scholarly journals Tolerance limits and tolerance intervals for ratios of normal random variables using a bootstrap calibration

2017 ◽  
Vol 59 (3) ◽  
pp. 550-566 ◽  
Author(s):  
Marilena Flouri ◽  
Shuyan Zhai ◽  
Thomas Mathew ◽  
Ionut Bebu
Keyword(s):  
Random Variables ◽  
Tolerance Limits ◽  
Tolerance Intervals ◽  
Bootstrap Calibration

Tolerance Limits for a Ratio of Normal Random Variables

2009 ◽  
Vol 20 (1) ◽  
pp. 172-184 ◽  
Author(s):  
Lanju Zhang ◽  
Thomas Mathew ◽  
Harry Yang ◽  
K. Krishnamoorthy ◽  
Iksung Cho
Keyword(s):  
Random Variables ◽  
Tolerance Limits

Content‐adjusted tolerance intervals via bootstrap calibration

Stat ◽  
2021 ◽  
Author(s):  
Junjun Jiao ◽  
Xu Zhao ◽  
Weihu Cheng
Keyword(s):  
Tolerance Intervals ◽  
Bootstrap Calibration

scholarly journals On The Quotient of a Centralized and a Non-centralized Complex Gaussian Random Variable

2020 ◽  
Vol 125 ◽  
Author(s):  
Dazhen Gu
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Complex Signal ◽  
Tolerance Intervals ◽  
Practical Applications ◽  
Second Moments ◽  
Complex Gaussian Random Variable ◽  
The Mean ◽  
Theoretical Developments ◽  
Straightforward Extension

A detailed investigation of the quotient of two independent complex random variables is presented. The numerator has a zero mean, and the denominator has a non-zero mean. A normalization step is taken prior to the theoretical developments in order to simplify the formulation. Next, an indirect approach is taken to derive the statistics of the modulus and phase angle of the quotient. That in turn enables a straightforward extension of the statistical results to real and imaginary parts. After the normalization procedure, the probability density function of the quotient is found as a function of only the mean of the random variable that corresponds to the denominator term. Asymptotic analysis shows that the quotient closely resembles a normally-distributed complex random variable as the mean becomes large. In addition, the first and second moments, as well as the approximate of the second moment of the clipped random variable, are derived, which are closely related to practical applications in complex-signal processing such as microwave metrology of scattering-parameters. Tolerance intervals associated with the ratio of complex random variables are presented.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

On the Mathematical Basis of Zelen’s Prerandomized Designs

1985 ◽  
Vol 24 (03) ◽  
pp. 120-130 ◽  
Author(s):  
E. Brunner ◽  
N. Neumann
Keyword(s):  
Clinical Trial ◽  
Normal Distribution ◽  
Random Variables ◽  
Small Samples ◽  
Selection Effects ◽  
Sample Sizes ◽  
Mathematical Basis ◽  
Additional Assumptions

SummaryThe mathematical basis of Zelen’s suggestion [4] of pre randomizing patients in a clinical trial and then asking them for their consent is investigated. The first problem is to estimate the therapy and selection effects. In the simple prerandomized design (PRD) this is possible without any problems. Similar observations have been made by Anbar [1] and McHugh [3]. However, for the double PRD additional assumptions are needed in order to render therapy and selection effects estimable. The second problem is to determine the distribution of the statistics. It has to be taken into consideration that the sample sizes are random variables in the PRDs. This is why the distribution of the statistics can only be determined asymptotically, even under the assumption of normal distribution. The behaviour of the statistics for small samples is investigated by means of simulations, where the statistics considered in the present paper are compared with the statistics suggested by Ihm [2]. It turns out that the statistics suggested in [2] may lead to anticonservative decisions, whereas the “canonical statistics” suggested by Zelen [4] and considered in the present paper keep the level quite well or may lead to slightly conservative decisions, if there are considerable selection effects.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

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