Assessing Univariate and Multivariate Normality, A Guide For Non-Statisticians

Most parametric methods rely on the assumption of normality. Results obtained from these methods are more powerful compared to their non-parametric counterparts. However for valid inference, the assumptions underlying the use of these methods should be satisfied. Many published statistical articles that make use of the assumption of normality fail to guarantee it. Hence, quite a number of published statistical results are presented with errors. As a way to reduce this, various approaches used in assessing the assumption of normality are presented and illustrated in this paper.   In assessing both univariate and multivariate normality, several methods have been proposed. In the univariate setting, the Q-Q plot, histogram, box plot, stem-and-leaf plot or dot plot are some graphical methods that can be used. Also, the properties of the normal distribution provide an alternative approach to assess normality. The Kolmogorov-Smirnov (K-S) test, Lilliefors corrected K-S test, Shapiro-Wilk test, Anderson-Darling test, Cramer-von Mises test, D'Agostino skewness test, Anscombe-Glynn kurtosis test, D'Agostino-Pearson omnibus test, and the Jarque-Bera test are also used to test for normality. However, Kolmogorov-Smirnov (K-S) test, Shapiro-Wilk test, Anderson-Darling test, and Cramer-von Mises test are widely used in practice and implemented in many statistical applications. For multivariate normal data, marginal distribution and linear combinations should also be normal. This provides a starting point for assessing normality in the multivariate setting. A scatter plot for each pair of variables together with a Gamma plot (Chi-squared Q-Q plot) is used in assessing bivariate normality. For more than two variables, a Gamma plot can still be used to check the assumption of multivariate normality. Among the many test proposed for testing multivariate normality, Royston's and Mardia's tests are used more often and are implemented in many statistical packages. When the normality assumption is not justifiable, techniques for non-normal data can be used. Likewise, transformation to near normality is another alternative. Keywords: Univariate normal, Multivariate normal, Q-Q plot, Gamma plot, Kolmogorov-Smirnov test, Shapiro-Wilk test, Mardia's test, Royston's test

A One-Sample Test for Normality with Kernel Methods

We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure

Financiamiento por el Programa Reactiva Perú y el efecto en la rentabilidad en las pequeñas empresas de textiles, Gamarra 2020

La presente investigación tiene como objetivo principal determinar el efecto del financiamiento en la rentabilidad, beneficio economico con la rentabilidad en las pequeñas empresas de textiles. Es una investigación de método descriptivo correlacional, diseño no experimental, enfoque cuantitativo y de tipo aplicada, la población de estudio se ha determinado a través del censo la cual está conformada por 33 trabajadores del sector textil, se empleó la encuesta como instrumento validado con el Alfa de Cronbach para recoger la información que permite medir las variables en estudio. Se procesó la información en el programa estadístico IBM SPSS Statistics 25, determinando que el financiamiento tiene efecto en la rentabilidad, y para determinar el efecto se utilizó la prueba Eta para determinar la dependencia de las variables. Finalmente, se concluye que existe efecto entre el financiamiento y la rentabilidad ya que las empresas textiles han identificado las ventajas que tiene el proceso de financiamiento si se lleva una correcta gestión financiera, sin embargo, los gestores de estas empresas no utilizan estas alternativas

Non-Linear Optimization Applied to Angle-of-Arrival Satellite Based Geo-Localization for Biased and Time Drifting Sensors

Multiple sensors are used in a variety of geolocation systems. Many use Time Difference of Arrival (TDOA) or Received Signal Strength (RSS) measurements to locate the most likely location of a signal. When an object does not emit a classical RF signal, Angle of Arrival (AOA) measurements become more feasible than TDOA or RSS measurements. AOA measurements can be created from any sensor platform with any sort of camera. When location and attitude knowledge of the sensor passive objects can be tracked. A Non-Linear Optimization (NLO) method for calculating the most likely estimate from AOA measurements has been created in previous work. This thesis, modifies that algorithm to automatically correct AOA measurement errors by estimating the inherent bias and timedrift in the Inertial Measurement Unit (IMU) of the AOA sensing platform. Two methods are created to correct the sensor bias. One method corrects the sensor bias in post processing while treating the previous NLO method as a module. The other method directly corrects the sensor bias within the NLO algorithm by incorporating the bias parameters as a state vector in the estimation process. These two methods are analyzed using various Monte-Carlo simulations to check the general performance of the two modifications in comparison to the original NLO algorithm. These methods appear to improve performance by 10 − 60% depending on the data

Signal Processing in Arrayed MIMO Systems

Multiple-Input Multiple-Output (MIMO) systems, using antenna arrays at both receiver and transmitter, have shown great potential to provide high bandwidth utilization efficiency. Unlike other reported research on MIMO systems which often assumes independent antennas, in this thesis an arrayed MIMO system framework is proposed, which provides a richer description of the channel charac- teristics and additional degrees of freedom in designing communication systems. Firstly, the spatial correlated MIMO system is studied as an array-to-array system with each array (Tx or Rx) having predefined constrained aperture. The MIMO system is completely characterized by its transmit and receive array man- ifolds and a new spatial correlation model other than Kronecker-based model is proposed. As this model is based on array manifolds, it enables the study of the effect of array geometry on the capacity of correlated MIMO channels. Secondly, to generalize the proposed arrayed MIMO model to a frequency selective fading scenario, the framework of uplink MIMO DS-CDMA (Direct- Sequence Code Division Multiple Access) systems is developed. DOD estimation is developed based on transmit beamrotation. A subspace-based joint DOA/TOA estimation scheme as well as various spatial temporal reception algorithms is also proposed. Finally, the downlink MIMO-CDMA systems in multiple-access multipath fading channels are investigated. Linear precoder and decoder optimization problems are studied under different criterions. Optimization approaches with different power allocation schemes are investigated. Sub-optimization approaches with close-form solution and thus less computation complexity are also proposed

Influence of Transformational Leadership on Regional Commissioners’ Offices Performance in Tanzania

This study aimed at examining the influence of transformational leadership (TL) on regional commissioners` offices performance in Tanzania (RCOP). Specifically, the study focused: i) to examine the influence of inspirational motivation on RCOP performance in Tanzania; ii) to determine the influence of individualized consideration on RCOP in Tanzania; iii) to analyze the influence of intellectual stimulation on RCOP in Tanzania and; iv) to investigate the influence of idealized influence on RCOP in Tanzania. The study employed quantitative research methods where survey design was used. The targeted sample size of the study was 360 respondents. Multi stage sampling technique applied to draw the study subjects. Data were collected in eight regional commissioners` offices whereby each zone was represented by one region. Data analyses were done quantitatively using structural equation modeling. The findings of the study indicate that individualized consideration and idealized influence have significant influence on RCOP in Tanzania. Contrary to that, inspirational motivation and intellectual stimulation found with insignificant influence on RCOP in Tanzania. The study concludes that transformational leadership partially influences RCOP in Tanzania. The study recommends that individualized consideration and idealized influence should be used to the maximum potential in order to improve RCOP in Tanzanian context. The key words in this study are leadership, transformational leadership and Regional commissioners` offices performance