In the recurrent-event model, SMPLE is robust under non-Poisson misspecification through the sandwich covariance PRESERVED_PLACEHOLDER_^^^^3 OR \3relevance3\3^^^^, yet becomes semiparametrically efficient when the NHPP working model is valid (&&&^^^^3 OR \3^^^^&&&). This suggests that the “pseudo” qualifier refers to the construction of the criterion, not to an inevitable loss of first-order optimality.

A second misconception is that sieve methods are purely asymptotic devices. The single-index paper directly counters this by establishing finite-sample Wilks and Fisher results, nonasymptotic confidence sets, and convergence guarantees for alternating maximization (&&&^^^^3\3^^^^&&&). The recurrent-event paper complements this with a scalable implementation based on a shared ODE solve, sensitivity equations, and resampling-based covariance estimation (&&&^^^^3 OR \3^^^^&&&). Taken together, these works place SMPLE at the intersection of semiparametric efficiency theory, sieve approximation, and computationally tractable pseudo-likelihood optimization.