Aging Intensity for Step-Stress Accelerated Life Testing Experiments

The aging intensity (AI), defined as the ratio of the instantaneous hazard rate and a baseline hazard rate, is a useful tool for the describing reliability properties of a random variable corresponding to a lifetime. In this work, the concept of AI is introduced in step-stress accelerated life testing (SSALT) experiments, providing new insights to the model and enabling the further clarification of the differences between the two commonly employed cumulative exposure (CE) and tampered failure rate (TFR) models. New AI-based estimators for the parameters of a SSALT model are proposed and compared to the MLEs in terms of examples and a simulation study.

Why Does Cross-Sectional Analyst Coverage Incorporate Market-Wide Information?

This paper shows that the empirical distribution of cross-sectional analyst coverage in China's stock markets follows an exponential law in a given month from 2011 to 2020. The findings hold in both the emerging (Shanghai) and the developed market (Hong Kong). Moreover, the unique distribution parameter (i.e., mean) is directly related to the amount of market-wide information. Average analyst coverage exhibits a significant negative predictive power for stock-market uncertainty, highlighting the role of security analysts in diminishing the total uncertainty. The exponential law can be derived from the maximum entropy principle (MEP). When analysts, who are constrained by average ability in generating information (i.e., the first-order moment), strive to maximize the amount of market-wide information, this objective yields the exponential distribution. Contrary to the conventional wisdom that security analysts specialize in the generation of firm-specific information, empirical findings suggest that analysts primarily produce market-wide information for 25 countries. Nevertheless, it remains unclear why cross-sectional analyst coverage reflects market-wide information, this paper provides an entropy-based explanation.

Simulating survival data when one subgroup lacks information

In this paper, we aim to show the process of simulating survival data when the distribution of the overall population and one subgroup (called "positive subgroup") as well as the proportion of the subgroup is known, while the distribution of the other subgroup (called "negative subgroup") is unknown. We propose a combination method which generates survival data of the positive subgroup and negative subgroup, respectively, and survival data of the overall population are the combination of the two subgroups. The parameters of the overall population and the positive subgroup need to satisfy certain constraints, otherwise the parameters may lead to contradictions. From simulation, we show that our proposed combination method can reflect the correlation between the test statistics of overall population and positive subgroup, which makes the simulated data more realistic and the results of simulation more reliable. Moreover, for a multiplicity control in trial design, the combination method can help to determine the α splitting strategy between primary endpoints, and is helpful in designs of clinical trials as shown in three applications.

A stationary Weibull process and its applications

In this paper we introduce a discrete-time and continuous state-space Markov stationary process { X n ; n = 1 , 2 , … } , where X n has a two-parameter Weibull distribution, X n 's are dependent and there is a positive probability that X n = X n + 1 . The motivation came from the gold price data where there are several instances for which X n = X n + 1 . Hence, the existing methods cannot be used to analyze this data. We derive different properties of the proposed Weibull process. It is observed that the joint cumulative distribution function of X n and X n + 1 has a very convenient copula structure. Hence, different dependence properties and dependence measures can be obtained. The maximum likelihood estimators cannot be obtained in explicit forms, we have proposed a simple profile likelihood method to compute these estimators. We have used this model to analyze two synthetic data sets and one gold price data set of the Indian market, and it is observed that the proposed model fits quite well with the data set.

Minimum Message Length Inference of the Exponential Distribution with Type I Censoring

Data with censoring is common in many areas of science and the associated statistical models are generally estimated with the method of maximum likelihood combined with a model selection criterion such as Akaike’s information criterion. This manuscript demonstrates how the information theoretic minimum message length principle can be used to estimate statistical models in the presence of type I random and fixed censoring data. The exponential distribution with fixed and random censoring is used as an example to demonstrate the process where we observe that the minimum message length estimate of mean survival time has some advantages over the standard maximum likelihood estimate.

Analyzing the outcomes of skeletal trauma within a forensic population: Potential issues and implications in inferential modeling of blunt force trauma

Analyzing complex skeletal trauma can present a challenge for forensic practitioners to reliably determine the causes and circumstances of traumatic injury. The forensic value of skeletal fracture pattern analysis can be diminished due to obscuration and similarity between injuries associated with various impact mechanisms (e.g., vertical vs horizontal blunt force) and can provide issues when questions arise surrounding circumstances of traumatic injuries. Using the Python coding language, code was written that segments traumatized regions of interest from post-mortem computed tomography (PMCT) scans, allowing the user to calculate percentage of fragmentation in the context of extreme trauma events. Using cases of known trauma which resulted in fragmentation of the femur, the objective was to assess if there were statistical differences in the fragmentation resulting from horizontal pedestrian motor vehicle impacts (PMVIs; n = 44) compared to vertical high impact falls (>3 m; n = 41). Results indicated that percentage differences between the PMVI group and high impact falls group were statistically significant. Although it was possible to develop a standardized method that records fragmentation, and results were significant in distinguishing between the two groups, the outcomes of the data follow an exponential distribution which has implications for how skeletal trauma is modeled.

Evaluation of the lifetime performance index on first failure progressive censored data based on Topp Leone Alpha power exponential model applied on HPLC data

In this paper, the statistical inference of the lifetime performance index for the first failure progressive censoring schemes for the Topp Leone Alpha power exponential distribution (TLAPE) which excluded from a new structure of distribution models called T-Alpha Power X (T-APX) family is introduced. The proposed statistical inference of the lifetime performance index is applied on the High-Performance Liquid Chromatography data of blood samples from organ transplant recipients which is known as HPLC. The goodness of fit criteria of TLAPE for HPLC data proved the potentiality of TLAPE compared with other well-known distributions. This result has an effect on achieving the required results in testing procedure of the Lifetime performance index. Moreover, the statistical and reliability characteristics of TLAPE are studied. A simulation study is performed to examine the performance of the ML parameter estimates in terms of bias and mean square error. Two real data sets for survival and breast cancer are modeled using the TLAPE distribution and compared with other well-known distributions, to illustrate its performance. The results emphasize that the TLAPE distribution has a superior fitting performance to cancer data than the compared distributions.

Estimating the sample variance from the sample size and range

For meta-analysis studies and systematic reviews, it is important to pool the data from a set of similar clinical trials. To pool the data, one needs to know their SD. Many trial reports, however, contain only the median, the minimum and maximum values, and the sample size. It is therefore important to be able to estimate the SD S from the sample size n and range r. For small n ≤ 100, we improve existing estimators of r/S, the "divisor," denoted by ξ ( n ) . This in turn yields improved estimators of the SD in the form S ^ = r / ξ ^ ( n ) on simulated as well as real datasets. We provide numerical values of the proposed estimator as well as approximation by a simple formula 3 ln ( n ) - 1 . 4025 . Furthermore, for large n, we provide estimators ξ ^ ( n ) of the divisor ξ ( n ) for the normal, exponential, and other bounded and unbounded distributions.

Expected Logarithm and Negative Integer Moments of a Noncentral χ2-Distributed Random Variable

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral χ2-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

On Leader-Following Consensus in Multi-Agent Systems with Discrete Updates at Random Times

This paper studies the leader-following consensus problem in continuous-time multi-agent networks with communications/updates occurring only at random times. The time between two consecutive controller updates is exponentially distributed. Some sufficient conditions are derived to design the control law that ensures the leader-following consensus is asymptotically reached (in the sense of the expected value of a stochastic process). The numerical examples are worked out to demonstrate the effectiveness of our theoretical results.

Allometric analysis using the multivariate shifted exponential normal distribution

In allometric studies, the joint distribution of the log-transformed morphometric variables is typically elliptical and with heavy tails. To account for these peculiarities, we introduce the multivariate shifted exponential normal (MSEN) distribution , an elliptical heavy-tailed generalization of the multivariate normal (MN). The MSEN belongs to the family of MN scale mixtures (MNSMs) by choosing a convenient shifted exponential as mixing distribution. The probability density function of the MSEN has a simple closed-form characterized by only one additional parameter, with respect to the nested MN, governing the tail weight. The first four moments exist and the excess kurtosis can assume any positive value. The membership to the family of MNSMs allows us a simple computation of the maximum likelihood (ML) estimates of the parameters via the expectation-maximization (EM) algorithm; advantageously, the M-step is computationally simplified by closed-form updates of all the parameters. We also evaluate the existence of the ML estimates. Since the parameter governing the tail weight is estimated from the data, robust estimates of the mean vector of the nested MN distribution are automatically obtained by downweighting; we show this aspect theoretically but also by means of a simulation study. We fit the MSEN distribution to multivariate allometric data where we show its usefulness also in comparison with other well-established multivariate elliptical distributions.

Unique sleep-stage transitions determined by obstructive sleep apnea severity, age and gender

In obstructive sleep apnea, patients' sleep is fragmented leading to excessive daytime sleepiness and co-morbidities like arterial hypertension. However, traditional metrics are not always directly correlated with daytime sleepiness, and the association between traditional sleep quality metrics like sleep duration and arterial hypertension is still ambiguous. In a development cohort, we analysed hypnograms from mild (n = 209), moderate (n = 222) and severe (n = 272) obstructive sleep apnea patients as well as healthy controls (n = 105) from the European Sleep Apnea Database. We assessed sleep by the analysis of two-step transitions depending on obstructive sleep apnea severity and anthropometric factors. Two-step transition patterns were examined for an association to arterial hypertension or daytime sleepiness. We also tested cumulative distributions of wake as well as sleep-states for power-laws (exponent α) and exponential distributions (decay time τ) in dependency on obstructive sleep apnea severity and potential confounders. Independent of obstructive sleep apnea severity and potential confounders, wake-state durations followed a power-law distribution, while sleep-state durations were characterized by an exponential distribution. Sleep-stage transitions are influenced by obstructive sleep apnea severity, age and gender. N2 → N3 → wake transitions were associated with high diastolic blood pressure. We observed higher frequencies of alternating (symmetric) patterns (e.g. N2 → N1 → N2, N2 → wake → N2) in sleepy patients both in the development cohort and in a validation cohort (n = 425). In conclusion, effects of obstructive sleep apnea severity and potential confounders on sleep architecture are small, but transition patterns still link sleep fragmentation directly to obstructive sleep apnea-related clinical outcomes like arterial hypertension and daytime sleepiness.

A new variable interval schedule with constant hazard rate and finite time range

We propose a new variable interval (VI) schedule that achieves constant probability of reinforcement in time while using a bounded range of intervals. By sampling each trial duration from a uniform distribution ranging from 0 to 2 T seconds, and then applying a reinforcement rule that depends linearly on trial duration, the schedule alternates reinforced and unreinforced trials, each less than 2 T seconds, while preserving a constant hazard function.

Stability of the Distribution of Patient Health Questionnaire-9 Scores Against Age in the General Population: Data From the National Health and Nutrition Examination Survey

Background: Epidemiological studies using the nine-item Patient Health Questionnaire (PHQ-9) have reported inconsistencies regarding the relationship between age and total scores. To determine whether this discrepancy is due to the stability of the distribution of PHQ-9 total scores against age, we investigated whether the total score distribution remains stable during adulthood, and also investigated the mathematical patterns of the total score distribution. Methods: The present study utilized data from 15,847 participants of the 2009-2014 United States National Health and Nutrition Examination Survey, all of whom responded to all PHQ-9 items. The stability of the total score distribution among different age groups was examined using overlap coefficients and graphical analysis. Results: High overlap coefficients were observed between all age groups for the distributions of PHQ-9 total scores, suggesting that the distribution of PHQ-9 total scores remains stable against age. Graphical analysis demonstrated that distributions of PHQ-9 total scores were similar across age groups. In addition, distributions of PHQ-9 total scores exhibited an exponential pattern, except at the lower end of the distribution. Conclusions: Our findings indicate that the stability of the distribution of PHQ-9 total scores throughout adulthood may underlie inconsistencies in the evidence regarding age-related changes in total depression scores.

Non-normal Distributions Commonly Used in Health, Education, and Social Sciences: A Systematic Review

Statistical analysis is crucial for research and the choice of analytical technique should take into account the specific distribution of data. Although the data obtained from health, educational, and social sciences research are often not normally distributed, there are very few studies detailing which distributions are most likely to represent data in these disciplines. The aim of this systematic review was to determine the frequency of appearance of the most common non-normal distributions in the health, educational, and social sciences. The search was carried out in the Web of Science database, from which we retrieved the abstracts of papers published between 2010 and 2015. The selection was made on the basis of the title and the abstract, and was performed independently by two reviewers. The inter-rater reliability for article selection was high (Cohen’s kappa = 0.84), and agreement regarding the type of distribution reached 96.5%. A total of 262 abstracts were included in the final review. The distribution of the response variable was reported in 231 of these abstracts, while in the remaining 31 it was merely stated that the distribution was non-normal. In terms of their frequency of appearance, the most-common non-normal distributions can be ranked in descending order as follows: gamma, negative binomial, multinomial, binomial, lognormal, and exponential. In addition to identifying the distributions most commonly used in empirical studies these results will help researchers to decide which distributions should be included in simulation studies examining statistical procedures.

Limit properties for ratios of order statistics from exponentials

In this paper, we study the limit properties of the ratio for order statistics based on samples from an exponential distribution and obtain the expression of the density functions, the existence of the moments, the strong law of large numbers for [Formula: see text] with [Formula: see text]. We also discuss other limit theorems such as the central limit theorem, the law of iterated logarithm, the moderate deviation principle, the almost sure central limit theorem for self-normalized sums of [Formula: see text] with [Formula: see text].

Item Response Patterns on the Patient Health Questionnaire-8 in a Nationally Representative Sample of US Adults

Recent studies have shown that item responses on the Center for Epidemiologic Studies Depression Scale (CES-D) and Kessler Screening Scale for Psychological Distress (K6) exhibit the same characteristic item response patterns among the general population. However, the distributional patterns of responses on the Patient Health Questionnaire-8 (PHQ-8) among the general population have not been adequately studied. Thus, we conducted a pattern analysis of PHQ-8 item responses among US adults. Data (18,446 individuals) were obtained from the 2015 Behavioral Risk Factor Surveillance Survey (BRFSS). Item responses on the BRFSS version of the PHQ-8 were scored using the number of days response set and then converted to the original 4-point scale. The patterns of item responses were analyzed through graphical analysis. Lines of item responses scored using the number of days response set showed the same pattern among the eight items, characterized by crossing at a single point between "0 days" and "1 day," and parallel fluctuation from "1 day" to "14 days" on a semi-logarithmic scale. Lines of item responses converted to the 4-point scale also showed the same characteristic pattern among the eight items. The present results demonstrate that the item responses on the PHQ-8 show the same characteristic patterns among items, consistent with the CES-D and the K6.

Effective dose conversion coefficients for radionuclides exponentially distributed in the ground

In order to provide fundamental data required for dose evaluation due to environmental exposures, effective dose conversion coefficients, that is, the effective dose rate per unit activity per unit area, were calculated for a number of potentially important radionuclides, assuming an exponential distribution in ground, over a wide range of relaxation depths. The conversion coefficients were calculated for adults and a new-born baby on the basis of dosimetric methods that the authors and related researchers have previously developed, using Monte Carlo simulations and anthropomorphic computational phantoms. The differences in effective dose conversion coefficients due to body size between the adult and baby phantoms were found to lie within 50 %, for most cases; however, for some low energies, differences could amount to a factor of 3. The effective dose per unit source intensity per area was found to decrease by a factor of 2-5, for increasing relaxation depths from 0 to 5 g/cm(2), above a source energy of 50 keV. It is also shown that implementation of the calculated coefficients into the computation of the tissue weighting factors and the adult reference computational phantoms of ICRP Publication 103 does not significantly influence the effective dose conversion coefficients of the environment. Consequently, the coefficients shown in this paper could be applied for the evaluation of effective doses, as defined according to both recommendations of ICRP Publications 103 and 60.

Focused attention vs. crossmodal signals paradigm: deriving predictions from the time-window-of-integration model

In the crossmodal signals paradigm (CSP) participants are instructed to respond to a set of stimuli from different modalities, presented more or less simultaneously, as soon as a stimulus from any modality has been detected. In the focused attention paradigm (FAP), on the other hand, responses should only be made to a stimulus from a pre-defined target modality and stimuli from non-target modalities should be ignored. Whichever paradigm is being applied, a typical result is that responses tend to be faster to crossmodal stimuli than to unimodal stimuli, a phenomenon often referred to as “crossmodal interaction.” Here, we investigate predictions of the time-window-of-integration (TWIN) modeling framework previously proposed by the authors. It is shown that TWIN makes specific qualitative and quantitative predictions on how the two paradigms differ with respect to the probability of multisensory integration and the amount of response enhancement, including the effect of stimulus intensity (“inverse effectiveness”). Introducing a decision-theoretic framework for TWIN further allows comparing the two paradigms with respect to the predicted optimal time window size and its dependence on the prior probability that the crossmodal stimulus information refers to the same event. In order to test these predictions, experimental studies that systematically compare crossmodal effects under stimulus conditions that are identical except for the CSP-FAP instruction should be performed in the future.

Boltzmann-type distribution of side-chain conformation in proteins

We analyze packing imperfections in globular proteins as reflected in deviations of torsion angles from the equilibrium values for the isolated side chains. The distribution of conformations of methionine and lysine residues in a database of high-resolution structures is compared with energies of model compounds calculated with high-level quantum-mechanics. The distribution of the C-C and C-S torsion angles (chi(3)) correlates well with the Boltzmann factor of the torsion energy, exp(-betaE) of the model compounds C(2)H(5)-C(2)H(5) and C(2)H(5)-S-CH(3). An exponential relation was again found between the relative occurrence of g+, g- and t conformations for C(alpha)-C(beta) bonds in long side chains and the energy differences of rotamers of alpha-amino n-butyric acid, when dependence on backbone conformation was taken into account. The distribution of all 27 rotamers of methionine was correlated with the energy differences between the model's rotamers, corrected for clashes with nearby residues, the correlation being good for a set with backbone in the beta-conformation, but less clear for backbone alpha-conformation. In all correlations, the value of the coefficient beta corresponds to a temperature of circa 300 K. These results can be interpreted with a model that considers the structure of a folded protein as resulting from packing imperfectly complementary parts, with a requirement of an overall low energy. Compromises are required to optimize the fit of nonbonded contacts with surrounding groups, and side chains assume conformations away from the energy minimum. An exponential distribution is a most probable distribution, and this can be established easily under conditions other than thermal equilibrium.