Likelihood Ratio-Type Test

• Likelihood ratio-type tests are statistical methods that compare maximized likelihoods of nested models to assess parameter constraints and test hypotheses.
• They employ both smoothed and raw MLE approaches, achieving normality or cube-root asymptotics under conditions such as censoring and high-dimensional data.
• Bootstrap calibration and careful kernel smoothing improve test accuracy, ensuring robust power and type I error control in complex observational settings.

The likelihood ratio-type test is a central methodology for comparing statistical models, assessing parameter constraints, and handling complex censoring or latent structures, with widespread application across biomedical, engineering, social, and natural sciences research. It is fundamentally grounded in comparing the maximized likelihoods of two competing hypotheses, typically a null (restricted) and an alternative (full or less constrained) model, using the ratio of these maximum likelihoods as the test statistic. As illustrated in applications ranging from censored event time data to high-dimensional covariance testing and network community detection, likelihood ratio-type tests provide a flexible and theoretically principled approach for hypothesis testing in both classical and modern data regimes.

1. Canonical Framework and Construction

The likelihood ratio-type test is typically formulated as follows: for data $X_1, \ldots, X_n$ with likelihood function $L(\theta; X_1^n)$, consider two parameter spaces, $\Theta_0\subset \Theta$, corresponding to the null and full models. The test statistic is

$$\mathrm{LR}_n = -2 \log \frac{\sup_{\theta \in \Theta_0} L(\theta; X_1^n)}{\sup_{\theta \in \Theta} L(\theta; X_1^n)}.$$

Under standard regularity conditions (the parameter $\theta^*$ is an interior point, the Fisher information is positive-definite), Wilks’ theorem guarantees that, under the null, the asymptotic distribution of $\mathrm{LR}_n$ is $\chi^2_{d}$, where $d$ is the difference in the number of free parameters between the models.

However, in many contemporary settings—such as high-dimensional statistics, censored data, or boundary/latent constraints—these regularity conditions are violated. As a result, the limiting distribution may deviate from $\chi^2$, requiring refined asymptotic analysis using tools such as tangent cones, mixture distributions, and empirical process theory.

2. Nonparametric Two-Sample Likelihood Ratio Tests under Current Status Censoring

When data are observed under current status censoring—each subject is seen at a single random inspection time $T$ and only the indicator $\delta = \mathbf{1}_{\{T \geq X\}}$ is recorded—the problem arises of testing equality of two hidden event time distributions ($F_1, F_2$), potentially with non-identical observation (inspection) time distributions ($g_1, g_2$). The standard approach exploits the likelihood function involving these hidden distributions, but the highly indirect way in which $F_1, F_2$ enter the observed data likelihood complicates classical testing strategies.

The methodology developed for this setting comprises two likelihood ratio-type statistics:

• Smoothed MLE (MSLE)-based Test: Here, the key idea is to construct nonparametric kernel-smoothed maximum likelihood estimators for $F_1, F_2$ (denoted $\widetilde{F}_{n,1}$, $\widetilde{F}_{n,2}$), and for the pooled distribution, $\widetilde{F}_n$. The test statistic $V_N$ is defined via integrals of the type

$$V_N = (2m/N) \int_{a}^{b} \left\{ \widetilde{h}_{n,1}(t)\log\frac{\widetilde{F}_{n,1}(t)}{\widetilde{F}_n(t)} + [\widetilde{g}_{n,1}(t) - \widetilde{h}_{n,1}(t)]\log\frac{1-\widetilde{F}_{n,1}(t)}{1-\widetilde{F}_n(t)} \right\} dt + (2n/N)\{\cdots\},$$

with $\widetilde{g}_{n,j}(t)$ and $\widetilde{h}_{n,j}(t)$ being kernel smoothers for the density of inspection times and the joint density with $\delta=1$, respectively.

For suitable kernel bandwidth $b_N \sim N^{-\alpha}$, with $2/9 < \alpha < 1/3$, it is shown that $V_N$ admits an asymptotically normal distribution after normalization. This is in contrast to the classical (non-smoothed) NPMLE, which has a non-normal, cube-root limiting distribution.

• Raw MLE-based Test: The alternative approach uses the isotonic NPMLEs for $F_{n,1}^{\rm mle}$, $F_{n,2}^{\rm mle}$, and $F_n^{\rm mle}$ to build the likelihood ratio, but the limit theory is nonstandard due to the Grenander-type behavior; the limiting law is not normal but follows the cube-root asymptotics.

The kernel-smoothing in the MSLE delivers both faster convergence rate ($N^{-2/5}$) and normality, facilitating critical value calculations and boosting test power.

3. Bootstrapping and Calibration

Due to slow convergence rates and the presence of nuisance parameters, the critical values for the likelihood ratio-type tests—particularly for smoothed statistics—are reliably estimated by a data-adaptive bootstrap. The procedure is:

1. Compute the pooled MSLE with a comparatively large bandwidth $\bar{b}_N \sim N^{-1/5}$.
2. Generate bootstrap indicators $\delta_i^*$ at observed inspection times $T_i$ as Bernoulli random variables with success probability equal to the estimated pooled MSLE.
3. Recompute MSLEs from the bootstrap data and obtain the corresponding test statistic $V_N^*$.
4. The empirical distribution of $V_N^*$ (over many resamples) is used to obtain bootstrap-calibrated critical values (e.g., the 95th percentile for level 5% test).

A central theoretical result guarantees that—conditional on the observed sample—the normalized bootstrap statistic converges to the same normal limit as the original statistic, thus validating this calibration strategy.

4. Theoretical Properties, Optimal Rates, and Power

Important theoretical developments include:

• Under regularity assumptions (smoothness, boundedness away from 0,1 for $F$, and for $g_1, g_2$), the asymptotic normality of the MSLE-based statistic is established for bandwidths in $2/9 < \alpha < 1/3$ unless additional bias terms appear, in which case $1/5 < \alpha \leq 2/9$.
• The variance of the asymptotic normal distribution is derived in terms of the kernel (e.g., $\sigma_K^2 = 2 \int [\int K(u+v)K(u)du]^2dv$).
• When $g_1 = g_2$, the statistic is an asymptotic pivot (its null distribution does not depend on unknown parameters). If $g_1 \neq g_2$, an $\mathcal{O}_p(1)$ bias term needs to be considered.
• Empirical process and martingale central limit theories underpin the proofs, addressing the intricacies of kernel smoothing under current status censoring.

Extensive simulations indicate:

• The proposed LR-type tests—both MSLE- and MLE-based—deliver high rejection power, even under alternatives with crossing hazards or shape differences where cumulative function-based tests (e.g., those of Sun or Andersen & Rønning) have near-zero power.
• When observation time distributions differ between groups ($g_1 \neq g_2$), alternative simple-function-based tests exhibit anti-conservative behavior (inflated type I error), whereas the LR-type tests maintain accurate levels due to their accommodation of non-identical $g_j$.

For moderate sample sizes ($m = n = 50$ up to $250$), bootstrap-calibrated LR-type tests achieve nominal significance levels and near-perfect power for well-separated alternatives.

5. Implementation, Computational Considerations, and Limitations

Implementation of the smoothed LR-type test involves:

• Choice of kernel and bandwidth for estimation. Optimal theoretical rates are prescribed, but practical data-driven or cross-validation bandwidth selection may be used subject to constraints to ensure valid inference.
• Bootstrap computation, which requires repeated kernel estimations and maximization of smoothed likelihoods in each resample.
• Numerical integration over the interval $[a, b]$—arising in the definition of test statistics—must be handled via quadrature or fine discretization. Computational cost is moderate for moderate $N$.

Limitations:

• The approach assumes continuous and strictly monotone survival functions bounded away from 0,1 and continuously differentiable $g_j$, which may not precisely hold in some practical data.
• Very small samples may challenge the accurate estimation of kernel-smoothed functions.
• If the two observation time densities $g_1, g_2$ differ dramatically, and their difference alters at the boundaries of $[a, b]$, bias terms may be non-negligible, complicating theoretical calibration.

6. Comparison with Alternative Methods and Practical Impact

Relative to traditional function-based two-sample tests in the current status model:

Method	Valid if $g_1 \neq g_2$	Power for crossing alternatives	Size (null) control
LR-type (MSLE-based)	Yes	High	Accurate
LR-type (MLE-based, bootstr.)	Yes	High	Accurate
Sun (2006), Andersen & Rønning	No	Can be low/zero	Anti-conservative

The nonparametric LR-type tests provide strict robustness to variance in the observation schemes, do not require moment functionals (which miss complex alternatives), and are fully nonparametric in approach.

In summary, likelihood ratio-type two-sample tests for current status data—particularly the bootstrap-calibrated MSLE version—provide a theoretically justified, computationally feasible, and statistically powerful solution for equality testing of distributions under interval-censored and potentially heterogeneous observation time settings. This approach establishes the standard for robust inference in current status comparison problems, outperforming alternative nonparametric methods in scenarios with complex alternatives or differing inspection regimes (Groeneboom, 2011).