Asymptotic Loss Research Articles

Asymptotic loss of the MLE of a truncation parameter in the presence of a nuisance parameter for a one-sided truncated family of distributions

For a truncated family of distributions with a truncation parameter γ and a parameter θ as a nuisance parameter, we derive the stochastic expansions of bias-adjusted maximum likelihood estimators γ ̂ M L ∗ θ and γ ̂ M L ∗ of γ based on a sample of size n when θ is known and when θ is unknown, respectively. The asymptotic loss of γ ̂ M L ∗ relative to γ ̂ M L ∗ θ is obtained up to the second order, that is the order n −1. The results are a generalization of those for a one-sided truncated exponential family of distributions. Its application to truncated t-distributions is also given.

How bad is the merger paradox?

The merger paradox is a classic, counter-intuitive result from the literature of Industrial Organization saying that merging firms typically experience a decline in their overall profit compared to their total pre-merger profit. This phenomenon is more striking in small oligopolistic markets, where mergers increase market concentration and may hence trigger a substantial increase in prices. In this paper, we investigate the severity of the merger paradox in Cournot oligopoly markets. Namely, we study the worst-case magnitude of this profit loss in quantity-setting market games. We consider convex, asymmetric production costs for the firms, and we show that the profit loss can be substantial even in small markets. That is, two merging firms can lose half of their pre-merger profit, but no more than half in markets with concave demand functions. On the positive side, we show that in markets with affine demand two firms can never lose more than 1/9 of their profit when merging, and this bound is tight. We also study the asymptotic loss in larger markets, where it is easy to show that the profit loss can be arbitrarily large when multiple firms merge; we give bounds that characterize the profit loss from a merger as a function of the market size and the number of merging firms.

A new type of generalized information criterion for regularization parameter selection in penalized regression with application to treatment process data

ABSTRACT We propose a new approach to select the regularization parameter using a new version of the generalized information criterion ( GIC ) in the subject of penalized regression. We prove the identifiability of bridge regression model as a prerequisite of statistical modeling. Then, we propose asymptotically efficient generalized information criterion ( AGIC ) and prove that it has asymptotic loss efficiency. Also, we verified the better performance of AGIC in comparison to the older versions of GIC . Furthermore, we propose MSE search paths to order the selected features by lasso regression based on numerical studies. The MSE search paths provide a way to cover the lack of feature ordering in lasso regression model. The performance of AGIC with other types of GIC is compared using MSE and model utility in simulation study. We exert AGIC and other criteria to analyze breast and prostate cancer and Parkinson disease datasets. The results confirm the superiority of AGIC in almost all situations.

Information and Energy Transmission With Wavelet-Reconstructed Harvesting Functions

In practical simultaneous information and energy transmission (SIET), the exact energy harvesting function is usually unavailable because an energy harvesting circuit is nonlinear and nonideal. In this work, we consider a SIET problem where the harvesting function is accessible only at experimentally-taken sample points and study how close we can design SIET to the optimal system with such sampled knowledge. Assuming that the harvesting function is of bounded variation that may have discontinuities, we separately consider two settings where samples are taken without and with additive noise. For these settings, we propose to design a SIET system as if a wavelet-reconstructed harvesting function is the true one and study its asymptotic performance loss of energy and information delivery from the true optimal one. Specifically, for noiseless samples, it is shown that designing SIET as if the wavelet-reconstructed harvesting function is the truth incurs asymptotically vanishing energy and information delivery loss with the number of samples. For noisy samples, we propose to reconstruct wavelet coefficients via soft-thresholding estimation. Then, we not only obtain similar asymptotic losses to the noiseless case but also show that the energy loss by wavelets is asymptotically optimal up to a logarithmic factor.

Selection of linear mixed‐effects models for clustered data

AbstractWe consider model selection for linear mixed‐effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, , of clusters does not go to infinity with the sample size. Hence the random‐effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.

Maximum likelihood estimation for a one-sided truncated family of distributions

For a one-sided truncated family of distributions with an interest parameter $$\theta $$ and a truncation parameter $$\gamma $$ as a nuisance parameter, we consider the maximum likelihood estimators (MLEs) $${\hat{\theta }}_\mathrm{ML}^{\gamma} $$ and $${\hat{\theta }}_{\mathrm{ML}}$$ of $$\theta $$ for known $$\gamma $$ and unknown $$\gamma $$ , respectively. In this paper, the stochastic expansions of $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ and $${\hat{\theta }}_\mathrm{ML}$$ are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted $${\hat{\theta }}_{\mathrm{ML}^*}$$ relative to $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ is also given. The results are a generalization of those for a one-sided truncated exponential family of distributions. Examples on a one-sided truncated Cauchy distribution, a general truncated exponential family, etc. are also given.

Asymptotic Outage Analysis on Dual-Branch Diversity Receptions Over Non-Identically Distributed Correlated Lognormal Channels

It is challenging to quantify the left tail behavior of the cumulative distribution function of a sum of lognormal random variables, especially when they are correlated and non-identically distributed. The mathematical intractability hampers the analysis on diversity receptions over lognormal fading channels. In this paper, a novel asymptotic technique is exploited to study non-identically distributed dual-branch lognormal fading channels with correlation. Closed-form asymptotic outage probability expressions are derived for maximum-ratio combining (MRC), equal-gain combining (EGC), and selection combining (SC). Three counterintuitive facts are revealed: 1) one of the two channels contributes no asymptotic performance gain to the diversity reception systems under certain conditions, implying that a link can be discarded without causing asymptotic performance loss; 2) SC outperforms EGC under certain conditions; and 3) the outage probability of SC is arbitrarily close to that of MRC under certain conditions. The new result reveals insights into the long-standing problem of asymptotic analysis for correlated lognormal fading channels, paving the way for analysis on more general channel models. This paper also provides an efficient approach to evaluate the non-identically distributed lognormal fading channels at a high signal-to-noise ratio. Practically, the analytical results prevent some redundant units that contribute no performance gain.

The first passage problem for stable linear delay equations perturbed by power law Lévy noise.

This article studies a linear scalar delay differential equation subject to small multiplicative power tail Lévy noise. We solve the first passage (the Kramers) problem with probabilistic methods and discover an asymptotic loss of memory in this non-Markovian system. Furthermore, the mean exit time increases with the power of the small noise amplitude, whereas the prefactor accounts for memory effects. In particular, we discover a nonlinear delay-induced exit acceleration due to a non-normal growth phenomenon. Our results are illustrated by the example of a linear delay oscillator driven by α-stable Lévy flights.

Estimation of two high-dimensional covariance matrices and the spectrum of their ratio

Let Sp,1, Sp,2 be two independent p×p sample covariance matrices with degrees of freedom n1 and n2, respectively, whose corresponding population covariance matrices are Σp,1 and Σp,2, respectively. Knowing Sp,1, Sp,2, this article proposes a class of estimators for the spectrum (eigenvalues) of the matrix Σp,2Σp,1−1 as well as the pair of the whole matrices (Σp,1,Σp,2). The estimators are created based on Random Matrix Theory. Under mild conditions, our estimator for the spectrum of Σp,2Σp,1−1 is shown to be weakly consistent and the estimator for (Σp,1,Σp,2) is shown to be optimal in the sense of minimizing the asymptotic loss within the class of equivariant estimators as n1,n2,p→∞ with p∕n1→c1∈(0,1), p∕n2→c2∈(0,1)∪(1,∞). Also, our estimators are easy to implement. Even when p is 1000, our estimators can be computed in seconds using a personallaptop.

Direct simulation of thermo‐coupled fatigue failure for metals

AbstractA new finite strain elastoplastic flow model with thermo‐coupled effects is proposed for the purpose of simulating cyclic and non‐cyclic thermo‐coupled fatigue failure behavior of metals. Novelties in four respects are incorporated in the proposed model: (i) the usual notion of yielding is rendered irrelevant with a smooth transition from the elastic to the plastic state; (ii) the thermodynamic consistency stipulated by the second law is identically ensured; (iii) asymptotic loss of the strength in cyclic and non‐cyclic thermomechanical processes is incorporated as inherent constitutive features; and, accordingly, (iv) the fatigue failure behavior may be derived as direct consequences of the proposed model, without involving any additional failure criteria and any additional variables. Thermo‐coupled responses of a metal bar with two fixed ends under cyclic heating and cooling over various temperature ranges are studied based on a simple case of the proposed model. Results are obtained for the maximum tensile and compressive stresses changing with the cycle number and, in particular, for the thermal fatigue life versus the temperature amplitude. Model predictions compare well with test data.

A direct approach toward simulating cyclic and non-cyclic fatigue failure of metals

A new finite \(J_2-\)flow rate-dependent elastoplastic model is proposed toward directly simulating cyclic and non-cyclic fatigue failure behavior of metals. Novel results in four respects are presented: (1) the concept of yielding is rendered irrelevant with smooth transitions to both the plastic state and the rate-independent case; (2) asymptotic loss of the strength is incorporated as inherent constitutive feature; (3) the failure behavior may accordingly be derived as a direct consequence of the proposed model, without involving any additional failure criteria and any additional variables; and (4) direct and explicit procedures are established for identifying each rate-dependent parameter based on suitable test data. Numerical examples are provided for various uniaxial loading cases up to failure, including monotone and cyclic strain cases from low to high strain rates and cyclic loading cases as well as non-cyclic loading cases with either variable stress amplitudes or variable strain rates. Model predictions compare well with test data.

Optimal Shrinkage of Singular Values

We consider the recovery of low-rank matrices from noisy data by shrinkage of singular values, in which a single, univariate nonlinearity is applied to each of the empirical singular values. We adopt an asymptotic framework, in which the matrix size is much larger than the rank of the signal matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. For a variety of loss functions, including Mean Square Error (MSE) - (square Frobenius norm), the nuclear norm loss and the operator norm loss, we show that in this framework there is a well-defined asymptotic loss that we evaluate precisely in each case. In fact, each of the loss functions we study admits a unique admissible shrinkage nonlinearity dominating all other nonlinearities. We provide a general method for evaluating these optimal nonlinearities, and demonstrate our framework by working out simple, explicit formulas for the optimal nonlinearities in the Frobenius, nuclear and operator norm cases. For example, for a square low-rank n-by-n matrix observed in white noise with level σ, the optimal nonlinearity for MSE loss simply shrinks each data singular value y to √(y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -4nσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) (or to 0 if y <; 2√nσ). This optimal value y to nonlinearity guarantees an asymptotic MSE of 2nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , which compares favorably with optimally tuned hard thresholding and optimally tuned soft thresholding, providing guarantees of 3nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and 6nrσ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , respectively. Our general method also allows one to evaluate optimal shrinkers numerically to arbitrary precision. As an example, we compute optimal shrinkers for the Schatten-p norm loss, for any p > 0.

Local checkability, no strings attached: (A)cyclicity, reachability, loop free updates in SDNs

In this work we study local checkability of network properties like s–t reachability, or whether the network is acyclic or contains a cycle. A structural property S of a graph G is locally checkable, if there is a prover-and-verifier pair (P,V) as follows. The proverP assigns a label to each node in graphs satisfying S. The verifierV is a constant time distributed algorithm that returns Yes at all nodes if G satisfies S and was labeled by P, and No for at least one node if G does not satisfy S, regardless of the node labels. The quality of (P,V) is measured in terms of the label size.Our model has no strings attached, i.e., we do not assume any identifiers or port numbers: All we allow is a single exchange of labels between neighbors.We obtain (asymptotically) tight bounds for the bit complexity of the latter two problems for undirected as well as directed networks, where in the directed case we consider one-way and two-way communication, i.e., we distinguish whether communication is possible only in the edge direction or not. For the one-way case we obtain a new asymptotically tight lower bound for the bit complexity of s–t reachability, which also extends to distributed algorithms with constant time execution. For the two-way case we devise an emulation technique that allows us to transfer a previously known s–t reachability upper bound without asymptotic loss in the bit complexity.Lastly, we also study how to apply the concept of local checkability to updating spanning trees in a loop free manner in the context of asynchronous networking, by exploring the similarities between prover-and-verifier pairs and Software Defined Networks (SDNs).

Second-order asymptotic comparison of the MLE and MCLE for a two-sided truncated exponential family of distributions

ABSTRACTFor a one-sided truncated exponential family of distributions with a natural parameter θ and a truncation parameter γ as a nuisance parameter, it is shown by Akahira (2013) that the second-order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE) of θ for unknown γ relative to the MLE of θ for known γ is given and and the maximum conditional likelihood estimator (MCLE) are second-order asymptotically equivalent. In this paper, in a similar way to Akahira (2013), for a two-sided truncated exponential family of distributions with a natural parameter θ and two truncation parameters γ and ν as nuisance ones, the stochastic expansions of the MLE of θ for known γ and ν and the MLE and the MCLE of θ for unknown γ and ν are derived, their second-order asymptotic means and variances are given, a bias-adjusted MLE and are shown to be second-order asymptotically equivalent, and the second-order asymptotic losses of and relative to are also obtained. Further, some examples including an upper-truncated Pareto case are given.

Second-order asymptotic loss of the MLE of a truncation parameter for a truncated exponential family of distributions

ABSTRACTFor a truncated exponential family of distributions with a truncation parameter γ and a natural parameter θ as a nuisance parameter, the stochastic expansions of bias-adjusted maximum likelihood estimators and of γ when θ is known and when θ is unknown, respectively, are derived. The second-order asymptotic loss of relative to is also obtained through their asymptotic variances. Further, some examples are given.

Second Order Asymptotic Loss of the MLE of a Truncation Parameter for a Two-Sided Truncated Exponential Family of Distributions

For a one-sided truncated exponential family of distributions with a truncation parameter and a natural parameter as a nuisance parameter, it is shown by Akahira and Ohyauchi (2016) that the second order asymptotic loss of a bias-adjusted maximum likelihood estimator (MLE) of a truncation parameter for unknown natural parameter relative to a bias-adjusted MLE of a truncation parameter for known natural parameter is obtained. In this paper, in a similar way to Akahira and Ohyauchi (2016), for a two-sided truncated exponential family of distributions with a natural parameter and lower and upper truncation parameters, the stochastic expansions of the bias-adjusted MLE of an upper truncation parameter for known natural and lower truncation parameters, the bias-adjusted MLE of an upper truncation parameter for unknown natural parameter and known lower truncation parameter and the bias-adjusted MLE of an upper truncation parameter for unknown natural and lower truncation parameters are derived, their asymptotic variances are given, and the second order asymptotic losses of the MLEs of an upper truncation parameter for unknown natural parameter and known/unknown lower truncation parameter relative to the MLE of an upper truncation parameter for known natural and lower truncation parameters are also obtained. Further, some examples including an upper-truncated Pareto case are given.

Second-order asymptotic comparison of the MLE and MCLE of a natural parameter for a truncated exponential family of distributions

For a truncated exponential family of distributions with a natural parameter \(\theta \) and a truncation parameter \(\gamma \) as a nuisance parameter, it is known that the maximum likelihood estimators (MLEs) \(\hat{\theta }_{\mathrm{ML}}^{\gamma }\) and \(\hat{\theta }_{\mathrm{ML}}\) of \(\theta \) for known \(\gamma \) and unknown \(\gamma \), respectively, and the maximum conditional likelihood estimator \(\hat{\theta }_{\mathrm{MCL}}\) of \(\theta \) are asymptotically equivalent. In this paper, the stochastic expansions of \(\hat{\theta }_{\mathrm{ML}}^{\gamma }\), \(\hat{\theta }_{\mathrm{ML}}\) and \(\hat{\theta }_{\mathrm{MCL}}\) are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted MLE \(\hat{\theta }_{\mathrm{ML}}^{*}\) relative to \(\hat{\theta }_{\mathrm{ML}}^{\gamma }\) is also given, and \(\hat{\theta }_{\mathrm{ML}}^{*}\) and \(\hat{\theta }_{\mathrm{MCL}}\) are shown to be second-order asymptotically equivalent. Further, some examples are given.

Evaluating panel data forecasts under independent realization

Independent realization is a commonly used shortcut for deriving forecast properties. It is also an unrealistic assumption in many empirical applications. In this paper we consider the effect of the assumption when deriving the properties of panel data forecasts. To do so, we derive and compare the asymptotic forecast loss of a set of vector autoregressions (VARs) under both independent and same-sample realization. We show that adopting the independent realization assumption can result in an overstatement of forecast loss when common forms of parameterized heterogeneity (such as fixed effects) are included in the VAR. Because these parameters are often used in dynamic panel forecasting applications, our results imply that the independent realization shortcut should only be used with caution when working with panel data.

BER of Subcarrier MPSK and MDPSK Systems in Atmospheric Turbulence

Bit-error rate (BER) performance of subcarrier $M$ -ary phase-shift keying (MPSK) and $M$ -ary differential PSK (MDPSK) is analyzed for optical wireless communications over Gamma–Gamma and lognormal turbulence channels. We study the relation between the exact BER and the approximate BER, which is obtained by dividing the symbol-error rate by the number of bits per symbol, for subcarrier MPSK and MDPSK modulations. The asymptotic BER performance gap between the exact and the approximate BERs is quantified analytically through our asymptotic analyses. The accuracy of the approximate BER of both MPSK and MDPSK depends on the channel conditions. Under weak turbulence conditions, the approximate BER expression can be used to predict the system performance with high accuracy, while under strong turbulence conditions the approximate BER becomes inaccurate and can only serve as a loose lower bound of the exact BER. The asymptotic BER performance loss of MDPSK with respect to MPSK is also quantified analytically.

Asymptotic theory of generalized information criterion for geostatistical regression model selection

Information criteria, such as Akaike's information criterion and Bayesian information criterion are often applied in model selection. However, their asymptotic behaviors for selecting geostatistical regression models have not been well studied, particularly under the fixed domain asymptotic framework with more and more data observed in a bounded fixed region. In this article, we study the generalized information criterion (GIC) for selecting geostatistical regression models under a more general mixed domain asymptotic framework. Via uniform convergence developments of some statistics, we establish the selection consistency and the asymptotic loss efficiency of GIC under some regularity conditions, regardless of whether the covariance model is correctly or wrongly specified. We further provide specific examples with different types of explanatory variables that satisfy the conditions. For example, in some situations, GIC is selection consistent, even when some spatial covariance parameters cannot be estimated consistently. On the other hand, GIC fails to select the true polynomial order consistently under the fixed domain asymptotic framework. Moreover, the growth rate of the domain and the degree of smoothness of candidate regressors in space are shown to play key roles for model selection.