Linear Forms in the Order Statistics from an Exponential Distribution

In this paper, a new continuous probability distribution is proposed for fitting real data using Biweight kernel function and the exponential distribution. The suggested distribution is named the Biweight-exponential distribution (BiEd). Some statistical properties of this distribution are derived and illustrated mathematically. The probability density function and the cumulative distribution function are derived. Some reliability analysis functions are defined. The moments and moment generating function are derived. Re’nyi entropy is derived. The maximum likelihood method of estimation is used to derive the parameter estimates. The Bonferroni and Lorenz curves and Gini index equations are derived. The distribution of the order statistic and the quantile function are derived as well. The mean and median absolute deviations of the new distribution are derived. A numerical study was conducted to the quantile equation. An application to real data set is conducted to investigate the usefulness of the suggested distribution. In the real data application, the values of Cramer-von misses (W), Anderson Darling statistic (A), Kolmogorov-Smirnov (D) statistic, the p-value, the maximum likelihood estimates (MLE), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information criterion (HQIC) and minimum value of the log-likelihood function are obtained. The distribution is compared with the exponential (base) distribution based on these criteria. The results showed that the BiED fits better than the exponential distribution. This means that the suggested distribution can replace the exponential distribution for analysis of some real data sets.

The electronics industry has heavily prioritized enhancing the quality, lifetime and conforming rate (conforming to specifications) of electronic components. Various methods have been developed for assessing quality performance. In practice, process capability indices (PCIs) are used as a means of measuring process potential and performance. Moreover, most PCIs have been developed or investigated under the assumption that electronic components have a lifetime with a normal distribution. However, PCIs for non‐normal distributions have seldom been discussed. Nevertheless, the lifetime of electronic components generally may possess an exponential, gamma or Weibull distribution and so forth. Under an exponential distribution, some properties of the PCIs and their estimators differ from those in a normal distribution. To utilize the PCIs more reasonably and accurately in assessing the lifetime performance of electronic components, this study constructs a uniformly minimum variance unbiased (UMVU) estimator of their lifetime performance index under an exponential distribution. The UMVU estimator of the lifetime performance index is then utilized to develop the hypothesis testing procedure. The purchasers can then employ the testing procedure to determine whether the lifetime of the electronic components adheres to the required level. Manufacturers can also utilize this procedure to enhance process capability.

A return on investment analysis of applying health monitoring to LED lighting systems

Abstract. Exponential family statistical distributions, including the well-known normal, binomial, Poisson, and exponential distributions, are overwhelmingly used in data analysis. In the presence of covariates, an exponential family distributional assumption for the response random variables results in a generalized linear model. However, it is rarely ensured that the parameters of the assumed distributions are stable through the entire duration of the data collection process. A failure of stability leads to nonsmoothness and nonlinearity in the physical processes that result in the data. In this paper, we propose testing for stability of parameters of exponential family distributions and generalized linear models. A rejection of the hypothesis of stable parameters leads to change detection. We derive the related likelihood ratio test statistic. We compare the performance of this test statistic to the popular normal distributional assumption dependent cumulative sum (Gaussian CUSUM) statistic in change detection problems. We study Atlantic tropical storms using the techniques developed here, so to understand whether the nature of these tropical storms has remained stable over the last few decades.

Some Thoughts About Data Type, Distribution, and Statistical Significance

We examine the accuracy and precision of parameter estimates for both the exponential and normal distributions when using only a collection of sample extremes. That is, we consider a collection of random variables, where each of the random variables is either the minimum or maximum of a sample of nj independent, identically distributed random variables drawn from a normal or exponential distribution with unknown parameters. Previous work derived estimators for the population parameters assuming the nj sample sizes are constant. Since sample sizes are often not constant in applications, we derive new unbiased estimators that take into account the varying sample sizes. We also perform simulations to assess how the previously derived estimators perform when the constant sample size is simply replaced with the average sample size. We explore how varying the mean, standard deviation, and probability distribution of the sample sizes affects the estimation error. Overall, our results demonstrate that using the average sample size in place of the constant sample size still results in reliable estimates for the population parameters, especially when the average sample size is large. Our estimation framework is applied to a biological example involving plant pollination.

1 Order statistics in exponential distribution

Theoretical analysis of particle number density in steady aeolian saltation

Shape adaptive estimation using two normal distributions ECG

The exponential distribution remains a cornerstone of probability theory, particularly valued in reliability engineering, survival analysis, and risk management for its computational efficiency and distinctive memoryless property. While mathematically elegant, this distribution's constrained single-parameter framework presents significant limitations when modeling complex empirical phenomena, especially those displaying non-monotonic failure patterns. Academic literature documents numerous extensions to address these constraints, including the inverse exponential distribution and its exponentiated variants, which achieve greater modeling flexibility by introducing additional shape parameters. Building upon these developments, the current study presents an enhanced probability model through the innovative synthesis of the Inverse Exponential distribution with the Sine Topp-Leone Exponentiated-G (STLEG) family, yielding the novel Sine Topp-Leone Exponentiated Inverse Exponential (STLEIE) Distribution. Our investigation encompasses a rigorous analysis of the model's fundamental statistical properties, including its survival characteristics, hazard function behavior, moment structure, quantile properties, and order statistics. For parameter estimation, we implement and compare two established methodologies: the widely-used Maximum Likelihood Estimation (MLE) approach and the Maximum Product of Spacing (MPS) technique, both renowned for their statistical efficiency. Comprehensive Monte Carlo simulations demonstrate the consistent performance of these estimators, with both bias and Root Mean Squared Error (RMSE) metrics showing progressive improvement as sample sizes increase. Empirical validation through application to two real-world datasets reveals the STLEIE distribution's superior modeling capabilities compared to existing alternatives, as evidenced by multiple goodness-of-fit criteria. The concluding comparative analysis substantiates the practical utility of the proposed model, confirming its enhanced flexibility and improved performance across diverse data scenarios, thereby offering researchers and practitioners a more robust analytical tool for complex statistical modeling applications.

For i.i.d. random variables x1,x2,… with positive mean, finite variance and exponential right tail distribution, asymptotic expansions up to vanishing terms will be derived for the variance of first passage times of the form T = T(b) = inf{n ≧ 1: sn > nf(b/n)}, b ≧ 0, where sn = x1 +…+ xn and f is a strictly increasing, positive and three times continuously differentiable function on (0,∞). In particular, it will be shown that the excess over the boundary sT − Tf(b/T) is exponentially distributed and independent of T extending a result which is known when f(x) = x.

Bounds for mixtures of order statistics from exponentials and applications

Research in event cognition has shown that how people segment events plays an important role in how they are perceived and remembered. One potential source of information for anticipating upcoming event boundaries is the typical duration of the event. The small amount of prior work that exists suggests that the durations of daily events have normal distributions and that people have accurate prior beliefs about these durations (Griffiths & Tenenbaum, 2006). This research study aims to further explore the temporal distribution of self-reported events from daily life. Forty-eight participants provided information regarding the duration of events from their daily life for 14 days. Event durations for various activity types were characterised using duration modelling with truncated normal, exponential, and gamma models. Predictions based on mean and median values, including ± 5% and ± 10% error margins, were obtained, and the cumulative hazard functions were derived. Exponential and gamma distributions fit most event categories better than the truncated normal distribution. Mean-based predictions of boundaries based on duration showed only 4–5% accuracy, indicating limited usefulness. This aligns with the cumulative hazard results, where most events followed the exponential distribution, suggesting no optimal time for predicting the end of the event. This suggests that when daily events are studied in an ecological context, most of them have little sign of a typical duration. Consequently, it is unlikely that duration estimation plays a large role in the anticipation of event boundaries.

The normal distribution is one of the most popular probability distributions with applications to real life data. In this research paper, an extension of this distribution together with Weibull distribution called the Weimal distribution which is believed to provide greater flexibility to model scenarios involving skewed data was proposed. The probability density function and cumulative distribution function of the new distribution can be represented as a linear combination of exponential normal density functions. Analytical expressions for some mathematical quantities comprising of moments, moment generating function, characteristic function and order statistics were presented. The estimation of the proposed distribution’s parameters was undertaken using the method of maximum likelihood estimation. Two data sets were used for illustration and performance evaluation of the proposed model. The results of the comparative analysis to other baseline models show that the proposed distribution would be more appropriate when dealing with skewed data.

For a non-negative continuous random variable , Chaudhry and Zubair (2002, p. 19) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed by the proposed characterization techniques.