arXiv subjects

Chris A. J. Klaassen

Publications and source records attributed to Chris A. J. Klaassen.

At least 19 recordsLinked to original sources

arXiv2023

Semiparametric Statistics and the Spread Inequality

Classical mathematical statistics deals with models that are parametrized by a Euclidean, i.e. finite dimensional, parameter. Quite often such models have been and still are chosen in practical situations for their mathematical simplicity and tractability. However, these models are typically inappropriate since the implied distributional assumptions cannot be supported by hard evidence. It is natural then to relax these assumptions. This leads to the class of semiparametric models. These models have been studied in a local asymptotic setting, in which the Convolution Theorem yields bounds on the performance of regular estimators. Alternatively, local asymptotics can be based on the Local Asymptotic Minimax Theorem and on the Local Asymptotic Spread Theorem, both valid for any sequence of estimators. This Local Asymptotic Spread Theorem is a straightforward consequence of a Finite Sample Spread Inequality, which has some intrinsic value for estimation theory in general. We will discuss both the Finite Sample and Local Asymptotic Spread Theorem, as well as the Convolution Theorem.

math.ST↗

arXiv2023

Inference via the Skewness-Kurtosis Set

Kurtosis minus squared skewness is bounded from below by 1, but for unimodal distributions this parameter is bounded by 189/125. In some applications it is natural to compare distributions by comparing their kurtosis-minus-squared-skewness parameters. The asymptotic behavior of the empirical version of this parameter is studied here for i.i.d. random variables. The result may be used to test the hypothesis of unimodality against the alternative that the kurtosis-minus-squared-skewness parameter is less than 189/125. However, such a test has to be applied with care, since this parameter can take arbitrarily large values, also for multimodal distributions. Numerical results are presented and for three classes of distributions the skewness-kurtosis sets are described in detail.

math.ST↗

arXiv2023

A Markovian Gauß\, Inequality for Asymmetric Deviations from the Mode of Symmetric Unimodal Distributions

For a random variable with a unimodal distribution and finite second moment Gauß\, (1823) proved a sharp bound on the probability of the random variable to be outside a symmetric interval around its mode. An alternative proof for it is given based on Khintchine's representation of unimodal random variables. Analogously, a sharp inequality is proved for the slightly broader class of unimodal distributions with finite first absolute moment, which might be called a Markovian Gauß inequality. For symmetric unimodal distributions with finite second moment Semenikhin (2019) generalized Gauß's inequality to arbitrary intervals. For the class of symmetric unimodal distributions with finite first absolute moment we construct a Markovian version of it. Related inequalities of Volkov (1969) and Sellke and Sellke (1997) will be discussed as well.

math.PR↗

arXiv2023

Sharp unifying generalizations of Opial's inequality

Opial's inequality and its ramifications play an important role in the theory of differential and difference equations. A sharp unifying generalization of Opial's inequality is presented that contains both its continuous and discrete version. This generalization based on distribution functions is extended to the case of derivatives of arbitrary order. This extension optimizes and improves the constant as given in the literature. The special case of derivatives of second order is studied in more detail. Two closely related Opial inequalities with a weight function are presented as well. The associated Wirtinger inequality is studied briefly.

math.CA↗

arXiv2022

Sharp Inequalities of Bienaymé-Chebyshev and GaußType for Possibly Asymmetric Intervals around the Mean

Gauß(1823) proved a sharp upper bound on the probability that a random variable falls outside a symmetric interval around zero when its distribution is unimodal with mode at zero. For the class of all distributions with mean at zero, Bienaymé (1853) and Chebyshev (1867) independently provided another, simpler sharp upper bound on this probability. For the same class of distributions, Cantelli (1928) obtained a strict upper bound for intervals that are a half line. We extend these results to arbitrary intervals for six classes of distributions, namely the general class of `distributions', the class of `symmetric distributions', of `concave distributions', of `unimodal distributions', of `unimodal distributions with coinciding mode and mean', and of `symmetric unimodal distributions'. For some of the known inequalities, such as the Gauß\, inequality, an alternative proof is given.

math.PR↗

arXiv2022

Probability Theoretic Generalizations of Hardy's and Copson's Inequality

A short proof of the classic Hardy inequality is presented for $p$-norms with $p>1$. Along the lines of this proof a sharpened version is proved of a recent generalization of Hardy's inequality in the terminology of probability theory. A probability theoretic version of Copson's inequality is discussed as well. Also for $0<p<1$ probability theoretic generalizations of the Hardy and the Copson inequality are proved.

math.PR↗

arXiv2021

Hardy's Inequality and Its Descendants

We formulate and prove a generalization of Hardy's inequality (Hardy,1925) in terms of random variables and show that it contains the usual (or familiar) continuous and discrete forms of Hardy's inequality. Next we improve the recent version by Li and Mao of Hardy's inequality with weights for general Borel measures and mixed norms so that it implies the discrete version of Liao and the Hardy inequality with weights of Muckenhoupt as well as the mixed norm versions due to Hardy and Littlewood, Bliss, and Bradley. An equivalent formulation in terms of random variables is given as well. We also formulate a reverse version of Hardy's inequality, the closely related Copson inequality, a reverse Copson inequality and a Carleman-Pólya-Knopp inequality via random variables. Finally we connect our Copson inequality with counting process martingales and survival analysis, and briefly discuss other applications.

math.PR↗

arXiv2016

Semiparametrically Efficient Estimation of Euclidean Parameters under Equality Constraints

Assume a (semi)parametrically efficient estimator is given of the Euclidean parameter in a (semi)parametric model. A submodel is obtained by constraining this model in that a continuously differentiable function of the Euclidean parameter vanishes. We present an explicit method to construct (semi)parametrically efficient estimators of the Euclidean parameter in such equality constrained submodels and prove their efficiency. Our construction is based solely on the original efficient estimator and the constraining function. Only the parametric case of this estimation problem and a nonparametric version of it have been considered in literature.

math.ST↗

arXiv2015

Semiparametrically Efficient Estimation of Constrained Euclidean Parameters

Consider a quite arbitrary (semi)parametric model with a Euclidean parameter of interest and assume that an asymptotically (semi)parametrically efficient estimator of it is given. If the parameter of interest is known to lie on a general surface (image of a continuously differentiable vector valued function), we have a submodel in which this constrained Euclidean parameter may be rewritten in terms of a lower-dimensional Euclidean parameter of interest. An estimator of this underlying parameter is constructed based on the original estimator, and it is shown to be (semi)parametrically efficient. It is proved that the efficient score function for the underlying parameter is determined by the efficient score function for the original parameter and the Jacobian of the function defining the general surface, via a chain rule for score functions. Efficient estimation of the constrained Euclidean parameter itself is considered as well. Our general estimation method is applied to location-scale, Gaussian copula and semiparametric regression models, and to parametric models under linear restrictions.

math.ST↗

arXiv2015

Optimal Rate of Direct Estimators in Systems of Ordinary Differential Equations Linear in Functions of the Parameters

Many processes in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their 'true' value is thus required. In this paper we focus on the quite common systems for which the derivatives of the states may be written as sums of products of a function of the states and a function of the parameters. For such a system linear in functions of the unknown parameters we present a necessary and sufficient condition for identifiability of the parameters. We develop an estimation approach that bypasses the heavy computational burden of numerical integration and avoids the estimation of system states derivatives, drawbacks from which many classic estimation methods suffer. We also suggest an experimental design for which smoothing can be circumvented. The optimal rate of the proposed estimators, i.e., their $\sqrt n$-consistency, is proved and simulation results illustrate their excellent finite sample performance and compare it to other estimation approaches.

math.ST↗

arXiv2015

Evidential Value in ANOVA-Regression Results in Scientific Integrity Studies

Some scientific publications are under suspicion of fabrication of data. Since humans are bad random number generators, there might be some evidential value in favor of fabrication in the statistical results as presented in such papers. In case of ANOVA-Regression studies we present the evidential value of the results of such a study in favor of the hypothesis of a dependence structure in the underlying data, which indicates fabrication, versus the hypothesis of independence, which is the ANOVA model assumption. Applications of this approach are also presented.

stat.ME↗

arXiv2013

Information weighted sampling for detecting rare items in finite populations with a focus on security

Frequently one has to search within a finite population for a single particular individual or item with a rare characteristic. Whether an item possesses the characteristic can only be determined by close inspection. The availability of additional information about the items in the population opens the way to a more effective search strategy than just random sampling or complete inspection of the population. We will assume that the available information allows for the assignment to all items within the population of a prior probability on whether or not it possesses the rare characteristic. This is consistent with the practice of using profiling to select high risk items for inspection. The objective is to find the specific item with the minimum number of inspections. We will determine the optimal search strategies for several models according to the average number of inspections needed to find the specific item. Using these respective optimal strategies we show that we can order the numbers of inspections needed for the different models partially with respect to the usual stochastic ordering. This entails also a partial ordering of the averages of the number of inspections. Finally, the use, some discussion, extensions, and examples of these results, and conclusions about them are presented.

math.PR↗

arXiv2013

Evidential Value in ANOVA Results in Favor of Fabrication

Some scientific publications are under suspicion of fabrication of data. Since humans are bad random number generators, there might be some evidential value in favor of fabrication in the statistical results as presented in such papers. In line with Uri Simonsohn (2012, 2013) we study the evidential value of the results of an ANOVA study in favor of the hypothesis of a dependence structure in the underlying data.

stat.ME↗

arXiv2012

$\sqrt{n}$-consistent parameter estimation for systems of ordinary differential equations: bypassing numerical integration via smoothing

We consider the problem of parameter estimation for a system of ordinary differential equations from noisy observations on a solution of the system. In case the system is nonlinear, as it typically is in practical applications, an analytic solution to it usually does not exist. Consequently, straightforward estimation methods like the ordinary least squares method depend on repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimises the objective function. This induces a huge computational load to such estimation methods. We study the consistency of an alternative estimator that is defined as a minimiser of an appropriate distance between a nonparametrically estimated derivative of the solution and the right-hand side of the system applied to a nonparametrically estimated solution. This smooth and match estimator (SME) bypasses numerical integration altogether and reduces the amount of computational time drastically compared to ordinary least squares. Moreover, we show that under suitable regularity conditions this smooth and match estimation procedure leads to a $\sqrt{n}$-consistent estimator of the parameter of interest.

math.ST↗

arXiv2008

Why stratification may hurt, & how much

There are circumstances under which stratified sampling is worse than simple random sampling, even if the allocation of the sample sizes is optimal. This phenomenon was discovered more than sixty years ago, but is not as widely known as one might expect. We provide it with lower and upper bounds for its badness as well as with an explanation.

stat.ME↗

arXiv2006

A Comparison of Information Concerning the Regression Parameter in The Accelerated Failure Time Model under Current Duration and Length Biased Sampling: Does it Pay to be Patient?

Longitudinal observations are sometimes costly or not available. Cross sectional sampling can be an alternative. Observations are drawn then at a specific point in time from a population of durations whose distributions satisfy a {\em core model}. Subsequently, one has a choice. One may process the data immediately, obtaining so called current duration data. Or one waits until the sampled durations are known completely obtaining the full durations via length biased sampling. We compare the Fisher information for the Euclidean parameter corresponding to an Accelerated Failure Time core model when the observations are obtained by either current duration or length biased sampling.

math.ST↗

arXiv2005

Efficient estimation of Banach parameters in semiparametric models

Consider a semiparametric model with a Euclidean parameter and an infinite-dimensional parameter, to be called a Banach parameter. Assume: (a) There exists an efficient estimator of the Euclidean parameter. (b) When the value of the Euclidean parameter is known, there exists an estimator of the Banach parameter, which depends on this value and is efficient within this restricted model. Substituting the efficient estimator of the Euclidean parameter for the value of this parameter in the estimator of the Banach parameter, one obtains an efficient estimator of the Banach parameter for the full semiparametric model with the Euclidean parameter unknown. This hereditary property of efficiency completes estimation in semiparametric models in which the Euclidean parameter has been estimated efficiently. Typically, estimation of both the Euclidean and the Banach parameter is necessary in order to describe the random phenomenon under study to a sufficient extent. Since efficient estimators are asymptotically linear, the above substitution method is a particular case of substituting asymptotically linear estimators of a Euclidean parameter into estimators that are asymptotically linear themselves and that depend on this Euclidean parameter. This more general substitution case is studied for its own sake as well, and a hereditary property for asymptotic linearity is proved.

math.ST↗

arXiv2003

Efficient estimation in the accelerated failure time model under cross sectional sampling

Consider estimation of the regression parameter in the accelerated failure time model, when data are obtained by cross sectional sampling. It is shown that it is possible under regularity of the model to construct an efficient estimator of the unknown Euclidean regression parameter if the distribution of the covariate vector is known and also if it is unknown with vanishing mean.

math.ST↗