Hubbry Logo
Nuisance parameterNuisance parameterMain
Open search
Nuisance parameter
Community hub
Nuisance parameter
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Nuisance parameter
Nuisance parameter
Nuisance parameter
View on Wikipedia
from Wikipedia

In statistics, a nuisance parameter is any parameter which is unspecified[1] but which must be accounted for in the hypothesis testing of the parameters which are of interest.

The classic example of a nuisance parameter comes from the normal distribution, a member of the location–scale family. In the case of normal distribution, the variance(s), σ2 is often not specified or known, but one desires to hypothesis test on the mean(s). Another example might be linear regression with unknown variance in the explanatory variable (the independent variable): its variance is a nuisance parameter that must be accounted for to derive an accurate interval estimate of the regression slope, calculate p-values, hypothesis test on the slope's value; see regression dilution.

Nuisance parameters are often scale parameters, but not always; for example in errors-in-variables models, the unknown true location of each observation is a nuisance parameter. A parameter may also cease to be a "nuisance" if it becomes the object of study, is estimated from data, or known.

Theoretical statistics

[edit]

The general treatment of nuisance parameters can be broadly similar between frequentist and Bayesian approaches to theoretical statistics. It relies on an attempt to partition the likelihood function into components representing information about the parameters of interest and information about the other (nuisance) parameters. This can involve ideas about sufficient statistics and ancillary statistics. When this partition can be achieved it may be possible to complete a Bayesian analysis for the parameters of interest by determining their joint posterior distribution algebraically. The partition allows frequentist theory to develop general estimation approaches in the presence of nuisance parameters. If the partition cannot be achieved it may still be possible to make use of an approximate partition.

In some special cases, it is possible to formulate methods that circumvent the presences of nuisance parameters. The t-test provides a practically useful test because the test statistic does not depend on the unknown variance but only the sample variance. It is a case where use can be made of a pivotal quantity. However, in other cases no such circumvention is known.

Practical statistics

[edit]

Practical approaches to statistical analysis treat nuisance parameters somewhat differently in frequentist and Bayesian methodologies.

A general approach in a frequentist analysis can be based on maximum likelihood-ratio tests. These provide both significance tests and confidence intervals for the parameters of interest which are approximately valid for moderate to large sample sizes and which take account of the presence of nuisance parameters. See Basu (1977) for some general discussion and Spall and Garner (1990) for some discussion relative to the identification of parameters in linear dynamic (i.e., state space representation) models.

In Bayesian analysis, a generally applicable approach creates random samples from the joint posterior distribution of all the parameters: see Markov chain Monte Carlo. Given these, the joint distribution of only the parameters of interest can be readily found by marginalizing over the nuisance parameters. However, this approach may not always be computationally efficient if some or all of the nuisance parameters can be eliminated on a theoretical basis.

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Nuisance parameter

View on Grokipedia
from Grokipedia
In , a nuisance parameter is a component of a that is not the primary focus of but nonetheless affects the likelihood or distribution of the observed , requiring it to be estimated or integrated out to draw valid conclusions about the parameter of interest. These parameters often arise in models with systematic uncertainties, such as unknown variances in normal distributions or detector errors in experimental , where they represent effects that must be modeled accurately without being the target of the analysis. Nuisance parameters complicate by increasing the variance of estimates for the parameter of interest and potentially biasing results if mishandled, thereby reducing the overall sensitivity and power of tests or intervals. For instance, in high-energy physics experiments like those at the , nuisance parameters might capture uncertainties in jet energy scales or simulation statistics, influencing the evaluation of signal yields for particles like the . Their presence underscores the need for robust methods to separate effects of interest from extraneous influences, a challenge addressed through principles like sufficiency and conditionality in classical statistics. Common approaches to handling nuisance parameters include profiling, where the likelihood is maximized over the nuisance parameters for fixed values of the interest parameter to form profile likelihood ratios or intervals; marginalization, particularly in Bayesian frameworks, which integrates the nuisance parameters out using priors to obtain a marginal posterior; and conditioning, which relies on ancillary or sufficient statistics to eliminate their influence under specific hypotheses. Constraints from subsidiary measurements, such as Gaussian or Poisson terms on background rates, further limit nuisance parameter variability in complex models. These techniques ensure that inferences remain valid across diverse applications, from econometric modeling to discovery claims.

Fundamentals

Definition

In statistics, a nuisance parameter is any parameter in a statistical model that is not the primary focus of inference but must be accounted for to ensure valid estimation or testing of the parameters of interest. These parameters arise in models where the full specification requires additional components beyond those directly relevant to the scientific question at hand, yet ignoring them would lead to incorrect conclusions. The role of a nuisance parameter is to influence the sampling distribution of statistics computed for the parameter of interest, potentially introducing bias or distorting inference if not properly addressed. For instance, in a normal distribution N(μ,σ2)N(\mu, \sigma^2), when testing hypotheses about the mean μ\mu, the variance σ2\sigma^2 serves as a nuisance parameter because it affects the distribution of the sample mean but is not itself the target of the analysis. This highlights how nuisance parameters are conceptually distinct from parameters of interest: they are often unspecified or deemed uninteresting for the current purpose, yet essential for accurate model specification and valid statistical procedures.

Historical Development

The concept of nuisance parameters emerged implicitly in the early through developments in hypothesis testing for composite hypotheses, where parameters of secondary interest complicate inference about primary ones. In their seminal 1933 paper, and addressed the challenge of constructing optimal tests when hypotheses involve multiple unknown parameters, laying the groundwork for recognizing the role of such "nuisance" elements in limiting the power and uniformity of tests. This work highlighted the need to account for extraneous parameters to achieve reliable statistical procedures, though the term itself was not yet in use. Ronald A. Fisher contributed foundational ideas in the and by introducing concepts like sufficiency and ancillarity, which directly relate to isolating parameters of interest from irrelevant ones. Fisher's 1925 paper on statistical estimation formalized sufficiency as a means to summarize data without loss of information, while his later discussions of ancillary statistics—those whose distribution does not depend on unknown parameters—underscored the difficulties posed by parameters in conditioning inference. The term "" was first explicitly coined by in 1940, in the context of selecting predictors while marginalizing over irrelevant variates, emphasizing their interference in predictive modeling. Concurrently, Dennis V. Lindley advanced Bayesian treatments in the , integrating prior distributions to handle parameters through marginalization, as explored in his work on fiducial inference and information measures. A key formalization came with Debabrata Basu's 1955 theorem, which established the independence between a complete sufficient statistic and any ancillary statistic, providing a rigorous tool to separate inferences involving nuisance parameters from those of interest. This result illuminated the structural challenges nuisance parameters introduce in frequentist settings, influencing subsequent theoretical developments. By the mid-20th century, computational limitations prompted a shift from exact methods to approximations for eliminating or adjusting for nuisance parameters, as computational resources restricted full likelihood evaluations in complex models. This evolution is synthesized in David R. Cox and David V. Hinkley's 1974 text Theoretical Statistics, which reviews historical approaches and advocates practical strategies for nuisance parameter adjustment across frequentist and likelihood-based frameworks.

Theoretical Foundations

Frequentist Perspective

In , nuisance parameters are handled by focusing on procedures that achieve desirable long-run frequency properties, such as unbiasedness, consistency, and controlled error rates, without assigning prior probabilities to the s. The parameter space is typically partitioned into the parameter of interest, denoted as ψ, and the nuisance , denoted as λ, with inference conditioned or adjusted to eliminate the influence of λ while preserving the properties for ψ. This approach relies on foundational principles like sufficiency and ancillarity to reduce the data without loss of about ψ. A key tool for partitioning involves sufficient statistics for the full parameter vector (ψ, λ) and that provide information solely about λ. An A is a function of the whose distribution does not depend on (ψ, λ), allowing conditioning on A to tailor to the observed value of λ and yield exact pivotal quantities for ψ. For example, in models where the minimal can be factored into components S (sufficient for ψ given A) and A (), the conditional distribution of S given A eliminates λ, enabling based on the conditional likelihood. This conditioning principle, formalized in works on , ensures that the resulting tests and intervals have exact frequentist coverage regardless of λ. Likelihood ratio tests (LRTs) are central for hypothesis testing in the presence of nuisance parameters, where the test statistic accounts for maximization over λ. For testing H₀: ψ = ψ₀ against H₁: ψ ≠ ψ₀, the LRT statistic is given by Λ=sup⁡λL(ψ0,λ)sup⁡ψ,λL(ψ,λ),\Lambda = \frac{\sup_{\lambda} L(\psi_0, \lambda)}{\sup_{\psi, \lambda} L(\psi, \lambda)}, where L denotes the likelihood function. Under H₀, -2 log Λ follows a known distribution asymptotically or exactly in certain models, providing a uniform p-value calibrated for all values of λ. This maximization over nuisance parameters ensures the test's validity by profiling out λ at both the null and alternative, maintaining the desired size. Confidence intervals for ψ are constructed by inverting such tests using the profile likelihood, which maximizes the likelihood over λ for each fixed ψ. The profile log-likelihood is pl(ψ) = sup_λ log L(ψ, λ), and an approximate (1 - α) confidence interval consists of values ψ where -2 [log L(ψ, \hat{λ}(ψ)) - log L(\hat{ψ}, \hat{λ})] ≤ χ²_{1, 1-α}, with \hat{ψ} and \hat{λ} the maximum likelihood estimates. This method yields intervals with good frequentist coverage properties, particularly in large samples, by effectively concentrating the likelihood on ψ while adjusting for λ. Exact methods exist when pivotal quantities can be derived, often via conditioning or . A classic example is the one-sample t-test for the μ of a N(μ, σ²) with unknown variance σ² as the nuisance parameter. The test statistic t = \bar{X} / (S / \sqrt{n}), where \bar{X} is the sample and S² the sample variance, follows a with n-1 under the null, independent of σ². This arises from the joint sufficiency of (\bar{X}, S²) and the ancillarity of the standardized residuals, allowing for μ without estimating σ² directly. Asymptotically, provides the distribution for the profiled likelihood ratio in the presence of parameters. Under regularity conditions, for testing a about ψ with λ profiled out, -2 log Λ converges in distribution to χ²_d, where d is the difference in the number of free parameters between the full and restricted models (typically 1 for point nulls on ψ). This holds even when λ is high-dimensional, as the profiling maximizes over it, justifying approximate confidence regions and tests for moderate sample sizes. The theorem's applicability underscores the robustness of likelihood-based procedures in frequentist settings with parameters.

Bayesian Perspective

In , nuisance parameters are incorporated into the full parameter space and eliminated through marginalization over their posterior distribution, allowing inferences to focus solely on the parameter of interest without ad hoc adjustments. This approach leverages the joint posterior distribution, which combines the likelihood with a prior over all parameters. Specifically, if θ\theta denotes the parameter of interest and ν\nu the nuisance parameter(s), the joint posterior is given by p(θ,ν∣data)∝L(data∣θ,ν) π(θ,ν),p(\theta, \nu \mid \text{data}) \propto L(\text{data} \mid \theta, \nu) \, \pi(\theta, \nu), where LL is the and π\pi is the joint prior distribution. This formulation naturally accounts for uncertainty in ν\nu by treating it probabilistically rather than fixing it via estimation. To obtain inferences about θ\theta alone, the marginal posterior is computed as p(θ∣data)=∫p(θ,ν∣data) dν,p(\theta \mid \text{data}) = \int p(\theta, \nu \mid \text{data}) \, d\nu, which integrates out the nuisance parameters and yields a distribution free of their direct influence. This marginalization process is a core strength of the Bayesian paradigm, as it fully propagates the uncertainty from ν\nu into the of θ\theta, often requiring numerical methods like for high-dimensional cases. A key challenge in this framework lies in specifying the prior π(ν)\pi(\nu) for the nuisance parameters, as subjective or ill-chosen priors can distort the marginal posterior for θ\theta, introducing unintended bias. To address this, non-informative priors—such as those proposed by Jeffreys in the 1960s for multiparameter models—are frequently employed; these priors, derived from the matrix, aim to be minimally informative and invariant under reparameterization, thereby avoiding undue influence on θ\theta. Reference priors, an extension of Jeffreys' approach, further refine this by sequentially prioritizing non-informativity for θ\theta before ν\nu, ensuring the marginal posterior approximates objective inference when prior knowledge is lacking. Under certain independence assumptions, Bayesian analysis can exploit ancillary statistics—quantities whose distribution does not depend on θ\theta—to condition and simplify the posterior. By conditioning on such statistics, the joint posterior factors into components that isolate the influence of ν\nu, facilitating easier marginalization or exact computation in models where full integration is intractable. This conditioning preserves the posterior for θ\theta while leveraging model structure to reduce computational demands, though it requires verifying ancillarity in the Bayesian sense, where the of the ancillary remains independent of θ\theta under the chosen prior.

Handling Methods

Profiling and Conditioning

In frequentist statistics, the profile likelihood method addresses nuisance parameters by concentrating the full likelihood onto the parameter of interest. The profile likelihood is constructed as Lp(θ)=sup⁡νL(θ,ν)L_p(\theta) = \sup_{\nu} L(\theta, \nu), where L(θ,ν)L(\theta, \nu) denotes the joint likelihood function, θ\theta represents the parameter of interest, and ν\nu encompasses the nuisance parameters. This maximization over ν\nu for each fixed θ\theta effectively eliminates the nuisance parameters from the inference procedure while preserving the structure of the original likelihood. To obtain the profiled maximum likelihood estimate θ^\hat{\theta}, one maximizes Lp(θ)L_p(\theta) with respect to θ\theta, often by jointly solving the score equations or using numerical optimization to find ν^(θ)=arg⁡max⁡νL(θ,ν)\hat{\nu}(\theta) = \arg\max_{\nu} L(\theta, \nu) across a range of θ\theta values and substituting back into the likelihood. For inference, the profile likelihood ratio statistic −2log⁡(Lp(θ)Lp(θ^))-2 \log \left( \frac{L_p(\theta)}{L_p(\hat{\theta})} \right) is used, which asymptotically follows a χ2\chi^2 distribution with degrees of freedom equal to the dimension of θ\theta under the null hypothesis θ=θ0\theta = \theta_0. In the scalar case, this reduces to a χ2(1)\chi^2(1) distribution, enabling the construction of profile likelihood confidence intervals by inverting the test. Conditioning on ancillary statistics provides an alternative frequentist approach to removing the effects of . An has a sampling distribution independent of all parameters, and establishes that a boundedly complete TT for θ\theta is independent of any AA. This independence implies that the conditional distribution of TT given AA depends only on θ\theta and not on ν\nu, allowing conditional —such as tests or intervals—free from nuisance parameter influence. These methods offer key advantages in retaining the full of the data for on θ\theta, with no loss due to approximation under regularity conditions; in models, where complete sufficient statistics exist, profiling or conditioning can yield exact results without relying on asymptotic approximations. However, the profile likelihood ratio statistic can suffer from in small samples, potentially leading to intervals with coverage probabilities deviating from the nominal level; adjustments to the profile likelihood, such as those incorporating higher-order terms, mitigate this O(1/n)O(1/n) .

Marginalization Techniques

Marginalization techniques address nuisance parameters by integrating them out of the joint distribution, thereby obtaining a focused solely on the parameters of interest. In the Bayesian framework, this approach yields the for the parameter of interest θ\theta, defined as m(θ)=∫L(data∣θ,ν)π(ν∣θ) dνm(\theta) = \int L(\text{data} \mid \theta, \nu) \pi(\nu \mid \theta) \, d\nu, where LL is the , ν\nu denotes the nuisance parameters, and π(ν∣θ)\pi(\nu \mid \theta) is the conditional prior on ν\nu. This facilitates posterior on θ\theta by propagating the full from ν\nu, as the posterior for θ\theta is proportional to m(θ)π(θ)m(\theta) \pi(\theta). Exact marginalization is feasible in cases with conjugate priors, where the prior structure aligns with the likelihood to produce a closed-form . A example arises in the normal model with unknown μ\mu (parameter of interest) and variance σ2\sigma^2 (nuisance parameter), using the . Here, the marginal posterior for μ\mu is a , explicitly integrating out σ2\sigma^2 and capturing the uncertainty in both parameters without approximation. When exact integration is intractable, approximation methods such as the Laplace approximation provide a practical alternative for estimating the . This method approximates the ∫exp⁡(l(θ,ν)) dν≈exp⁡(l(θ,ν^))(2π)k/2∣I(ν^)∣−1/2\int \exp(l(\theta, \nu)) \, d\nu \approx \exp(l(\theta, \hat{\nu})) (2\pi)^{k/2} |\mathbf{I}(\hat{\nu})|^{-1/2}, where l(θ,ν)l(\theta, \nu) is the log-posterior, ν^\hat{\nu} maximizes ll for fixed θ\theta, kk is the dimension of ν\nu, and I(ν^)\mathbf{I}(\hat{\nu}) is the matrix (negative Hessian) at ν^\hat{\nu}. The approximation relies on a quadratic expansion around the mode, yielding Gaussian-like behavior for large samples or high . In frequentist contexts, marginalization analogs appear through integrated likelihood methods, where the nuisance parameters are eliminated via integration over a chosen measure, often to construct robust estimators or test statistics that account for uncertainty without maximization. These approaches, such as those using fractional likelihoods, parallel Bayesian marginalization by averaging over ν\nu rather than fixing it at a point estimate. Marginalization is particularly advantageous when prior distributions on nuisance parameters credibly reflect their , as it avoids the overconfidence that can arise from profiling methods, which maximize over ν\nu.

Alternative Approaches

In , concentrated likelihood methods address nuisance parameters by maximizing the joint likelihood over the nuisance parameters for fixed values of the parameter of interest, thereby obtaining a profile likelihood function in terms of the parameter of interest only. This approach is particularly effective in linear models, where nuisance parameters such as variance components can be "concentrated out" by maximizing the likelihood with respect to them, yielding a reduced-dimensional profile that focuses solely on the parameters of primary . For instance, in ordinary regression, the concentrated likelihood eliminates the need to handle intercept or scale parameters separately, leading to asymptotically equivalent for the coefficients. Score tests provide an alternative for hypothesis testing in the presence of nuisance parameters by evaluating the score function—the gradient of the log-likelihood—under the without requiring full estimation of the nuisance parameters. These tests partition the observed information matrix to account for the nuisance components, ensuring the test statistic remains valid even when the nuisance parameters are unidentified under the null, as in boundary problems or composite hypotheses. Developed as part of the Rao score test framework, this method is computationally efficient for large models, such as in generalized linear models, where it avoids iterative maximization by substituting consistent estimates only for the information adjustment. Empirical Bayes approaches treat nuisance parameters as hyperparameters drawn from a prior distribution estimated directly from the data, bridging frequentist and Bayesian paradigms to enhance efficiency before applying marginalization or other adjustments. In settings with many nuisance parameters, such as hierarchical models, this involves maximizing a over the hyperparameters to obtain shrinkage estimates, which then inform the posterior for the parameters of interest. This method, popularized in the context of exponential families, reduces bias from misspecified priors and improves conditional , as demonstrated in applications like empirical partially Bayes for increasing the number of nuisance parameters with sample size. Robust methods for handling parameters in semiparametric models adjust influence functions to mitigate from outliers or model misspecification affecting the nuisance components, ensuring stable of the parameters of interest. Bounded influence functions, optimized for , bound the contribution of individual observations while preserving semiparametric , as in partial linear models where the nonparametric nuisance is protected via Hampel-type neighborhoods. These techniques, including one-step robust estimators, are crucial in semi-parametric exponential mixtures, where they yield explicit optimal influence functions that balance robustness and asymptotic variance. Historically, early approaches to nuisance parameters often relied on ad-hoc adjustments, such as ignoring them in large-sample approximations under the assumption of consistency, a practice rooted in pre-1940s but now considered outdated due to its failure to maintain exact coverage or power in finite samples. These methods, exemplified in the Fieller-Creasy problem of ratio estimation, preceded systematic techniques like conditioning and were critiqued for inducing in small samples, paving the way for modern likelihood-based solutions.

Applications and Examples

In Parametric Inference

In parametric , nuisance parameters often arise when estimating or testing a parameter of interest within a specified distributional family, requiring adjustments to ensure valid . A classic example is the normal distribution, where observations X1,…,Xn\iidN(μ,σ2)X_1, \dots, X_n \iid \mathcal{N}(\mu, \sigma^2) and the goal is to test H0:μ=μ0H_0: \mu = \mu_0 with σ2\sigma^2 unknown and treated as a nuisance parameter. The likelihood function is L(μ,σ2)=(2πσ2)−n/2exp⁡(−12σ2∑(Xi−μ)2)L(\mu, \sigma^2) = (2\pi \sigma^2)^{-n/2} \exp\left( -\frac{1}{2\sigma^2} \sum (X_i - \mu)^2 \right). To eliminate the nuisance σ2\sigma^2, maximize over it for fixed μ\mu, yielding the profile likelihood proportional to (∑(Xi−μ)2)−n/2\left( \sum (X_i - \mu)^2 \right)^{-n/2}. Under H0H_0, this leads to the pivotal quantity t=n(Xˉ−μ0)/St = \sqrt{n} (\bar{X} - \mu_0) / S
Add your contribution
Related Hubs
User Avatar
No comments yet.