Two-sided Power Distribution Research Articles

Bootstrap-based confidence intervals for the standard two-sided power distribution

The two-parameter standard two-sided power family of distributions on (0, 1) is considered in this article. We propose bootstrap standard errors for the maximum likelihood estimators, as well as bootstrap confidence intervals for its parameters, once these important statistical measures of accuracy cannot be computed based on first-order asymptotic theory. We consider Monte Carlo simulation experiments to verify the performance of the bootstrap methods, and the numerical results are quite promising. Applications to real data are also considered to illustrate the proposed methodology in practice.

System reliability optimization with two-sided power distributed component failure times

Many systems connected in series are subject to failure earlier in their design life. The failure rate of such systems has been described by a two-sided power distribution, which enables modeling the entirety of the bathtub curve failure modes with a single distribution. We propose a time-dependent reliability optimization model for improving a system’s reliability through the optimal allocation of redundant components whose failure distribution follows a two-sided power distribution. To solve the multi-criteria optimization model, we propose an algorithm that (i) identifies the scale tolerance at which a selected component will have a higher mean residual life, (ii) identifies the level of redundancy for enhancing system reliability, and (iii) identifies other optimal system characteristics, including minimized cost and minimized volume.

Parametric Probabilistic Forecasting of Solar Power With Fat-Tailed Distributions and Deep Neural Networks

The need of solar power uncertainty quantification in the power system has inspired probabilistic solar power forecasting. This paper proposes a novel multi-step parametric method for intra-day probabilistic solar power forecasting. First, statistical analysis on solar power distribution is done using four forecasting methods in real-world data. Fat tails are clearly found in solar power distribution, which could not be modelled by the widely-used normal distribution. In light of this discovery, two fat-tailed distributions, i.e., Laplace and two-sided power distributions, along with their generalized variants are then proposed to better model the conditional distribution of solar power output. Second, a recently proposed DeepAR model for time series probabilistic forecasting based on deep recurrent neural network is used to map various predictors into parameters of the fat-tailed distribution. Moreover, a novel loss function based on the continuous ranked probability score is proposed, and its closed-form formula over the proposed fat-tailed distributions is derived for efficient model training. Numerical results on public real-world data show that our method is very effective and the proposed model can provide intra-day probabilistic solar power forecasting with high quality and reliability.

Multicomponent stress-strength reliability estimation for the standard two-sided power distribution

A system of $k$ components that functions as long as at least $s$ components survive is termed as $s$-out-of-$k$:G system, where G refers to "good". In this study, we consider the $s$-out-of-$k$:G system when $X_{1}, X_{2},\cdots, X_{k}$ are independent and identically distributed strength components and each component is exposed to common random stress $Y$ when the underlying distributions all belong to the standard two-sided power distribution. The system is regarded as surviving only if at least $s$ out of $k$ $1<s<k$ strengths exceed the stress. The reliability of such a system is the surviving probability and is estimated by using the maximum likelihood and Bayesian approaches. Parametric and nonparametric bootstrap confidence intervals for the maximum likelihood estimates and the highest posterior density confidence intervals for Bayes estimates by using the Markov Chain Monte Carlo technique are obtained. A real data set is also analyzed to illustrate the performances of the estimators.

On the reliability characteristics of the standard two-sided power distribution

In this study, the standard two-sided power (STSP) distribution is considered with regard to statistical reliability analysis in detail. For this purpose, along with the reliability and hazard functions of the distribution, particular reliability indices that are useful in maintenance and replacement policies are obtained and they are evaluated with their plots. The STSP distribution is classified based on aging according to various cases of its parameters. Then, we studied the classical and Bayesian estimations of the reliability and hazard functions. In Bayesian estimation, symmetric and different asymmetric loss functions are considered. For obtaining the Bayes estimates, Monte Carlo Markov Chain simulation using the Gibbs algorithm is performed. Various simulation schemes are performed for comparing the performances of the estimators. Further, the Bayesian predictions of the future observations based on the observed samples are obtained. A real data example is used to illustrate the theoretical outcomes.

TWO-SIDED GENERALIZED HYPERBOLIC SECANT DISTRIBUTION WITH REAL DATA APPLICATIONS AND MODEL COMPARISONS

In this paper, we use the structure of the standard two-sided power distribution to implement a new version of the hyperbolic secant distribution. This new distribution is more flexible than the hyperbolic secant distribution when it comes to interpreting data presenting an abrupt change in values. In the first part of the paper, we show some of its properties, such as the shape behavior of the probability density and hazard rate functions, and the analysis of moment-type measures. Then, the statistical side of the underlying model is explored. We provide the maximum likelihood estimates for the model parameters, as well as an efficient algorithm to calculate them. After this, to demonstrate the potential of the proposed modeling strategy, we present three real data applications. The beta-normal, power-normal, Kuramaswamy-normal and two-sided generalized normal distribution models are considered as competitors. The results are favorable to the proposed model.

The correlation of cost and schedule variance in satellite programs: level of effort versus discrete cost accounts

This research analyzes cost account-level detail from ten similar satellite programs to assess the relationship between cost and schedule variances. A model is defined to break activity into two types: level of effort (LOE), in which cost is directly proportional to the cost account’s duration and discrete, where the cost is independent of the schedule. Primary data collected by the authors from the ten satellite programs are split into these two categories and analyzed separately. Marginal distributions for schedule and cost for each of the two databases are built using the generalized two-sided power distribution (GTSP). Correlations between cost and schedule variance are calculated for the LOE and discrete cost accounts. Finally, joint distributions are created for each database using generalized diagonal band copulas. The results show that the GTSP effectively modeled the marginal distributions and the generalized diagonal band copula with a slope generating function successfully representing the observed joint distribution. The analysis, resulting from the correlation values, show that the cost and schedule variance for discrete cost accounts are not correlated. However, the LOE cost accounts show a correlation well below the projected perfect correlation. The models were validated by removing one program’s data from the database, regenerating the models, and assessing the accuracy of the model against the program excluded from the model. The model derived from nine programs successfully modeled the results from the tenth program. These results provide both a modeling method and guidance for modeling parameters for joint cost and schedule risk analysis.

Matrix Variate Two-Sided Power Distribution

A matrix variate generalization of the two-sided power distribution is proposed. Several properties of this distribution including probability density function, cumulative distribution function and characteristic function are derived.

Determining reasonable bid ratios under two-sided power distribution assumption for public procurement under capped lowest-cost bidding system

This paper proposes a new definition of reasonable bid ratios - a ratio of bid price to published owner cost estimate - under the capped lowest-cost bidding system and provides a closed-form solution to determine the ratios for a given number of bidders. Bid ratios are assumed to be random, obeying two-sided power distribution (TSP). A reasonable bid ratio is found to decrease with increased number of bidders. The Monte Carlo simulations demonstrate that the computed reasonable bid ratios under the assumption of TSP distribution are proven to be very close to those under beta distribution and those from empirical data and are more accurate than those estimated from triangular distribution. Using Indonesian public procurement as a case study, this paper shows that reasonable bid ratios with uncertain numbers of bidders are lower than the traditional cut-off point. To make the proposed model operational, this paper also presents a simple protocol including the determination of minimum sample size required for TSP approximation. This paper concludes with model limitations and directions for future research.

Determining reasonable bid ratios under two-sided power distribution assumption for public procurement under capped lowest-cost bidding system

This paper proposes a new definition of reasonable bid ratios - a ratio of bid price to published owner cost estimate - under the capped lowest-cost bidding system and provides a closed-form solution to determine the ratios for a given number of bidders. Bid ratios are assumed to be random, obeying two-sided power distribution (TSP). A reasonable bid ratio is found to decrease with increased number of bidders. The Monte Carlo simulations demonstrate that the computed reasonable bid ratios under the assumption of TSP distribution are proven to be very close to those under beta distribution and those from empirical data and are more accurate than those estimated from triangular distribution. Using Indonesian public procurement as a case study, this paper shows that reasonable bid ratios with uncertain numbers of bidders are lower than the traditional cut-off point. To make the proposed model operational, this paper also presents a simple protocol including the determination of minimum sample size required for TSP approximation. This paper concludes with model limitations and directions for future research.

The two-sided power-binomial distribution for modeling binary count data

The two-sided power (TSP) distribution is a flexible two-parameter distribution having uniform, power function and triangular as sub-distributions, and it is a reasonable alternative to beta distribution in some cases. In this work, we introduce the TSP-binomial model which is defined as a mixture of binomial distributions, with the binomial parameter p having a TSP distribution. We study its distributional properties and demonstrate its use on some data. It is shown that the newly defined model is a useful candidate for overdispersed binomial data.

Stress–strength reliability estimation under the standard two-sided power distribution

• The stress–strength model based on the two-sided power distribution is considered. • An expression for the stress-strength reliability is obtained exactly. • Methods for estimating of the stress–strength reliability are given. • Comparison of estimation methods are made through simulation studies. • A real data application to see the performances of the estimators is given. In this report, the distribution for setting up a system reliability exposed to some stress is studied. The standard two-sided power distribution is assumed to be the underlying distribution. We obtained the exact expressions and estimates for the reliability by applying different methods such as maximum likelihood and Bayesian estimators. Three different scenarios were examined: known and equal reflection parameters, known but unequal reflection parameters, and all parameters are unknown, providing practical guidance and recommendations for the estimator design. For large samples, we recommend use of the parametric bootstrap method with the maximum likelihood estimate . Real data sets were used to illustrate the performances of the estimators.

Parametric methods for probabilistic forecasting of solar irradiance

Abstract This paper proposes two parametric probabilistic forecast methods using beta and two-sided power distributions to predict solar irradiance. It also evaluates their performance. To improve their performance metrics a hybrid procedure based on the beta transformed linear opinion pool is utilized. Our simulations show that these methods – despite their simple structure – can effectively forecast solar irradiance and accurately describe its stochastic characteristics. The proposed approach is flexible and could be extended to many different point forecast methods which otherwise minimize RMSE or MSE.

Moments of order statistics of the standard two-sided power distribution

ABSTRACTThe standard two-sided power distribution is a flexible distribution having uniform, power function and triangular as subdistributions, and it is a reasonable alternative to the Laplace distribution in some cases. In this work, computationally efficient expressions for moments of order statistics, expressions for L-moments, and asymptotic results for sample extrema are derived. Then a simulation study is performed for the location-scale estimation problem of a small data set by considering the maximum likelihood estimation method and the best linear unbiased estimation method based on the moments of order statistics.

Modeling reliability with a two-sided power distribution

ABSTRACTModeling the reliability of a system typically involves the piecewise combination of mathematically disparate functions covering distinct periods (e.g., decreasing, constant, and increasing time-dependent failure rates defining the traditional bathtub curve) in the life of an engineered system. This work explores the two-sided power distribution, a nonlinear extension of the triangular distribution, for describing failure time and rendering a statistically valid bathtub curve with a single probability density function. The parameters of the two-sided power distribution also provide some flexibility to describe decreasing and increasing failure rates with a single distribution that are not available with similar distributions.

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

ABSTRACTThe ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.

On General Continuous Triangular and Two-sided Power Distributions

In this article, we directly introduce the continuous version of the general discrete triangular distributions (Kokonendji and Zocchi, 2010). It is bounded and, in general, unimodal with pike. It contains thus a very useful class of two-sided power distributions (van Dorp and Kotz, 2002a,b, 2003). Moments, particular cases, limit distributions, and relations between parameters are straightforwardly derived.

Distribution of the product of a pair of independent two-sided power variates

ABSTRACTThe distributions of algebraic functions of random variables are important in theory of probability and statistics and other areas such as engineering, reliability, and actuarial applications, and many results based on various distributions are available in the literature. The two-sided power distribution is defined on a bounded range, and it is a generalization of the uniform, triangular, and power-function probability distributions. This paper gives the exact distribution of the product of two independent two-sided power-distributed random variables in a computable representation. The percentiles of the product are then computed, and a real data application is given.

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.

The Bayes Premium in an Aggregate Loss Poisson-Lindley Model with Structure Function STSP

Many premium calculating problems in actuarial science consider the number of claims, denoted as K, as the variable risk. Traditionally, this random variable is modelled by the Poisson distribution. However, it is well known that automobile insurance portfolios are characterized by zero-inflation (high percentage of zero values in the sample) and overdispersion (the variance is greater than the mean), and the Poisson distribution does not properly reflect the last phenomenon. In this paper we determine the Bayes premium considering that K follows a Poisson-Lindley distribution, with parameter $\theta_{1}$ in $[0,1]$, which is a potential alternative to describe these situations. As the structure function for $\theta_{1}$ we elicit the standardized two-sided power distribution, which is a reasonable alternative to the usual beta distribution. In addition, an aggregate loss model is considered with primary distribution given by the Poisson-Lindley distribution. A Bayesian analysis is developed to obtain the Bayes premium. The conclusion is that the STSP is not an adequate alternative in the problem in question because it is more informative and less dispersed than the Beta distribution.