In this article, we study a novel extension of the classic balancing numbers, referred to as the higher-order balancing numbers and denoted by. This sequence is analogous to the higher-order Fibonacci numbers and follows the same recurrence relation as the balancing sequence itself. The case k=1 gives the classic balancing numbers (A001109) and for k=2 gives the sequence A029547, thus establishing a direct link to existing number sequences. Here, we first establish the Binet-like formula and then, with its help, present various algebraic properties of this newly introduced sequence, such as recurrence relations, generating functions (both ordinary and exponential), partial sums, binomial sums, combined identities, and more. We also obtain the limiting ratio and establish several well-known identities, including Catalan’s identity, d’Ocagane’s identity, Vajda’s identity, Honsberger’s identity, using the Binet-like formula. Finally, we give some mixed identity and series sum formulae. In this study, the obtained identities and algebraic properties are expressed in terms of the existing balancing and Lucas-balancing numbers.
We study a fast local-global window-based attention method to accelerate Informer for long sequence time-series forecasting (LSTF) in a robust manner. While window attention being local is a considerable computational saving, it lacks the ability to capture global token information which is compensated by a subsequent Fourier transform block. Our method, named FWin, does not rely on query sparsity hypothesis and an empirical approximation underlying the ProbSparse attention of Informer. Experiments on univariate and multivariate datasets show that FWin transformers improve the overall prediction accuracies of Informer while accelerating its inference speeds by 1.6 to 2 times. On strongly non-stationary data (power grid and dengue disease data), FWin outperforms Informer and recent SOTAs thereby demonstrating its superior robustness. We give mathematical definition of FWin attention, and prove its equivalency to the canonical full attention under the block diagonal invertibility (BDI) condition of the attention matrix. The BDI is verified to hold with high probability on benchmark datasets experimentally.
Denis Veliu, Aranit Shkurti, Antonio Luciano Martire
In minimal risk portfolios, costs associated with transactions are essential in calculating the net performance. The transaction costs of maintaining such portfolios are predominantly negative due to the fact that traditional portfolio optimization strategies, which focus solely on risk and return, neglecting transaction costs typically incurred through rebalancing. This research analyzes the impact of costs associated with transactions while constructing minimum risk portfolios centered around risk parity models and provides a way to control those costs. We investigate the performance of portfolios under fixed and flexible costs and include these parameters in the optimization model to achieve more realistic results. Applying real-world data on conventional stock portfolios and highly volatile cryptocurrency markets, we demonstrate the performance of mean–variance optimization (M-V), risk parity with standard deviation (RP-Std), and risk parity with Conditional Value at Risk (RP-CVaR) through empirical data for both stock portfolios and cryptocurrencies. We found that potential transaction costs can cause portfolio returns to change by anywhere between 0.5 to 2% per year depending on how often one trades, and market conditions. By highlighting how crucial it is to incorporate transaction costs into the decision-making process, this study contributes to the expanding literature of research on portfolio optimization. For investors looking to create and manage their portfolios in a way that balances risk, return, and cost effectiveness, our findings offer useful insights. Future studies might investigate adaptive models that dynamically adapt to shifting cost structures and market situations, or they could generalize these findings to other asset classes.
This paper proposes a new control chart based on the two-parameter standard two-sided power distribution for monitoring rates and proportions, that is, when the quality characteristic of interest belongs to the unit interval (0,1). Control charts based on the well-known beta and Kumaraswamy distributions are usually considered to deal with this kind of data. The standard two-sided power distribution
has many similarities to the beta and Kumaraswamy distributions and a number of advantages in terms of tractability. We evaluate and compare the performance of the new control chart with the beta and Kumaraswamy control charts through Monte Carlo simulation experiments. The simulation results reveal that the control chart based on the standard two-sided power distribution outperforms the beta and Kumaraswamy control charts in terms of run length analysis. An empirical application to a real data set is considered to
illustrate the new control chart in practice, and comparisons with the two most traditional control charts for rates and proportions (beta and Kumaraswamy) are made.
Changes in the Consumer Price Index (CPI) over time reflect the rate of increase (inflation) or decrease (deflation) of goods and services for daily household needs. The CPI and inflation serve as barometers for economic growth stability, as controlled inflation can increase people's purchasing power over time. According to the Central Statistics Agency (2023), in December, the year-on-year (y-o-y) inflation for five cities in South Sulawesi (Bulukumba, Watampone, Makassar, Parepare, and Palopo) was 2.81 percent, with a CPI of 117.35. Of the five cities, the highest y-o-y inflation occurred in Makassar at 2.89 percent, with a CPI of 117.49, while the lowest y-o-y inflation occurred in Palopo at 2.21 percent, with a CPI of 115.60. CPI forecasting is one way to predict future inflation values. This study aims to develop the best GSTAR model for forecasting CPI data for five cities in South Sulawesi, a topic that has not been extensively covered in previous research. The goal is to provide valuable information for maintaining CPI stability in South Sulawesi and to support the formulation of better economic policies. The study focuses on five cities within South Sulawesi, where direct relationships between cities are possible, allowing the spatial model to be limited to the first-order. The data used in this study consists of monthly CPI data from January 2014 to March 2023. The location weights used in the model include uniform weights, inverse distances, and normalized cross-correlations. The model development steps include testing for data stationarity, determining the space-time sequence, calculating location weights, estimating parameters, testing model adequacy, comparing Root Mean Square Error (RMSE), and selecting the best model for forecasting. The best GSTAR model found is GSTAR (1;1)-I(2) with inverse distance weighting, which yielded the smallest RMSE value. The results show that the forecasted values closely match the actual values for each city from March to September 2023.
<p>In this study we detect and quantify changes in the distribution of the annual maximum daily maximum temperature (TXx)
in a large observation-based gridded data set of European daily temperature during the years 1950–2018. Several statistical models are considered, each of which analyses TXx using a generalized extreme-value (GEV) distribution with the GEV parameters varying smoothly over space.
In contrast to several previous studies which fit independent GEV models at the grid-box level, our models pull information from neighbouring grid boxes for more efficient parameter estimation. The GEV location and scale parameters are allowed to
vary in time using the log of atmospheric <span class="inline-formula">CO<sub>2</sub></span> as a covariate.
Changes are detected most strongly in the GEV location parameter, with the TXx distributions generally shifting towards hotter temperatures. Averaged across our spatial domain, the 100-year return level of TXx based on the 2018 climate
is approximately 2 <span class="inline-formula"><sup>∘</sup></span>C (95 % confidence interval of <span class="inline-formula">[2.03,2.12]</span> <span class="inline-formula"><sup>∘</sup></span>C) hotter than that based on the 1950 climate. Moreover, averaged across our spatial domain, the 100-year return level of TXx based on the 1950 climate corresponds approximately to a 6-year return level in the 2018 climate.</p>
We provide a computational exercise suitable for early introduction in an undergraduate statistics or data science course that allows students to “play the whole game” of data science: performing both data collection and data analysis. While many teaching resources exist for data analysis, such resources are not as abundant for data collection given the inherent difficulty of the task. Our proposed exercise centers around student use of Google Calendar to collect data with the goal of answering the question “How do I spend my time?” On the one hand, the exercise involves answering a question with near universal appeal, but on the other hand, the data collection mechanism is not beyond the reach of a typical undergraduate student. A further benefit of the exercise is that it provides an opportunity for discussions on ethical questions and considerations that data providers and data analysts face in today’s age of large-scale internet-based data collection.
Probabilities. Mathematical statistics, Special aspects of education
This paper concerns kernel-type ridge estimators of parameters in a semiparametric model. These estimators are a generalization of the well-known Speckman’s approach based on kernel smoothing method. The most important factor in achieving this smoothing method is the selection of the smoothing parameter. In the literature, many selection criteria for comparing regression models have been produced. We will focus on six selection criterion improved version of Akaike information criterion (AICc), generalized cross-validation (GCV), Mallows’ Cp criterion, risk estimation using classical pilots (RECP), Bayes information criterion (BIC), and restricted maximum likelihood (REML). Real and simulated data sets are considered to illustrate the key ideas in the paper. Thus, suitable selection criterion are provided for optimum smoothing parameter selection.
The term survival analysis has been used in a broad sense to describe collection of statistical procedures for data analysis for which the outcome variable of interest is time until an event occurs, the time to failure of an experimental unit might be censored and this can be right, left, interval, and Partly Interval Censored data (PIC). In this paper, the analysis of this model is conducted based on parametric Weibull model via PIC data. Moreover, two imputation techniques are used, which are: left point and right point. The effectiveness of the proposed model is tested through numerical analysis on simulated and secondary data sets.
Gauss M. Cordeiro , Emrah Altun , Mustafa Ç. Korkmaz
et al.
In this paper, a new family of distributions with one extra shape parameter, called the xgamma-G, is proposed. comprehensive treatment of some of its mathematical properties including ordinary and incomplete moments and quantile and generating functions are derived. The unknown model parameters are estimated by the maximum likelihood method and the performance of the maximum likelihood estimators are assessed via two extensive simulation studies. Additionally, the log-location-scale regression model for censored data based on a special member of the family is introduced. The usefulness of the proposed models is illustrated utilizing three real data sets.
G.A. Yessenbayeva, D.N. Yesbayeva, N.K. Syzdykova
et al.
The article is devoted to the application of the collocation method to solving differential equations, which are the basis for calculating many problems of mechanics. In this article the structure of this method is presented, its main components are highlighted; its types are characterized, as well as its classical approaches. The research of the problem of rectangular plates bending is carried out by the method of collocations in this article. The collocation method, like all numerical - analytical approximate methods, has a number of advantages and disadvantages, which are also noted in this article. The article is focused mainly on mechanics, engineers and technical specialists.
Export contributes to a large extent to economic growth of an island-type economylikeTaiwan. The scientific forecasting on the total value of imports and exports of top traded commodities in Taiwan are needed as the essential inputs to determine whether new top traded commodities should be imported or exported, and to make right decision toward activities in various functional areas such as building new container terminals, operation plans, marketing strategies, as well as finance and accounting [1]. Taking the original data of the amount of import and export commodity during the years from 2007 to 2013, the author tries to establish a mathematical model of Grey forecasting to make a prediction of the total value of imports and exports of top commodities in Taiwan for the next 05 coming years from 2014 to 2018.The analysis results show that the usage of Grey forecasting models resulted in a very low mean absolute percentage error, which demonstrate its applicability in practice to provide accurate forecasts. This research also indicates that for the future period of time (2014-2017), there will be a steady increase in both exports and imports value of all top commodities. The current study may offer a good idea for the control and scheduling for the terminal operators in decision making and planning.
In this paper, we study the estimation problems for the generalized inverted exponential distribution based on progressively type-II censored order statistics and record values. We establish some theorems to construct the exact confidence intervals and regions for the parameters. Monte Carlo simulation studies are used to assess the performance of our proposed methods. Simulation results show that the coverage probabilities of the exact confidence interval and the exact confidence region are all close to the desired level. Finally, two numerical examples are presented to illustrate the methods developed here.
Recently, Mahmoudi and Mahmoodian [7] introduced a new class of distributions which contains univariate normal–geometric distribution as a special case. This class of distributions are very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the normal–geometric distribution marginals. Different properties of this new bivariate distribution have been studied. This distribution has five unknown parameters. The EM algorithm is used to determine the maximum likelihood estimates of the parameters. We analyze one series of real data set for illustrative purposes.
Based on the goodness of fit approach, a new test is presented for testing exponentiality against "exponential Better than Used in moment generating function ordering class" (<i>EBU<sub>mgf</sub></i>). The critical values and the powers of this test are calculated. It is shown that the proposed test enjoys good power and performs better than previous tests in terms of power and Pitman’s asymptotic efficiencies for several alternative. Finally sets of real data are used as examples to elucidate the use of the proposed test in practical application.
Agung Waluyo, Moch. Abdul Mukid, Triastuti Wuryandari
Credit is the greatest asset managed the bank and also the most dominant contributor to the bank’s revenue. Debtors to pay their credit to the bank may smoothly or jammed. This study aims to identify the variables that affect a debtor’s credit status and compare the acuration of classification method both classification and regression trees (CART) and logistic regression. The variables used were debtor’s gender, education level, occupation, marital status, and income. By using logistic regression, it was known that only the debtor’s income effect their credit status with the classification accuration reach into 80%. By using CART, there were some variables affect the credit status and the classification accuration 80,9%. This paper showed that the performance of CART in classifying the credit status of debtors was better than logistic regression.
Keywords: Credit Status, Logistic Regression, CART
We extend the results of Gupta and Liang (1998), derived for location parameters, to obtain lower confidence bounds for the probability of correctly selecting the t best populations (PCSt) simultaneously for all t = 1, …, k − 1 for the general scale parameter models, where k is the number of populations involved in the selection problem. The application of the results to the exponential and normal probability models is discussed. The implementation of the simultaneous lower confidence bounds for PCSt is illustrated through real‐life datasets.
SummaryThe calculation of multivariate normal orthant probabilities is practically impossible when the number of variates is greater than five or six, except in very special cases. A transformation of the integral is obtained which enables quite accurate Monte Carlo estimates to be obtained for a fairly high number of dimensions, particularly if control variates are used.