Theoretical Statistics

We study the problem of estimating the number of faces 𝑁 of a fair die when 𝑁 is unknown and only a small sample of rolls is observed. Modeling the die rolls as independent draws from the discrete uniform distribution on {1, … , 𝑁}, we show that the sample maximum 𝑀 is both a sufficient and complete statistic for 𝑁. This allows us to construct the unique uniformly minimum variance unbiased estimator (UMVUE) after deriving an unbiased estimator based on 𝑀. We derive an explicit closed form for the UMVUE and prove its unbiasedness and optimality.

In a recent paper Muirhead (1986) derived certaiu useful ideutities iavolviag expectatione taken with respect to the wishart distribution. This uote generalizes the above results by taking expectations with respect to a geaeralized. version of the wishart distributioa, coosidered by Sutradhar and Ali (1ggg), based on a multivariate t-distribution.

For high-dimensional data sets two-step model building algorithms usually select more important active covariates with fewer noise variates than one-step model building algorithms in liner models. In most of the previous studies, "learning curve", a manual procedure, has been used to determine the size of the reduced set in two-step model building procedure. In this paper, we propose a computer based automatic generation procedure instead of using this manual procedure to determine the size of the "reduced set" in two-step model building procedure. That is, the "reduced set" will be created automatically by computer based automatic generation procedure whichever occurs first from either or number of target variables = 8, and final model is obtained from the reduced set. We also made a comparison between one-step model building procedure and two-step automated robust model building procedure for clean data as well as contaminated data through a simulation study and real data applications. The performance of two-step automated robust model building procedure is much better than the robust one-step model building procedure in contaminated data as well as clean data.

This study compares the performance of various calibration ratio estimators in estimating the mean weight of newborn babies at the maternity ward of Federal Medical Centre (FMC), Umuahia, Abia State Nigeria, under stratified random sampling. Data were collected on maternal age, height, weight, and baby-related variables such as weight, gestational age and height. The maternal body mass index (BMI) was utilized as the stratification variable. The aim is to improve the accuracy of estimating mean weight of babies by incorporating gestational age as an auxiliary variable. An empirical study was carried out through population data sets obtained as to ascertain the efficiency and performance of various calibration ratio estimators considered in the study, and the results revealed that the estimator of Audu et al. (2020) outperformed the other estimators, and has proven to be consistent in all cases of sample size selection and the tuning parameter.

In this chapter, we have suggested an improved estimator for estimating the population mean in stratified sampling in presence of auxiliary information. The mean square error (MSE) of the proposed estimator have been derived under large sample approximation. Besides, considering the minimum case of the MSE equation, the efficient conditions between the proposed and existing estimators are obtained. These theoretical findings are supported by a numerical example.

In this paper, an improved class of estimators for estimating the finite population mean is proposed using supplementary information on an auxiliary attribute. The mathematical expressions for the bias and mean squared error of the proposed estimator are derived under first order of the approximation using simple random sampling. An empirical study is conducted to investigate the performances of the proposed and the existing estimators. It turns out that the proposed estimator is considerably more efficient than the recent estimators suggested by Koyuncu [1] and Haq and Shabbir [2].

This course provides an advanced study of theoretical statistics. Topics include: an introduction probability and sample space, conditional probability and Bayes Theorem, probability distribution of a random variable, discrete and continuous random variables, functions of random variables, moments and moment generating functions, order statistics and their distributions, discrete distributions, continuous distributions, bivariate and multivariate normal distribution, modes of convergence, limiting moment generating functions, and the central limit theorems. 3 hours

Naik and Gupta (1996), Singh et al. (2007) and Abd-Elfattah et al. (2010) introduced some estimators for estimating population mean using available auxiliary attributes under simple random sampling scheme. We adapt these estimators under systematic random sampling scheme using available auxiliary attributes. Further, a new family of estimators is proposed for the estimation of population mean under systematic random sampling scheme. The properties such as bias and mean square error of the proposed estimators are derived. From numerical illustration it is shown that proposed estimators are more efficient than the reviewed ones.

In this paper we have suggested two modified estimators of population mean using power transformation. It has been shown that the modified estimators are more efficient than the sample mean estimator, usual ratio estimator, Sisodia and Dwivedi's (1981) estimator and Upadhyaya and Singh's (1999) estimator at their optimum conditions. Empirical illustrations are also given for examining the merits of the proposed estimators. Following Kadilar and Cingi (2003) the work has been extended to stratified random sampling, and the same data set has been studied to examine the performance in stratified random sampling.

In this paper, we will examine the application of the Monte Carlo method in the testing of the robustness of statistical tests. The very computation-intensive Monte Carlo testing was implemented using the so-called GP-GPU method, which utilises the video cards' GPU (Graphical Processing Unit) to perform the necessary calculations. The robustness of a statistical test is defined as its property to remain valid even if its assumptions are not met. (We call a test valid if its significance level is equal to its Type I Error Rate.) One way to investigate the robustness of a statistical test (especially useful if the test's complicated structure makes analytic handling infeasible or impossible) is to employ the so-called Monte Carlo method. Here, we generate many random samples (meeting or violating the assumptions, which we can arbitrarily set), perform the statistical test many times on them, and then check whether the empirically found Type I Error Rate converges to the specified significance level or not. This way, we give up exactness for the complete insensitivity to the complexity of the examined statistical test. This method (by requiring enormous amount of random number generations and statistical testings for reliable results) is very computation-intensive; traditionally only supercomputers could be used effectively, limiting the availability of this method. However a new approach, called GP-GPU, makes it possible to harness the extremely high computing performance of-even ordinary, widely available-video cards found in every modern PC. We implemented a framework that performs the abovementioned MC-testing of the robustness of statistical tests. We call our program "framework", because it can be easily expanded with virtually any statistical test (to which no GP-GPU knowledge is needed), and be tested with very high performance-even with widely available tools. As an example, we performed the analysis of the well known Student's t-test; andusing this as a starting point-we demonstrated the main advantages of our framework.

In sampling theory, the auxiliary information is widely used to increase precision of the estimators when the study variable is correlated with the auxiliary variable. In several practical situations, the auxiliary information is available in the form of the auxiliary attribute(s). In this paper, we proposed a generalized estimator for the estimation of finite population mean using the auxiliary attribute under stratified two-phase sampling. The mathematical expressions of approximate bias and mean square error (MSE) of the proposed estimator are obtained to the first-order. Many special cases of the proposed estimator are also obtained using the known parameters of the auxiliary attribute. The algebraic comparisons of the MSE of the proposed estimator have been made the MSE of the competing estimators. The performance of the proposed estimator is evaluated using real data-sets. The results illustrate the good performance of the proposed estimators and its sub-cases.

This article proposes some estimators based on an adaptation of the estimators developed by Bahl and Tuteja [1], Diana [2], Koyuncu and Kadilar [3], Koyuncu and Kadilar [4], Shabbir and Gupta [5], and Koyuncu [6] utilizing available supplementry attributes. Further, a new family of estimators is also developed by adapting Koyuncu [7] and Koyuncu [6]. Under stratified sampling scheme along side the nonresponse issue, the expressions for the the mean square errors (MSEs) of the adapted and proposed estimators have been determined. These theoretical findings are demonstrated by a numerical illustration with original data set.

In sampling theory, the auxiliary information is widely used to increase precision of the estimators when the study variable is correlated with the auxiliary variable. In several practical situations, the auxiliary information is available in the form of the auxiliary attribute(s). In this paper, we proposed a generalized estimator for the estimation of finite population mean using the auxiliary attribute under stratified two-phase sampling. The mathematical expressions of approximate bias and mean square error (MSE) of the proposed estimator are obtained to the first-order. Many special cases of the proposed estimator are also obtained using the known parameters of the auxiliary attribute. The algebraic comparisons of the MSE of the proposed estimator have been made the MSE of the competing estimators. The performance of the proposed estimator is evaluated using real data-sets. The results illustrate the good performance of the proposed estimators and its sub-cases.

This course provides an advanced study of theoretical statistics. Topics include: an introduction probability and sample space, conditional probability and Bayes Theorem, probability distribution of a random variable, discrete and continuous random variables, functions of random variables, moments and moment generating functions, order statistics and their distributions, discrete distributions, continuous distributions, bivariate and multivariate normal distribution, modes of convergence, limiting moment generating functions, and the central limit theorems. 3 hours

In sampling theory, the auxiliary information is widely used to increase precision of the estimators when the study variable is correlated with the auxiliary variable. In several practical situations, the auxiliary information is available in the form of the auxiliary attribute(s). In this paper, we proposed a generalized estimator for the estimation of finite population mean using the auxiliary attribute under stratified two-phase sampling. The mathematical expressions of approximate bias and mean square error (MSE) of the proposed estimator are obtained to the first-order. Many special cases of the proposed estimator are also obtained using the known parameters of the auxiliary attribute. The algebraic comparisons of the MSE of the proposed estimator have been made the MSE of the competing estimators. The performance of the proposed estimator is evaluated using real data-sets. The results illustrate the good performance of the proposed estimators and its sub-cases.

This article proposes some estimators based on an adaptation of the estimators developed by Bahl and Tuteja [1], Diana [2], Koyuncu and Kadilar [3], Koyuncu and Kadilar [4], Shabbir and Gupta [5], and Koyuncu [6] utilizing available supplementry attributes. Further, a new family of estimators is also developed by adapting Koyuncu [7] and Koyuncu [6]. Under stratified sampling scheme along side the nonresponse issue, the expressions for the the mean square errors (MSEs) of the adapted and proposed estimators have been determined. These theoretical findings are demonstrated by a numerical illustration with original data set.

This article proposes some estimators based on an adaptation of the estimators developed by Bahl and Tuteja [1], Diana [2], Koyuncu and Kadilar [3], Koyuncu and Kadilar [4], Shabbir and Gupta [5], and Koyuncu [6] utilizing available supplementry attributes. Further, a new family of estimators is also developed by adapting Koyuncu [7] and Koyuncu [6]. Under stratified sampling scheme along side the nonresponse issue, the expressions for the the mean square errors (MSEs) of the adapted and proposed estimators have been determined. These theoretical findings are demonstrated by a numerical illustration with original data set.

In this paper some new improved estimators are suggested using auxiliary attribute(s) in stratified random sampling. The expressions of the mean square error of the proposed estimators are derived. Also, the theoretical findings are supported by a numerical example.

U-statistics, in which ''U'' stands for ''unbiased,'' generalize common notions of unbiased estimation such as the sample mean and the sample variance and thus play a vital role in estimation. This chapter describes the basic description, martingale structure, and the projection of a U-statistic.

In this article we have considered the problem of estimating the population mean Y in the stratified random sampling using the information of an auxiliary variable x which is correlated with y and suggested improved exponential ratio estimators in the stratified random sampling. The mean square error (MSE) equations for the proposed estimators have been derived and it is shown that the proposed estimators under optimum condition performs better than estimators suggested by Singh et al. (2008). Theoretical and empirical findings are encouraging and support the soundness of the proposed estimators for mean estimation.

We propose ratio estimators that can be used under unequal probability designs by adapting Horvitz-Thompson estimators to ratio estimators in literature. Mean square error (MSE) of all proposed ratio estimators is obtained and compared with the MSE of classical estimators. By these comparisons the conditions, which make each proposed estimator more efficient than the classical one, are found. The theoretical results are supported by a numerical example.

In this paper, some ratio estimators for the population mean using known coefficient of skewness based on auxiliary attribute have been proposed and these have been developed by combining ratio estimators for estimating population mean of variate of study using the procedure given in Kadilar and Cingi (2006). The mean square error (MSE) equations for suggested estimators have been derived and it has been concluded that the suggested estimators always perform better than the ratio estimators, given in Sections 1 and 2. The results are also supported by the three original datasets.

This paper suggests new ratio estimators in stratified random sampling using the information of an auxiliary attribute. Theoretically, it is obtained the bias and mean square error (MSE) for these estimators and compare them with the MSE of classical combined ratio estimator. By these comparisons, it is demonstrated that proposed estimators are more efficient than the considered estimators. In addition, these theoretical results are supported with the aid of numerical examples.

This paper suggested a new family of estimators for the population mean in the simple random sampling using the information of an auxiliary attribute. Theoretically, the mean square error (MSE) equations were obtained and it was shown that all the suggested ratio estimators are more efficient than some known estimators. These results were also supported by two original data sets.

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts and ideas on axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Beleivability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

In this paper, we have considered stratified two-phase sampling design with sub-sampling the non-respondents in the presence of non-response for estimating population mean considering the information of two auxiliary variables. The proposed estimators are the exponential function of two auxiliary variables when means of the two auxiliary variables are not known in prior. Further the proposed estimators are provided with their generalized form. The bias and mean square error expressions of the proposed estimators have been derived in two different cases of non-response. The conditions for which proposed estimators are more efficient as compared to some other estimators, have also been discussed in each case of non-response. It is shown that the proposed estimators are more efficient as compared to Hansen and Hurwitz (J Am Stat Assoc 41:517–529, 1946) unbiased estimator and Tabasum and Khan (Assam Stat Rev 20(1):73–83, 2006) two-phase ratio and product estimators modified to the stratified sampling. An empirical study has also been carried out to demonstrate the performances of the estimators.

Some ratio estimators for estimating the population mean of the variable under study, which make use of information regarding the population proportion possessing certain attribute, are proposed. Under simple random sampling without replacement (SRSWOR) scheme, the expressions of bias and mean-squared error (MSE) up to the first order of approximation are derived. The results obtained have been illustrated numerically by taking some empirical population considered in the literature.

2 In the front cover table the percent relative efficiency (PRE) of ys, t1, t2, tr (in optimum case) and tp (in optimum case) are computed with respect to 2 y s. This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of ...

1] suggested a class of estimators for estimating population mean in stratified random sampling using auxiliary attributes. This paper proposes a family of estimators based on adaptation of the [2] estimator and introduces a new family of exponential estimators in stratified random sampling using auxiliary attributes. The expressions for mean square errors of the adapted and proposed families are derived to the first order of approximation. It is shown that the adapted version of [2] is always more efficient than usual estimator, ratio, product, exponential ratio and exponential product type estimators. Moreover, the new exponential estimators based on auxiliary attributes are more efficient than of [1] and regression estimator using in stratified random sampling auxiliary attribute. In addition, theoretical findings are supported by an empirical study to show the superiority of the constructed estimators over others.

Some ratio estimators for estimating the population mean of the variable under study, which make use of information regarding the population proportion possessing certain attribute, are proposed. Under simple random sampling without replacement (SRSWOR) scheme, the expressions of bias and mean-squared error (MSE) up to the first order of approximation are derived. The results obtained have been illustrated numerically by taking some empirical population considered in the literature.

In this paper, we will examine the application of the Monte Carlo method in the testing of the robustness of statistical tests. The very computation-intensive Monte Carlo testing was implemented
using the so-called GP-GPU method, which utilises the video cards’ GPU (Graphical Processing Unit) to perform the necessary calculations.
The robustness of a statistical test is defined as its property to remain valid even if its assumptions are not met. (We call a test valid if its significance level is equal to its Type I Error Rate.) One way to investigate the robustness of a statistical test (especially useful if the test’s complicated structure makes analytic handling infeasible or impossible) is to employ the so-called Monte Carlo method. Here, we generate many random samples (meeting or
violating the assumptions, which we can arbitrarily set), perform the statistical test many times on them, and then check whether the empirically found Type I Error Rate converges to the specified significance level or not. This way, we give up exactness for the complete insensitivity to the complexity of the examined statistical test.
This method (by requiring enormous amount of random number generations and statistical testings for reliable results) is very computation-intensive; traditionally only supercomputers
could be used effectively, limiting the availability of this method. However a new approach, called GP-GPU, makes it possible to harness the extremely high computing performance of – even ordinary, widely available – video cards found in every modern PC.
We implemented a framework that performs the abovementioned MC-testing of the robustness of statistical tests. We call our program “framework”, because it can be easily expanded with virtually any statistical test (to which no GP-GPU knowledge is needed), and be tested with very high performance – even with widely available tools.
As an example, we performed the analysis of the well known Student’s t-test; and – using this as a starting point – we demonstrated the main advantages of our framework.