Parent Distribution Research Articles

Black hole masses of tidal disruption event host galaxies II

ABSTRACT We present new medium resolution, optical long-slit spectra of a sample of six ultraviolet (UV)/optical and 17 X-ray-selected tidal disruption event candidate host galaxies. We measure emission line ratios from the optical spectra, finding that the large majority of hosts are quiescent galaxies, while those displaying emission lines are generally consistent with star formation dominated environments; only three sources show clear evidence of nuclear activity. We measure bulge velocity dispersions using absorption lines and infer host black hole (BH) masses using the M – σ relation. While the optical and X-ray host BH masses are statistically consistent with coming from the same parent distribution, the optical host distribution has a visible peak near $M_{\rm BH} \sim 10^6 \, \mathrm{M}_\odot$, whereas the X-ray host distribution appears flat in MBH. We find a subset of X-ray-selected candidates that are hosted in galaxies significantly less luminous (Mg ∼ −16) and less massive (stellar mass ∼ 108.5–9 M⊙) than those of optical events. Using statistical tests we find suggestive evidence that, in terms of BH mass, stellar mass, and absolute magnitude, the hard X-ray hosts differ from the UV/optical and soft X-ray samples. Similar to individual studies, we find that the size of the emission region for the soft X-ray sample is much smaller than the optical emission region, consistent with a compact accretion disc. We find that the typical Eddington ratio of the soft X-ray emission is ∼ 0.01, as opposed to the optical events which have LBB ∼ LEdd. The latter seems artificial if the radiation is produced by self-intersection shocks, and instead suggests a connection to the supermassive black hole.

Measures of ageing tendency

AbstractA family of generalized ageing intensity functions of univariate absolutely continuous lifetime random variables is introduced and studied. They allow the analysis and measurement of the ageing tendency from various points of view. Some of these generalized ageing intensities characterize families of distributions dependent on a single parameter, while others determine distributions uniquely. In particular, it is shown that the elasticity functions of various transformations of distributions that appear in lifetime analysis and reliability theory uniquely characterize the parent distribution. Moreover, the recognition of the shape of a properly chosen generalized ageing intensity estimate admits a simple identification of the data lifetime distribution.

Pooled frequency analysis for intensity–duration–frequency curve estimation

AbstractRainfall intensity–duration–frequency (IDF) curves are used in the design of urban infrastructure. Their estimation is based on rainfall frequency analysis, usually performed on rainfall records from a single gauged station. However, available at‐site record length is often too short to provide accurate estimates for long return periods. In the present study, a general framework for pooled rainfall frequency analysis based on the index‐event model is proposed for IDF estimation at gauged stations. Pooling group formation is defined by the region of influence approach on the basis of the geographical distance similarity measure. Several pooled approaches are defined and evaluated by a procedure through which quantile estimation and uncertainty are assessed. Alternate approaches for the definition of a pooling group are based on different criteria regarding initial pooling group size (and the relationship between size and return period), approaches for assessing pooling group homogeneity, and the use of macroregions in pooling group formation. The proposed framework is applied to identify the preferred approach for pooled rainfall intensity frequency analysis in Canada. Pooled approaches are found to provide more precise estimates than the at‐site approach, especially for long return periods. Pooled parent distribution selection supported the use of the generalized extreme value distribution across the country. Recommendations for pooling group formation include increasing the pooling group size with increases in return period and identifying an appropriate trade‐off between pooling group homogeneity and size for long return periods.

On the Mass Function, Multiplicity, and Origins of Wide-orbit Giant Planets

Abstract A major outstanding question regarding the formation of planetary systems is whether wide-orbit giant planets form differently than close-in giant planets. We aim to establish constraints on two key parameters that are relevant for understanding the formation of wide-orbit planets: (1) the relative mass function and (2) the fraction of systems hosting multiple companions. In this study, we focus on systems with directly imaged substellar companions and the detection limits on lower mass bodies within these systems. First, we uniformly derive the mass probability distributions of known companions. We then combine the information contained within the detections and detection limits into a survival analysis statistical framework to estimate the underlying mass function of the parent distribution. Finally, we calculate the probability that each system may host multiple substellar companions. We find that (1) the companion mass distribution is rising steeply toward smaller masses, with a functional form of N ∝ M −1.3±0.03, and consequently, (2) many of these systems likely host additional undetected substellar companions. Combined, these results strongly support the notion that wide-orbit giant planets are formed predominantly via core accretion, similar to the better studied close-in giant planets. Finally, given the steep rise in the relative mass function with decreasing mass, these results suggest that future deep observations should unveil a greater number of directly imaged planets.

Study on establishment of standardized load spectrum on bogie frames of high-speed trains

Establishing a structural load spectrum under actual operating conditions is a major problem in structural fatigue life analysis. This study introduces a load measuring method for the bogie frame structure. The quasi-static load-measuring frame can measure different load systems synchronously. The t test method is employed to evaluate the least test time to deduce the parent distribution. In order to fit the load spectrum distribution accurately, the kernel density estimation method is employed, which is based on the sample characteristics. The expansion factor method is used to deduce the maximum load. The formula for a standardized load spectrum is derived from the deduced maximum load, the linear factor between operating condition length and cumulative frequency, and the parent distribution of each load system. The damage consistency criterion is performed by solving the objective function with constraint conditions. The calibrated damage provides a suitable representation of the real damage under actual operating conditions. By processing and analyzing the load and stress spectral data of the measured lines, it is verified that the standardized load spectrum established in this paper is superior to the European specification and the Japanese specification in evaluating the fatigue reliability of the structure.

X-ray absorbing column densities of a complete sample of short gamma ray bursts

Context. X-ray absorbing column densities (NH) are used as a parameter to quantify the amount of absorbing material along the line of sight. The high values found for long gamma ray bursts (LGRBs) confirmed that these events take place in dense, star-forming environments, joining as an indirect proof the observation of supernovae associated to the bursts and the location in the brightest galaxy regions. Recently, the simultaneous detection of a short gamma ray burst (SGRB) and a gravitational wave signal occurred, strongly supporting the hypothesis that SGRBs instead originate from the merger of compact objects. The different predictions of the two progenitor scenarios for short and long GRBs should be reflected in a difference in the amount of absorbing matter between the two populations, with SGRBs occurring in less dense environments. Previous studies found that the two column density distributions were indistinguishable when compared in the same redshift range. The samples, though, were relatively small (10–12 SGRBs), and spanned a redshift range z ≲ 1. Aims. We update a flux-limited complete sample of Swift-based SGRBs, the SBAT4, bringing it to 25 events and doubling its previous redshift range. We then evaluate the column densities of the events in the updated sample, in order to compare them with the NH distribution of LGRBs, using the sample BAT6ext. Methods. We relied on Monte Carlo simulations of the two populations and compare the computed NH distributions with a two sample Kolmogorov–Smirnov (K–S) test. We then studied how the K–S probability varies with respect to the redshift range we consider. Results. We find that the K–S probability keeps decreasing as redshift increases until at z ∼ 1.8 the probability that short and long GRBs come from the same parent distribution drops below 1%. This testifies for an observational difference among the two populations. This difference may be due to the presence of highly absorbed LGRBs above z ∼ 1.3, which have not been observed in the SGRB sample yet, although this may be due to our inability to detect them, or to the relatively small sample size.

Difference Distributions Applicable to Certain Health Physics Measurements.

This paper discusses calculational methods for the determination of the difference distributions associated with certain health physics measurements. These measurements include the check-source response counts relative to an initial reference count, the Albatross (i.e., HPI model 2080B) neutron tube counts relative to the gamma tube counts, and those that involve the use of the automatic background subtraction feature of portable health physics instrumentation. Examples are provided that illustrate the methods for a few specific measurements. For the comparison of a daily source count to its previously determined reference value, minimum counts for various scenarios were presented in order to reliably meet required tolerance limits of ±10% and ±20%. In either case, it was found beneficial that the initial reference readings be established using a counting interval of longer length than the daily interval. For the comparison of Albatross neutron counts to the gamma counts, it was seen that the relative error in the difference distribution was still related to that of the parent distribution. It was seen, therefore, that an effective way of reducing the gamma influence on the Albatross was to increase the counting interval used, hence yielding a significantly larger mean count per interval. For the automatic background subtraction feature, it was noted that net count values near 0 counts would almost always have the negative values of the difference distribution truncated to 0 counts by commercially available off-the-shelf instrumentation, whereas significant net count values would be displayed correctly but with a larger associated variance than the gross count itself. This paper therefore also provides a technical basis for the necessary source strength of a check source in order to meet daily limits, the gamma field limitations of the HPI 2080B Albatross, as well as the consequences of automatic background subtraction.

Order statistics from overlapping samples: bivariate densities and regression properties

ABSTRACTIn this paper, we are interested in the joint distribution of two order statistics from overlapping samples. We give an explicit formula for the distribution of such a pair of random variables under the assumption that the parent distribution is absolutely continuous. We are also interested in the question to what extent conditional expectation of one of such order statistic given another determines the parent distribution. In particular, we provide a new characterization by linearity of regression of an order statistic from the extended sample given the one from the original sample, special case of which solves a problem explicitly stated in the literature. It appears that to describe the correct parent distribution it is convenient to use quantile density functions. In several other cases of regressions of order statistics we provide new results regarding uniqueness of the distribution in the sample.

Modelling Annual Maximum River Flows with Generalized Extreme Value Distribution

A good understanding of probability distribution of annual maximum river flow is believed to improve water resources planning and design. Based on the annual maximum river flow record over 20-48 years at 9 individual river sites in Sabah, the data set are fitted into generalized extreme value (GEV) distribution with maximum likelihood estimator. Both stationary and non-stationary models are considered. Likelihood ratio test shows that most of the river flows are stationary. Over a homogeneous region, a parent distribution with common shape parameter is found well describing the behavior of selected annual maximum river flow. Hence, 10- and 100-year return levels are estimated using the single model.

Characterizing wind gusts in complex terrain

Abstract. Wind gusts are a key driver of aerodynamic loading, especially for tall structures such a bridges and wind turbines. However, gust characteristics in complex terrain are not well understood and common approximations used to describe wind gust behavior may not be appropriate at heights relevant to wind turbines and other structures. Data collected in the Perdigão experiment are analyzed herein to provide a foundation for improved wind gust characterization and process-level understanding of flow intermittency in complex terrain. High-resolution observations from sonic anemometers and vertically pointing Doppler lidars are used to conduct a detailed study of gust characteristics with a specific focus on the parent distributions of nine gust parameters (that describe velocity, time, and length scales), their joint distributions, height variation, and coherence in the vertical and horizontal planes. Best-fit distributional forms for varying gust properties show good agreement with those from previous experiments in moderately complex terrain but generate nonconservative estimates of the gust properties that are of key importance to structural loading. Probability distributions of gust magnitude derived from vertically pointing Doppler lidars exhibit good agreement with estimates from sonic anemometers despite differences arising from volumetric averaging and the terrain complexity. Wind speed coherence functions during gusty periods (which are important to structural wind loading) are similar to less complex sites for small vertical displacements (10 to 40 m), but do not exhibit an exponential form for larger horizontal displacements (800 to 1500 m).

Location charts based on ranked set sampling for normal and non‐normal processes

AbstractControl charts, based on ranked set sampling schemes, had been proposed recently for efficient monitoring of process location. All the proposals in the literature are based on the ideal assumption of normally distributed quality characteristics. No study as of yet investigated the performance of location charts based on ranked set sampling for non‐normal processes. In this study, we investigated the location chart based on simple random sampling (SRS) and three well‐known rank‐based schemes ie, ranked set sampling (RSS), median ranked set sampling (MRSS), and extreme ranked set sampling (ERSS), considering normal and a variety of non‐normal parent distributions. Both heavy‐tailed symmetric and skewed cases have been considered in this study. The performance of the charts is evaluated using average run length (ARL), extra quadratic loss (EQL), and relative ARL (RARL) measures. A real life example is also presented that details the monitoring of pH levels in water for an experiment conducted to study the reproduction of Mysids. The study will help quality practitioners to choose the chart based on an efficient sampling scheme for normal and non‐normal processes.

Improved estimators for stress-strength reliability using record ranked set sampling scheme

In this paper, the problem of estimating the stress-strength reliability based on upper record ranked set sampling is studied. In the exponential parent distribution, improved estimation strategies including the preliminary test estimator (PTE) and Stein-type shrinkage estimator (SE) are developed using the maximum likelihood estimator (MLE) and a prior information about R. Next, asymptotic distributional properties of the proposed estimator are exactly derived. Then we compare the proposed estimators in terms of asymptotic relative efficiency criteria. Furthermore, we construct some bootstrap confidence intervals (CI) in order to achieve more accurate inferences. The results show that our proposed improved estimators are superior to the MLE in wide range of parameter space. Specifically, the proposed shrinkage preliminary test estimator (SPTE) outperforms the MLE uniformity for small values of type one error. Performance analysis is also investigated through some numerical endeavors.

Difference in Restricted Mean Survival Time: Small Sample Distribution and Asymptotic Relative Efficiency

The difference between two arms in the restricted mean survival time is an alternative to the hazard ratio. It provides a more easily understood measure of the treatment effect of an intervention in a controlled clinical trial with a time to event endpoint. In this article, we consider the distribution of the estimate for small or moderate sample sizes or when the event rate is low and compare the asymptotic relative efficiency (ARE) and relative efficiency for practical sample sizes of the test to the standard logrank test for Weibull parent distributions. We find that for small sample sizes, the distribution can depart markedly from the normal distribution. We propose a method to approximate the correct null distribution using the Cornish–Fisher expansion and provide an example. For Weibull parent distributions, we consider two scenarios with either proportional hazards or nonproportional hazards where the hazard functions and survival distribution curves crossover. The ARE is compared for these two scenarios.

Measuring and testing for interval quantile dependence

In this article, we introduce the notion of interval quantile independence which generalizes the notions of statistical independence and quantile independence. We suggest an index to measure and test departure from interval quantile independence. The proposed index is invariant to monotone transformations, nonnegative and equals zero if and only if the interval quantile independence holds true. We suggest a moment estimate to implement the test. The resultant estimator is root-$n$-consistent if the index is positive and $n$-consistent otherwise, leading to a consistent test of interval quantile independence. The asymptotic distribution of the moment estimator is free of parent distribution, which facilitates to decide the critical values for tests of interval quantile independence. When our proposed index is used to perform feature screening for ultrahigh dimensional data, it has the desirable sure screening property.

A Study of Lognormal Model for Air Pollution Concentration

The Lognormal distribution has been widely used to represent the type of air pollutant concentration distribution. There are different methods to estimate the distribution parameters such as the method of moments, percentiles, maximum likelihood estimation (MLE) and Bayesian method of estimation. In this study, the theoretical distribution Lognormal is used to fit the parent distribution of PM 10 of Putalisadak of Kathmandu, Nepal. Two estimating methods namely method of maximum likelihood and method of moments are used to estimate the parameters of the theoretic distributions.

On the limiting distribution of sample central moments

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta-method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.

Stochastic Comparisons of Series and Parallel Systems with Kumaraswamy-G Distributed Components

SYNOPTIC ABSTRACTThis article deals with stochastic comparisons of the series and parallel systems comprising Kumaraswamy generalized family of distributions. The results are established in two directions. We consider systems with different model parameters, and obtain some ordering results under the condition of multivariate chain majorization. Next, we use the notion of vector majorization and other related orders to establish various stochastic orders, such as the usual stochastic order, the likelihood ratio order, and the dispersive order. Further, results are obtained when components of two systems have independent and heterogeneous Kumaraswamy generalized family of distributions with different parent distribution functions. Numerical examples and counterexamples are provided to illustrate the established results.

Model averaging versus model selection: estimating design floods with uncertain river flow data

ABSTRACTThis study compares model averaging and model selection methods to estimate design floods, while accounting for the observation error that is typically associated with annual maximum flow data. Model selection refers to methods where a single distribution function is chosen based on prior knowledge or by means of selection criteria. Model averaging refers to methods where the results of multiple distribution functions are combined. Numerical experiments were carried out by generating synthetic data using the Wakeby distribution function as the parent distribution. For this study, comparisons were made in terms of relative error and root mean square error (RMSE) referring to the 1-in-100 year flood. The experiments show that model averaging and model selection methods lead to similar results, especially when short samples are drawn from a highly asymmetric parent. Also, taking an arithmetic average of all design flood estimates gives estimated variances similar to those obtained with more complex weighted model averaging.

Modelling counts with state-dependent zero inflation

We introduce a state-dependent zero-inflation mechanism for count distributions with unbounded or bounded support. Instead of uniformly downweighting the parent distribution, this flexible approach allows us to generate most of the zeros from either low or high counts. We derive the stochastic properties of the inflated distributions and discuss special instances designed for zero inflation caused by, for example, excessive demand or underreporting. Furthermore, we apply the state-dependent zero-inflation mechanism to generalize existing models for count time series with bounded support.

Wind Gust Characterization at Wind Turbine Relevant Heights in Moderately Complex Terrain

AbstractImproved understanding of wind gusts in complex terrain is critically important to wind engineering and specifically the wind energy industry. Observational data from 3D sonic anemometers deployed at 3 and 65 m at a site in moderately complex terrain within the northeastern United States are used to calculate 10 descriptors of wind gusts and to determine the parent distributions that best describe these parameters. It is shown that the parent distributions exhibit consistency across different descriptors of the gust climate. Specifically, the parameters that describe the gust intensity (gust amplitude, rise magnitude, and lapse magnitude; i.e., properties that have units of length per time) fit the two-parameter Weibull distribution, those that are unitless ratios (gust factor and peak factor) are described by log-logistic distributions, and all other properties (peak gust, rise and lapse times, gust asymmetric factor, and gust length scale) are lognormally distributed. It is also shown that gust factors scale with turbulence intensity, but gusts are distinguishable in power spectra of the longitudinal wind component (i.e., they have demonstrably different length scales than the average eddy length scale). Gust periods at the lower measurement height (3 m) are consistent with shear production, whereas at 65 m they are not. At this site, there is only a weak directional dependence of gust properties on site terrain and land cover variability along sectorial transects, but large gust length scales and gust factors are more likely to be observed in unstable atmospheric conditions.