First-order Efficiency Research Articles

Sinusoidal transmission grating spectrometer for extreme ultraviolet measurement.

Spectral measurements play a vital role in understanding laser-plasma interactions. The ability to accurately measure the spectrum of radiation sources is crucial for unraveling the underlying physics. In this article, we introduce a novel approach that significantly enhances the efficiency of binary sinusoidal transmission grating spectrometers . The grating was tailored especially for Extreme Ultraviolet (EUV) measurements. The new design, High Contrast Sinusoidal Transmission Grating (HCSTG), not only suppresses high diffraction orders and retains the advantageous properties of previous designs but also exhibits a fourfold improvement in first-order efficiency. In addition, the HCSTG offers exceptional purity in the first order due to effectively eliminating half-order contributions from the diffraction pattern. The HCSTG spectrometer was employed to measure the emission of laser-produced Sn plasma in the 1-50nm spectral range, achieving a spectral resolution of λ/Δλ = 60. We provide a comprehensive analysis comparing the diffraction patterns of different STGs, highlighting the advantages offered by the HCSTG design. This novel, efficiency-enhanced HCSTG spectrometer opens up new possibilities for accurate and sensitive EUV spectral measurements.

Laser-Induced Fabrication of Micro-Optics on Bioresorbable Calcium Phosphate Glass for Implantable Devices.

In this study, a single-step nanosecond laser-induced generation of micro-optical features is demonstrated on an antibacterial bioresorbable Cu-doped calcium phosphate glass. The inverse Marangoni flow of the laser-generated melt is exploited for the fabrication of microlens arrays and diffraction gratings. The process is realized in a matter of few seconds and, by optimizing the laser parameters, micro-optical features with a smooth surface are obtained showing a good optical quality. The tunability of the microlens' dimensions is achieved by varying the laser power, allowing the obtaining of multi-focal microlenses that are of great interest for three-dimensional (3D) imaging. Furthermore, the microlens' shape can be tuned between hyperboloid and spherical. The fabricated microlenses exhibited good focusing and imaging performance and the variable focal lengths were measured experimentally, showing good agreement with the calculated values. The diffraction gratings obtained by this method showed the typical periodic pattern with a first-order efficiency of about 5.1%. Finally, the dissolution characteristics of the fabricated micropatterns were studied in a phosphate-buffered saline solution (PBS, pH = 7.4) demonstrating the bioresorbability of the micro-optical components. This study offers a new approach for the fabrication of micro-optics on bioresorbable glass, which could enable the manufacturing of new implantable optical sensing components for biomedical applications.

Chitosan from crab shell waste for soft lithography of bioplastic diffraction gratings.

Bioplastic diffraction gratings are fabricated from chitosan prepared from crab shells through a soft lithography replication process. Atomic force microscopy and diffraction experiments with the chitosan grating replicas indicate the successful copying of periodic nanoscale groove structures corresponding to densities of 600 and 1200lines/mm. First-order efficiency of bioplastic gratings is comparable to the output of elastomeric grating replicas.

Binary geometric-phase holograms.

Diffractive optics elements have exhibited many novel characteristics through various methods of employing Pancharatnam-Berry, or geometric, phase. One geometric-phase hologram (GPH) subset, consisting of a π-difference binary sampling, shows polarization-independent properties that are not present in the continuous GPH and the dynamic-phase binary analog. Here, we investigate the binary geometric-phase holograms (bin-GPHs) realized with anisotropic liquid crystal (LC) polymers. First, the optical properties of the ideal binary polarization grating are derived and simulated showing 81% cumulative first-order efficiency, polarization-independent diffraction when applying a π-switching scheme, innate odd (m = 2k + 1) diffractive orders, and variable polarization output. After, experimental results of two key bin-GPH elements, the binary polarization grating (Λ = 30μm) and binary geometric-phase lens (f/100), with π-offset regions and a 0.5μm transition pixel are presented. We found that the fabricated non-ideal bin-GPHs exhibit near-maximum theoretical polarization-insensitive diffraction efficiency and tunable polarization outputs. The simple, and scalable, fabrication of the anisotropic bin-GPH provides the potential for implementation within the next-generation near-eye displays for polarization-invariant beam-steering and waveguides.

Fixed-accuracy confidence interval estimation of P(X > c) for a two-parameter gamma population

The gamma distribution is a flexible right-skewed distribution widely used in many areas, and it is of great interest to estimate the probability of a random variable exceeding a specified value in survival and reliability analysis. Therefore, the study develops a fixed-accuracy confidence interval for P(X > c) when X follows a gamma distribution, Γ(α, β), and c is a preassigned positive constant through: 1) a purely sequential procedure with known shape parameter α and unknown rate parameter β; and 2) a nonparametric purely sequential procedure with both shape and rate parameters unknown. Both procedures enjoy appealing asymptotic first-order efficiency and asymptotic consistency properties. Extensive simulations validate the theoretical findings. Three real-life data examples from health studies and steel manufacturing study are discussed to illustrate the practical applicability of both procedures.

On general asymptotically second-order efficient purely sequential fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) strategies for a normal mean and optimality

We develop a generalized class of purely sequential sampling strategies associated with both fixed-width confidence interval (FWCI) and minimum risk point estimation (MRPE) problems for the unknown mean $$\mu $$ of a normally distributed population having its variance $$\sigma ^{2}$$ also unknown. Under this newly proposed general class of associated estimation strategies, we develop a variety of asymptotic first-order and asymptotic second-order properties such as asymptotic consistency, first-order efficiency, first-order risk efficiency, second-order efficiency, and second-order regret analysis. Next, we proceed to locate an optimal strategy within our newly built large class of possibilities. Such optimality is defined as having been associated with the minimal second-order asymptotic variance of a stopping time within the general class of proposed strategies. We follow through by exploring both the FWCI and MRPE problems with the help of data analysis from simulations.

On the Iterated Estimation of Dynamic Discrete Choice Games

Abstract We study the first-order asymptotic properties of a class of estimators of the structural parameters in dynamic discrete choice games. We consider $K$-stage policy iteration (PI) estimators, where $K$ denotes the number of PIs employed in the estimation. This class nests several estimators proposed in the literature. By considering a “pseudo likelihood” criterion function, our estimator becomes the $K$-pseudo maximum likelihood (PML) estimator in Aguirregabiria and Mira (2002, 2007). By considering a “minimum distance” criterion function, it defines a new $K$-minimum distance (MD) estimator, which is an iterative version of the estimators in Pesendorfer and Schmidt-Dengler (2008) and Pakes et al. (2007). First, we establish that the $K$-PML estimator is consistent and asymptotically normal for any $K \in \mathbb{N}$. This complements findings in Aguirregabiria and Mira (2007), who focus on $K=1$ and $K$ large enough to induce convergence of the estimator. Furthermore, we show under certain conditions that the asymptotic variance of the $K$-PML estimator can exhibit arbitrary patterns as a function of $K$. Second, we establish that the $K$-MD estimator is consistent and asymptotically normal for any $K \in \mathbb{N}$. For a specific weight matrix, the $K$-MD estimator has the same asymptotic distribution as the $K$-PML estimator. Our main result provides an optimal sequence of weight matrices for the $K$-MD estimator and shows that the optimally weighted $K$-MD estimator has an asymptotic distribution that is invariant to $K$. The invariance result is especially unexpected given the findings in Aguirregabiria and Mira (2007) for $K$-PML estimators. Our main result implies two new corollaries about the optimal $1$-MD estimator (derived by Pesendorfer and Schmidt-Dengler (2008)). First, the optimal $1$-MD estimator is efficient in the class of $K$-MD estimators for all $K \in \mathbb{N}$. In other words, additional PIs do not provide first-order efficiency gains relative to the optimal $1$-MD estimator. Second, the optimal $1$-MD estimator is more or equally efficient than any $K$-PML estimator for all $K \in \mathbb{N}$. Finally, the Appendix provides appropriate conditions under which the optimal $1$-MD estimator is efficient among regular estimators.

Key-block based analytical stability method for discontinuous rock slope subjected to toppling failure

Abstract This study introduces a simplified semi-distinct element algorithm for discontinuous rock slopes under toppling instability assessment based on block theory. The presented algorithm is developed to investigate block, flexural and block-flexural types of toppling failures which have been coded in the Python high-level programming language. In order to investigate toppling instabilities for different modes, the three modelling steps, namely, geometrical, behavioural and mechanical simulations were conducted to appoint high-order calculation trending loops. These loops were based on the analytical description of modified second-order reliability method with first-order efficiency to determine the factor of safety values for discontinuous rock slopes. Each type of toppling failure was described based on main assumptions that were modified by geometric qualifications such as the analytical procedure, single block slip, and the Voronoi diagram algorithm for the stability analysis of the three different failure types. According to the stability assessment results of the studied slopes which were verified by the distinct numerical method via UDEC software, it has been clarified that the results obtained from the algorithm and the numerical analysis were in an appropriate trend. The used algorithm is superior in pinpointing the critical sliding zones, safety factor estimation and progressive failure analyses as compared to the numerical analyses. Moreover, the block theory based algorithm has higher processing capability and speed with respect to the numerical method.

Numerical Simulation of a New Horizontal Spray Tower for Sintering Flue Gas Desulfurization

The flow rate and temperature of the flue gas from a sintering machine can fluctuate in a wide range; thus, the follow-up desulfurization facility should be operated stably and maintained on-line. A new type of horizontal spray tower, which can achieve high efficiency with low pressure loss and easy maintenance, was invented by the University of Science and Technology Beijing. A 3D multiphase computational fluid dynamics (CFD) model of the horizontal spray tower was constructed with Fluent software to study the internal flow field and the reaction of the horizontal spray tower. The standard k–ξ model and the Eulerian–Lagrangian model were adopted in this study. Results show that the flow field in the spray tower can be numerically simulated utilizing the CFD technology. The inlet angle exerts an important effect on the flue gas flow field of the horizontal tower. When the first-order efficiency and the economic indicators are taken into account, an inlet angle of 75° is recommended in the top inlet intake mode, whereas a 30° inlet angle performs the best in the side inlet intake mode. The spray configuration exerts an important influence. When the spray cone angle is 110° and the installation height is controlled at 0.8 m from the top surface, the smoke residence time is long, and the contact is sufficient. The results can provide theoretical support for the optimization of the structure of the spray tower.

Second-order analysis of regret for sequential estimation of the autoregressive parameter in a first-order autoregressive model

This article revisits the problem of sequential point estimation of the autogressive parameter in an autoregressive model of order 1, where the errors are independent and identically distributed with mean 0 and unknown variance . This problem was originally considered in Sriram (1988), where first-order efficiency properties and a second-order expansion for the expected value of a stopping rule were established. Here, we obtain an asymptotic expression for the so-called regret due to not knowing σ, as the cost of estimation error tends to infinity. Under suitable assumptions, our extensive analysis shows that all but one term in the regret are asymptotically bounded. If the errors have a bounded support, however, then the regret remains asymptotically bounded. Finally, we illustrate the performance of our sequential procedure and the associated regret for well-known blowfly data (Nicholson, 1950) and Internet traffic data using the residual bootstrap method for autoregressive models.

Robust estimation for longitudinal data based upon minimum Hellinger distance

ABSTRACTGeneralized linear mixed models have been widely used in the analysis of correlated data in a lot of research areas. The linear mixed model with normal errors has been a popular model for the analysis of repeated measures and longitudinal data. Outliers, however, can severely have an wrong influence on the linear mixed model. The aforementioned model has not fully taken those severe outliers into consideration. One of the popular robust estimation methods, M-estimator attains robustness at the expense of first-order or second-order efficiency whereas minimum Hellinger distance estimator is efficient and robust. In this paper, we propose more robust Bayesian version of parameter estimation via pseudo posterior distribution based on minimum Hellinger distance. It accommodates an appropriate nonparametric kernel density estimation for longitudinal data to require the proposed cross-validation estimator. We conduct simulation study and real data study with the orthodontic study data and the Alzheimers Disease (AD) study data. In simulation study, the proposed method shows smaller biases, mean squared errors, and standard errors than the (residual) maximum likelihood method (REML) in the presence of outliers or missing values. In real data analysis, standard errors and variance-covariance components for the proposed method in two data sets are shown to be lower than those for REML method.

Two-stage fixed-width and bounded-width confidence interval estimation methodologies for the common correlation in an equi-correlated multivariate normal distribution

In this article, two-stage and fixed sample size procedures are developed for constructing confidence intervals for the common correlation ρ of an equi-correlated multivariate normal distribution. Two different approaches for estimation are considered, namely, fixed-width and bounded-width interval estimation. In the fixed-width confidence interval estimation problem, a two-stage procedure is developed and the exact distribution of the corresponding stopping variable and the exact coverage probability of the interval estimator are derived. Asymptotic optimality properties such as asymptotic first-order efficiency and asymptotic consistency properties of the two-stage procedure are established. Bounded-width confidence intervals are obtained by applying fixed-accuracy estimation methodologies of Mukhopadhyay and Banerjee (2014, 2015a,b) and Banerjee and Mukhopadhyay (2016) using different transformations on ρ. Both fixed sample size and two-stage sampling methodologies for bounded-width confidence interval estimation of are developed incorporating such transformations. Finally, performances of all the procedures are compared via extensive sets of simulation studies.

Highly chromatic retardation via multi-twist liquid crystal films

Here we study birefringent films with highly customizable chromatic retardation spectra, using multi-twist liquid crystal (LC) films. These are made of two or more layers of chiral nematic LC polymer network materials, also known as reactive mesogens, which form a monolithic thin-film wherein the in-plane orientation of subsequent layers is automatically determined by the single alignment layer on the substrate. The multiple layer thicknesses and twists present many degrees of freedom to tailor the retardation. While prior work examined achromatic spectra, here we show how to use Mueller matrix analysis to create highly chromatic spectra. We experimentally demonstrate both a uniformly aligned retarder as a green/magenta color filter and a “hot” polarization grating (PG) that diffracts infrared while passing visible light. The three-twist color filter shows a contrast ratio in transmittance between polarizers as high as 10:1 between the half- and zero-wave retardation bands. The “hot” PG shows an average first-order efficiency of about 90% for 1000–2700 nm and an average zero-order efficiency of about 90% for 500–900 nm. The principles here can be extended to nearly any chromatic retardation spectra, including high/low/bandpass, and to nearly any LC orientation pattern, in general known as geometric-phase holograms.

Purely Sequential Fixed Accuracy Confidence Intervals for P(X < Y) under Bivariate Exponential Models

SYNOPTIC ABSTRACTThis article deals with the estimation of the reliability or the availability parameter θ = P(X < Y) in a stress-strength model. We consider separate cases where (X, Y) has a bivariate exponential distribution due to Marshall and Olkin (1976) and Freund (1961). We construct a purely sequential estimation methodology to find a fixed accuracy confidence interval for the unknown θ in each case. First-order asymptotic efficiency and consistency properties are also established of our proposed procedures. We then supplement extensive sets of simulation studies and real data analysis to show encouraging performances of our proposed stopping strategies.

Bragg polarization gratings for wide angular bandwidth and high efficiency at steep deflection angles

Optical films and surfaces using geometric phase are increasingly demonstrating unique and sometimes enhanced performance compared to traditional elements employing propagation phase. Here, we report on a diffraction grating with wider angular bandwidth and significantly higher average first-order efficiency than the nearest prior art of metasurfaces, volume holographic gratings, and surface-relief gratings configured to achieve a steep deflection angle. More specifically, we demonstrate a liquid crystal (LC) polymer Bragg polarization grating (PG) with large angular bandwidth and high efficiency in transmission-mode for 532 nm wavelength and 400 nm period. Angular bandwidth was significantly increased by arranging two slanted grating layers within the same monolithic film. First, we studied the optical properties with simulation and identified a structure with 48° angular bandwidth and 70% average first-order efficiency. Second, we fabricated a sample using a photo-aligned chiral nematic LC, where the two grating slants were controlled by the chiral dopants. We measured 40° angular bandwidth, 76% average efficiency, and 96% peak efficiency. Strong input polarization sensitivity (300:1 contrast) and spectral bandwidth (200 nm) mostly matched prior PGs. This approach is especially advantageous for augmented-reality systems and nonmechanical beam steering.

Nanoscale liquid crystal polymer Bragg polarization gratings.

We experimentally demonstrate nearly ideal liquid crystal (LC) polymer Bragg polarization gratings (PGs) operating at a visible wavelength of 450 nm and with a sub-wavelength period of 335 nm. Bragg PGs employ the geometric (Pancharatnam-Berry) phase, and have many properties fundamentally different than their isotropic analog. However, until now Bragg PGs with nanoscale periods (e.g., < 800 nm) have not been realized. Using photo-alignment polymers and high-birefringence LC materials, we employ multiple thin sublayers to overcome the critical thickness threshold, and use chiral dopants to induce a helical twist that effectively generates a slanted grating. These LC polymer Bragg PGs manifest 85-99% first-order efficiency, 19-29° field-of-view, Q ≈ 17, 200 nm spectral bandwidth, 84° deflection angle in air (in one case), and efficient waveguide-coupling (in another case). Compared to surface-relief and volume-holographic gratings, they show high efficiency with larger angular/spectral bandwidths and potentially simpler fabrication. These nanoscale Bragg PGs manifest a 6π rad/μm phase gradient, the largest reported for a geometric-phase hologram while maintaining a first-order efficiency near 100%.

Confidence intervals and point estimators for a normal mean under purely sequential strategies involving Gini’s mean difference and mean absolute deviation

ABSTRACTWe have developed purely sequential methodologies for problems associated with both fixed-width confidence interval estimation and minimum risk point estimation for the normal mean μ when the variance σ2 is assumed unknown. New stopping rules are constructed by replacing the sample variance with appropriate multiples of Gini’s mean difference (GMD) and mean absolute deviation (MAD) in defining the conditions for boundary crossing. A number of asymptotic first-order consistency, efficiency, and risk efficiency properties associated with these new estimation strategies have been investigated. These are followed by summaries obtained from extensive sets of simulations by drawing samples from (i) normal universes or (ii) mixture-normal universes where samples may be reasonably treated as observations from a normal universe in a large majority of simulations. We also include illustrations using sales data and horticulture data. Overall, we empirically feel confident that our newly developed GMD-based or MAD-based methodologies are more robust for practical purposes when we compare them with the sample variance–based methodologies respectively, especially when up to 20% suspect outliers may be expected.

A general sequential fixed-accuracy confidence interval estimation methodology for a positive parameter: illustrations using health and safety data

Estimation of positive parameters is important in areas including ecology, biology, medicine, nuclear power, and study of cell membranes. We develop a general structure for a fixed-accuracy sequential confidence interval estimation methodology in the spirit of Mukhopadhyay and Banerjee (Sequ Anal, 33:251–285, 2014a) for a positive parameter of an arbitrary distribution which may be discrete or continuous. The confidence interval is constructed using a maximum likelihood (ML) estimator of the unknown parameter. The methodology enjoys attractive properties such as asymptotic consistency and asymptotic first-order efficiency (Theorem 1). Three specific illustrations are included. Comprehensive data analyses from large-scale simulations have been incorporated which substantiate encouraging performances of the proposed estimation methodology. These are followed by real data analyses corresponding to the Bernoulli distribution (odds ratio of poisonous mushrooms), Poisson distribution (radioactive decay of isotopes), and a Normal distribution with the same mean and variance (real-time 911 calls dispatch).

Plug-In Two-Stage and Sequential Normal Density Estimation Under MISE Loss: Both Mean and Variance are Unknown

Consider independent observations having a common normal probability density function with x ∈ R, unknown mean μ(∈R), and unknown variance σ2(∈R +). We propose estimating f(x; μ, σ2) with both two-stage and purely sequential methodologies under the mean integrated squared error (MISE) loss function. Our goal is to make the associated risk not to exceed a preassigned positive number c, referred to as the risk bound. No fixed-sample-size methodology would handle this estimation problem. We show that both density estimation methodologies satisfy an asymptotic (a) first-order efficiency property and a (b) first-order risk-efficiency property. Interestingly, purely sequential density estimation methodology has a better second-order efficiency property than that associated with two-stage methodology. Some robustness issues have been addressed. Small, moderate, and large sample performances are examined with the help of simulations. Illustrations are given with real data sets.

Higher-order expansions and efficiencies of tests based on spacings

Statistics based on spacings, or the gaps between points, have been widely used in many contexts, primarily in testing goodness of fit. This paper derives Edgeworth-type asymptotic expansions for the sum of functions of s-step spacings where s, the order of spacings, may increase together with the sample size n. When s is fixed, it is known that only the Greenwood test, based on the sum of squares of these spacings, is first-order asymptotically efficient. In contrast, it is shown here that if s goes to infinity, there exist many other tests which are first-order efficient. We introduce and study the second-order efficiency of such tests and show that if s is sufficiently large relative to n, the Greenwood test is no longer second-order efficient. Interestingly, we see that the common phenomenon of first-order efficiency implying second-order efficiency does not hold true in this situation.