Constrained Bayesian Methods of Hypotheses Testing: A New Philosophy of Hypotheses Testing in Parallel and Sequential Experiments

Kartlos Kachiashvili
Tbilisi State University, Tbilisi, Georgia

Series: Mathematics Research Developments
BISAC: MAT027000

Clear

$270.00

eBook

Digitally watermarked, DRM-free.
Immediate eBook download after purchase.

Product price
Additional options total:
Order total:

Quantity:

Details

The problems of one of the basic branches of mathematical statistics – statistical hypotheses testing – are considered in this book. The intensive development of these methods began at the beginning of the last century. The basic results of modern theory of statistical hypotheses testing belong to the cohort of famous statisticians of this period: Fisher, Neyman-Pearson, Jeffreys and Wald (Fisher, 1925; Neyman and Pearson, 1928, 1933; Jeffreys, 1939; Wald, 1947a,b). Many other bright scientists have brought their invaluable contributions to the development of this theory and practice. As a result of their efforts, many brilliant methods for different suppositions about the character of random phenomena are under study, as well as their applications for solving very complicated and diverse modern problems.

Since the mid-1970s, the author of this book has been engaged in the development of the methods of statistical hypotheses testing and their applications for solving practical problems from different spheres of human activity. As a result of this activity, a new approach to the solution of the considered problem has been developed, which was later named the Constrained Bayesian Methods (CBM) of statistical hypotheses testing. Decades were dedicated to the description, investigation and applications of these methods for solving different problems. The results obtained for the current century are collected in seven chapters and three appendices of this book. The short descriptions of existing basic methods of statistical hypotheses testing in relation to different CBM are examined in Chapter One. The formulations and solutions of conventional (unconstrained) and new (constrained) Bayesian problems of hypotheses testing are described in Chapter Two.

The investigation of singularities of hypotheses acceptance regions in CBM and new opportunities in hypotheses testing are presented in Chapter Three. Chapter Four is devoted to the investigations for normal distribution. Sequential analysis approaches developed on the basis of CBM for different kinds of hypotheses are described in Chapter Five. The special software developed by the author for statistical hypotheses testing with CBM (along with other known methods) is described in Chapter Six. The detailed experimental investigation of the statistical hypotheses testing methods developed on the basis of CBM and the results of their comparison with other known methods are given in Chapter Seven. The formalizations of absolutely different problems of human activity such as hypotheses testing problems in the solution – of which the author was engaged in different periods of his life – and some additional information about CBM are given in the appendices.

Finally, it should be noted that, for understanding the materials given in the book, the knowledge of the basics of the probability theory and mathematical statistics is necessary. I think that this book will be useful for undergraduate and postgraduate students in the field of mathematics, mathematical statistics, applied statistics and other subfields for studying the modern methods of statistics and their application in research. It will also be useful for researchers and practitioners in the areas of hypotheses testing, as well as the estimation theory who develop these new methods and apply them to the solutions of different problems.

Acknowledgment

Preface

Reminder

Introduction

Chapter 1. Methods of Statistical Hypotheses Testing

Chapter 2. The Bayesian Problem of Hypotheses Testing

Chapter 3. Investigation of Hypotheses Acceptance Regions in Constrained Bayesian Tasks

Chapter 4. Investigations for Normal Distribution

Chapter 5. Sequential Analysis Approach

Chapter 6. Software for Statistical Hypotheses Testing

Chapter 7. Experimental Research

Concluding Remarks

Bibliography

Appendix A

Appendix B

Appendix C

To read the review, click here - Sergei Chobanyan, Visiting Professor of Department of Statistics, Michigan State University, USA.

To read the review, click here - Alexander Topchishvili, Professor, Dr. Habil. Ing., Marburg, Germany.

Aivazjan, S. A. (1959). A comparison of the optimal properties of the Neuman-Pearson and the Wald sequential probability ratio test. Teoria Verojatn. i Primenen. 4(1), 86-93.
Amster, S. J. (1963). A modified Bayes stopping rule. Ann. Math. Statist. 34, 4, 1404-1413.
Andersson S. (1982) Distributions of Maximal Invariants Using Quotient Measures. The Annals of Statistics, Vol. 10, No. 3, 955-961.
Anderson T. W. (2003). An introduction to Multivariate Statistical Analysis. Third Edition. Wiley & Sons, Inc. Hoboken, New Jersey.
Anscombe, F. J. (1963). Sequential medical trials. J. Amer. Statist. Assoc. 58, 365-383.
Arrow, K. J., Blackwell, D. &Girshick, M. A. (1949). Bayes and minimax solutions of sequential decision problems. Econometrica 17 (2), 213-244.
Bagui S. and Datta S. (1998) Some useful properties of the Bayes risk in classification. Calcutta Statist. Assoc. Bull. 48 1 83-91.
Bahadur, R. R. (1952). A property of the t-statistics. Sankhya, 12, 79-88.
Balasanova E. V. (1995) Asymptotic expansions for the statistic and risk function of a Bayesian classification rule, J. Math. Sci. 75(2), pp. 1552–1556.
Bansal, N. K., Hamedani, G. G. & Maadooliat, M. (2015). Testing Multiple Hypotheses with Skewed Alternatives.Biometrics, 72(2):494-502.
Bansal, N. K., Hamedani, G. G. & Sheng, R. (2012). Bayesian analysis of hypothesis testing problems for general population: A Kullback–Leibler alternative. J. of Statistical Planning and Inference, 142, 1991–1998.
Bansal, N. K. & Miescke, K. J. (2013). A Bayesian decision theoretic approach to directional multiple hypotheses problems, J. of Multivariate Analysis, 120, 205–215.
Bansal, N. K. & Sheng, R. (2010). Beyesian Decision Theoretic Approach to Hypothesis Problems with Skewed Alternatives. J. of Statistical Planning and Inference, 140, 2894-2903.
Barnard, G. A. (1946). Sequential test in industrial statistics. Suppl. J. Roy. Statist. Soc., 8, 1.
Barnard, G. A. (1947). A review of “Sequential Analysis” by Abraham Wald. J. Amer. Statist. Assoc. 42, 658-669.
Barnard, G. A. (1949). Statistical inference (with discussion). J. Roy. Statist. Soc. (Ser. B) 11, 115-139.
Barnett H. A. R. (1955 The variance of the product of two independent variables and its application to an investigation based on sample data,” Journal of the Institute of Actuaries, Vol. 81, No. 190, pp. 54-65.
Bartholomew, D. J. (1967). Hypothesis testing when the sample size is treated as a random variable. J. Roy. Statist. Soc. 29, 53-82.
Bartlett M. S. (1957) A comment on D. V. Lindley’s statistical paradox. Biometrica, 44, 533-534.
Bartroff J. and Lai Z. L. (2010) Multistage Tests of Multiple Hypotheses, Communications in Statistics - Theory & Methods 39: 1597-1607.
Basu, D., (1975). Statistical information and likelihood (with discussion). Sankhya (Ser. A) 37, 1, 1-71.
Bauer, P., Hommel G. and Sonnemann E. eds. (1988) Multiple Hypothesenprüfung, (Multiple Hypotheses Testing.) Berlin: Springer-Verlag (In German and English).
Bazaraa M. S. and Shetty C. M. (1979) Nonlinear Programming. Theory and Algorithms, Wiley and Sons, New York.
Bedbur S., Beutner E. and Kamps U. (2013) Multivariate testing and model-checking for generalized order statistics with applications.Statistics, 1-114.
Berger J. O. (1985) Statistical Decision Theory and Bayesian Analysis, Springer, New York.
Berger J. O. (2003) Could Fisher, Jeffreys and Neyman have Agreed on Testing? Statistical Science, 18: 1–32.
Berger J. O., Boukai B. and Wang Y. (1997) Unified Frequentist and Bayesian Testing of a Precise Hypothesis. Statistical Science, 12(3): 133-160.
Berger J. O., Boukai B. and Wang Y. (1999) Simultaneous Bayesian–frequentist sequential testing of nested hypotheses. Biometrika, 86:79–92.
Berger J. O., Brown L. D. and Wolpert R. L. (1994) A Unified Conditional Frequentist and Bayesian Test for Fixed and Sequential Simple Hypothesis Testing. The Annals of Statistics, 22(4): 1787-1807.
Berger J. O. and Delampady M. (1987) Testing precise hypothesis (with discussion). Statist. Sci., 2: 317-352.
Berger J. O. and Pericchi L. R. (1996) The Intrinsic Bayes Factor for Model Selection and Prediction, Journal of the American Statistical Association, 91, 109-122.
Berger J. O. and Sellke T. (1987) Testing a point null hypothesis. The irreconcilability of p-values and evidence (with discussion). J. Amer. Statist. Assoc., 82: 112-139.
Berger J. O. and Wolpert R. L. (1984) The Likelihood Principle. Institute of Mathematical Statistics Monograph Series (IMS), Hayward: CA.
Berger J. O. and Wolpert R. L. (1988) The Likelihood Principle, 2nd ed. (with discussion). IMS, Hayward: CA.
Berk RH, Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist., 1966; 37: 51-58.
Bernardo J. M. (1980) A Bayesian analysis of classical hypothesis testing. Universidad de Valencia, 605-617.
Bernardo J. M. and Rueda R. (2002) Bayesian Hypothesis Testing: A Reference Approach. International Statistical Review,1-22.
Blackwell D. and Girshick M. A. (1954) Theory of Games and Statistical Decisions. Wiley, New York.
Blum M. G. B. and François O. (2010) Non-linear regression models for Approximate Bayesian Computation. Stat. Comput., 20: 63–73.
Boll R. M., Connell J. H., Pankanti SH., Ratha N. K. and Senior A. W. (2004) Guide to Biometrics. Springer-Verlag New York, Inc.
Boratynska A. and Drozdowicz M. (1999) Robust Bayesian estimation in a normal model with asymmetric loss function. J. Appl. Math. (Warsaw), 26 1 85-92.
Braun H. I. (1994) The Collected Works of John W. Tukey. Vol. VIII: Multiple Comparisons: 1948-1983. New York: Chapman & Hall.
Breusch, T. S. (1986). Hypothesis Testing in Unidentified Models, The Review of Economic Studies, 53, 4, 635-651.
Brownie C, Keifer J. The Ideas of Conditional Confidence in the simplest setting. Comm. Statist. Theory Methods, 1977; 6: 691-751.
Bühl, A. and Zöfel, P. (2001) SPSS Version 10. Einführung in die moderne datenanalyse unter Windows, [SPSS Version 10. Introduction to advanced data analysis on Windows] Pearson Education Deutschland GmbH.
Cassela G. and Berger R. L. (1987) Reconciling evidence in the one-sided testing problem (with discussion). J. Amer. Statist. Assoc., 82: 106-112.
Casella G. and Wells M. T. (1990) Comparing p-values to Neyman-Pearson tests. Technical Report BU-1073-M, Biometrics Unit and Statistics Cent., Cornell Univ.
Cheng Y., Su F. and Berry D. A. (2003) Choosing sample size for a clinical trial using decision analysis. Biometrika, 90 4 923–936.
Choudhury, A. and Borthakur, A. C. (2008). Bayesian inference and prediction in the single server Markovian queue, Metrika, 67, 3, 371-383.
Christensen R. (2005) Testing Fisher, Neyman, Pearson, and Bayes,The American Statistician,59(2): 121-126.
Christophi C. A. and Modarres R. (2005) Approximating the distribution function of risk,Comput. Statist. Data Anal. 49(4), pp. 1053–1067.
Cook R. D. and Ni L. (2006) Using intraslice covariances for improved estimation of the central subspace in regression. Biometrika, 93, 1, 65–74.
Cox D. R. (1988) Some aspects of conditional and asymptotic inference: A review, Sankhya 50(3), pp. 314–337.
Cramer H. (1999) Mathematical Methods of Statistics, Princeton University Press, Princeton, New Jersey.
Dass S. C. and Berger J. O. (2003) Unified Conditional Frequentist and Bayesian Testing of Composite Hypotheses. Scandinavian Journal of Statistics, 30(1): 193-210.
David P. J. and Rabinovitz P. (1984) Methods of numerical Integration. Computer Science and Applied Mathematics, 2nd Edition, Academic Press Inc., Orlando, Florida.
De, S. K. and Baron, M. (2012a). Step-up and Step-down Methods for Testing Multiple Hypotheses in Sequential Experiments, Journal of Statistical Planning and Inference 142: 2059–2070.
De, S. K. and Baron, M. (2012b). Sequential Bonferroni Methods for Multiple Hypothesis Testing with Strong Control of Family-Wise Error Rates I and II,Sequential Analysis 31: 238–262.
De Groot, M. H. (1970). Optimal Statistical Decisions. New-York: McGraw-Hill Book Company.
Delampady M. and Berger J. O. (1990) Lower bounds on Bayes factors for the multinomial distribution, with application to chi-squared tests of fit. Ann. Statist., 18: 1295-1316.
Demster A. P. (1971) Model searching and estimation in the logic of inference. In Foundations of Statistical Inference (Godambe V.P. and Sprott D. A. eds.), 56-81. Toronto: Holt, Rinehart and Winston.
Dhrymes, P. J. (1978), Introducrory Econometrics. New York: Springer Verlag.
Dhar S. S. and Chaudhuri P. (2008) A comparison of robust estimators based on two types of trimming, AStA Advances in Statistical Analysis, 93, 2, 151-158.
Dickey J. (1977) Is the tail area useful as an approximate Bayes factor? J. American Statistical Association, 72, 138-142.
Dixon, W. J. (1992) BMDP Statistical Software Manual, Edition User's Digest.
Duchesne P.and Francq Ch. (2014) Multivariate hypothesis testing using generalized and {2}-inverses – with applications. Statistics, 2014, 1-22.
Dudoit, S., Shaffer, J. P., Boldrick, J. C., 2003. Multiple hypothesis testing in microarray experiment. Statistical Science 18, 71–103.
Edwards W., Lindman H. and Savage L. J. (1963) Bayesian statistical inference for psychological research. Psychological Review, 70: 193-242.
Efron B. (2004) Large-Scale Simultaneous Hypothesis Testing. Journal of the American Statistical Association,99(465): 96-104.
Efron B. (2005) Bayesians, Frequentists, and Scientists. Journal of the American Statistical Association. Vol. 100, Issue 469:1-5.
Finner, H. (1999). Stepwise multiple test procedures and control of directional errors. The Annals of Statistics, 27(1), 274-289.
Fisher R. A. (1925) Statistical Methods for Research Workers, London: Oliver and Boyd.
Genz A. (1992) Numerical Computation of Multivariate Normal Probabilities, Journal of Computational and Graphical Statistics, No. 1, pp. 141-149.
Genz A. (1993) Comparison of Methods for the Computation of Multivariate Normal Probabilities,” Computing Science and Statistics, Vol. 25, pp. 400-405.
Genz A. and Bretz F. (1999) Numerical Computation of Multivariate t-Probabilities with Application to Power Calculation of Multiple Contrasts,” Journal of Statistical Computation and Simulation, Vol. 63, pp. 361-378.
Girshick, M. A. (1946a). Contributions of the Theory of Sequential Analysis. Ann. Math. Statist. 17, 2, 123-143.
Girshick, M. A. (1946b). Contributions of the Theory of Sequential Analysis. Ann. Math. Statist. 17, 3, 282-298.
Ghosh, B. K. (1970). Sequential Tests of Statistical Hypotheses. Addison-Wesley, Reading, Massachusetts.
Ghosh, B. K. and Sen, P. K., (eds.) (1991). Handbook of Sequential Analysis. Dekker, NY.
Gómez-Villegas M. A. and González-Pérez B. (2011) A Bayesian Analysis for the Homogeneity Testing Problem Using –Contaminated Priors. Communications in Statistics - Theory and Methods, 40(6): 1049-1062.
Gómez–Villegas M. A., Maín P. and Sanz L. (2009) A Bayesian analysis for the multivariate point null testing problem. Statistics, 43(4): 379-391.
Good I. J. (1992) The Bayesian/Non-Bayesian compromise: a brief review. J Amer. Statist. Assoc., 87: 597-606.
Goodman L. A. (1960) On the exact variance of products, Journal of the American Statistical Association, Vol. 55, pp. 708-713.
Goodman L. A. (1962) The variance of the product of K random variables, Journal of the American Statistical Association, Vol. 57, No. 297, pp.54-60.
Goritskiy, Y. A., Kachiashvili K. J. and Datuashvili M. N. (1977). Application of the generalized Neyman-Pearson criterion for estimating the number of false decisions at isolation of true trajectories. Technical Cybernetics: Proceedings of Georgian Technical Institute, 7(198), Tbilisi, 79-85.
Grillenzoni C. (2008) Robust nonparametric estimation of the intensity function of point data, AStA Advances in Statistical Analysis, 92, 2, 117-134.
Gutmann, S. (1984). Loss functions for p-values and simulations inference. Technical Report, 43, Statistics Cent., MIT.
Hafner, Ch. M. and Herwartz, H. (2008). Analytical quasi maximum likelihood inference in multivariate volatility models, Metrika, 67, 2, 219-239.
Hajivassiliou V., McFadden D. and Ruud P. (1996) Simulation of multivariate normal rectangle probabilities and their derivatives: Theoretical and computational results,” Journal of Econometrics, Vol. 72, pp. 85-134.
Hamilton, D. C., Lesperance, M. L., 1991. A consulting problem involving bivariate acceptance sampling by variables. Canadian Journal of Statistics, 19, 109–117.
Hochberg Y. (1988) A Sharper Bonferroni Procedure for Multiple Tests of Significance. Biometrika, 75: 800-8003.
Hochberg Y. and Rom D. (1995) Extensions of Simes’ Test for Logically related Hypotheses. J. Statist. Planning Inference, 48: 141-152.
Hochberg Y. and Tamhance A. C. (1987) Multiple Comparison Procedures, New York: Wiley.
Hollander, M. and Wolfe, D. A. (1973) Nonparametric Statistical Methods, New York: John Wiley and Sons.
Holm S. (1979) A Simple Sequential Rejective Multiple Test Procedure. Scand. J. Stat. 6: 65-70.
Hoppe F. M. (1993) Multiple Comparisons. Selection and Applications in Biometry, New York: Dekker.
Hsu J. C. (1996) Multiple Comparisons: Theory and methods, New York: Chapman & Hall.
Hubbard R. and Bayarri M. J. (2003) Confusion over Measures of Evidence (p’s) Versus Errors (α’s) in Classical Statistical Testing. The American Statistician, 57: 171–177.
Hwang J. T., Casella G., Robert Ch., Wells M. T. and Farrell R. H. (1992) Estimation of Accuracy in Testing. The Annals of Statistics, 20(1): 490-509.
Hwang J. T. and Pemantle R. L. (1990) Evaluation of estimators of statistical significance under a class of loss functions. Technical report, Statistics Cent., Cornell Univ., Ithaca, NY.
Jeffreys H. (1939) Theory of Probability, 1st ed. Oxford: The Clarendon Press.
Jeffreys H. (1961) Theory of Probability. 3rd ed. Oxford Classic Texts in the Physical Sciences, Oxford: Oxford University Press.
Joe S. (S.) Approximations to multivariate normal rectangle probabilities based on conditional expectations, Journal of the American Statistical Association, Vol. 90, pp. 957-964.
Jones, L. V. & Tukey, J. W. (2000). A sensible formulation of the significance test. Psychological Methods, 5(4), 411-414.
Kachiashvili K.J. (1984) Identification of pollution sources by means of automatic stations of river water quality control. In the book: Regulation of the qualities of the environmental waters. Scientific Proceedings of VNIIVO, Kharkov, 114-119.
Kachiashvili K. J. (1989) Bayesian algorithms of many hypothesis testing, Tbilisi: Ganatleba.
Kachiashvili K. J. (1990) Restoration of polynomial regression on the basis of active experiment. Zavodskaya laboratoriya, No. 10, 87-90.
Kachiashvili K. J. (2003) Generalization of Bayesian Rule of Many Simple Hypotheses Testing. International Journal of Information Technology & Decision Making, 2(1): 41-70.
Kachiashvili K. J. (2011) Investigation and Computation of Unconditional and Conditional Bayesian Problems of Hypothesis Testing. ARPN Journal of Systems and Software, 1(2): 47-59.
Kachiashvili K. J. (2014a) The Methods of Sequential Analysis of Bayesian Type for theMultiple TestingProblem. Sequential Analysis, 33(1), 23-38 DOI: 10.1080/07474946.2013.843318.
Kachiashvili K. J. (2014b) Comparison of Some Methods of Testing Statistical Hypotheses. Part I. Parallel Methods. International Journal of Statistics in Medical Research, 3, 174-189.
Kachiashvili K. J. (2014c) Comparison of Some Methods of Testing Statistical Hypotheses. Part II. Sequential Methods. International Journal of Statistics in Medical Research, 3, 189-197.
Kachiashvili K. J. (2014d) Investigation of the method of sequential analysis of Bayesian type. Journal of Advances in Mathematics. Vol. 18, No. 1, p. 1367-1380.
Kachiashvili K. J. (2015) Constrained Bayesian Method for Testing Multiple Hypotheses in Sequential Experiments. Sequential Analysis, (in press) DOI: 1030973 DOI:10.1080/07474946.2015.1030973.
Kachiashvili K. J. (2016) Constrained Bayesian Method of Composite Hypotheses Testing: Singularities and Capabilities. International Journal of Statistics in Medical Research, Vol. 5, No. 3, pp. 135-167.
Kachiashvili K. J., Gordeziani D. G., Lazarov R. G. and Melikdzhanian D. I. (2007) Modeling and simulation of pollutants transport in rivers. International Journal of Applied Mathematical Modelling (AMM), 31, 1371-1396.
Kachiashvili G. K., Kachiashvili K. J. and Mueed A. (2012a) Specific Features of Regions of Acceptance of Hypotheses in Conditional Bayesian Problems of Statistical Hypotheses Testing. Sankhya: The Indian Journal of Statistics, 74(1): 112-125.
Kachiashvili K. J. and Hashmi M. A. (2010) About Using Sequential Analysis Approach for Testing Many Hypotheses. Bulletin of the Georgian Academy of Sciences, 4(2): 20-25.
Kachiashvili K. J. and Hashmi M. A. (2012) Computation of the Multivariate Normal Integral over a Complex Subspace, Applied Mathematics, Vol. 3 No. 5, 489-498.
Kachiashvili K. J., Hashmi M. A. and Mueed A. (2008) The statistical risk analysis as the basis of the sustainable development. Proceedings of the 4th IEEE International Conference on Management of Innovation & Technology (ICMIT2008), Bangkok, Thailand, 1210-1215.
Kachiashvili K. J., Hashmi M. A. and Mueed A. (2011) Comparison Analysis of Unconditional and Conditional Bayesian Problems of Testing Many Hypotheses, Transactions. “Automated Control Systems”. Georgian Technical University, No. 1(10), 89-100.
Kachiashvili K. J., Hashmi M. A. and Mueed A. (2012b) Sensitivity Analysis of Classical and Conditional Bayesian Problems of Many Hypotheses Testing. Communications in Statistics—Theory and Methods, 41(4): 591–605.
Kachiashvili K. J., Hashmi M. A. and Mueed A. (2012c) The Statistical Risk Analysis as the Basis of the Sustainable Development. Int. J. of Innovation and Technol. Management (World Scientific Publishing Company), Vol. 9, No. 3, 1250024 (2012) [10 pages] DOI: 10.1142/S0219877012500241
Kachiashvili K. J., Hashmi M. A. and Mueed A. (2013) Quasi-optimal Bayesian procedures of many hypotheses testing. Journal of Applied Statistics, Vol. 40, No. 1, 103–122.
Kachiashvili K. J., Hashmi M. A. and Mueed A. (2009) Bayesian Methods of Statistical Hypothesis Testing for Solving Different Problems of Human Activity. Applied Mathematics and Informatics (AMIM), Vol. 14, No. 2, pp. 3-17.
Kachiashvili K. J. and Melikdzhanian D. I. (2000) Methodology of nonlinear regressions identification by modified method of least squares. Zavadskaia Laboratoria, 5, 157-164.
Kachiashvili K. J. and Melikdzhanian D.I. (2005) The methods of the definite class non-linear functions interpolation with practical examples. Applied Mathematics and Informatics (AMIM), Vol. 10, No. 2, 37-52.
Kachiashvili K. J. and Melikdzhanian D. I. (2006) Identification of River Water Excessive Pollution Sources. International Journal of Information Technology & Decision Making, World Scientific Publishing Company, Vol.5, Issue 2, 397-417.
Kachiashvili K. J. and Melikdzhanian, D. Y. (2009a) Software for Determination of Biological Age,Current Bioinformatics 4: 41-47.
Kachiashvili K. J. and Melikdzhanian, D. Y. (2009b) Software Realization Problems of Mathematical Models of Pollutants Transport in Rivers, Advances in Engineering Software 40: 1063–1073.
Kachiashvili K. J. and Melikdzhanian D. I. (2010) SDpro – The Software Package for Statistical Processing of Experimental Information. International Journal Information Technology & Decision Making (IJITDM), Vol. 9, No 1, 115-144.
Kachiashvili K.J. and Melikdzhanian D.I. (2012) Advanced Modeling and Computer Technologies for Fluvial Water Quality Research and Control. Nova Science Publishers, Inc., New York.
Kachiashvili K. J. and Melikdzhanian D. I. (2015) Software for statistical hypotheses testing. International Journal of Modern Sciences and Engineering Technology (IJMSET), Volume 2, Issue 4, 2015, pp.33-52.
Kachiashvili K.J. and Melikdzhanian D.I. (2016) Software for Pollutants Transport in Rivers and for Identification of Excessive Pollution Sources. MOJ Ecology & Environmental Science, Volume 1 Issue 1, pp. 1-8.
Kachiashvili K. J. and Mueed A. (2013) Conditional Bayesian Task of Testing Many Hypotheses. Statistics, 47(2): 274-293.
Kachiashvili K. J. and Prangishvili A. I. (2016) Verification in biometric systems: problems and modern methods of their solution.Journal of Applied Statistics, DOI: 10.1080/02664763.2016.1267122, pp. 1-20.
Kaiser, H. F. (1960). Directional statistical decisions. Psychological Review, 67, 160-167.
Kartashov M.V. and Stroev O.M. (2006) Lundberg approximation for the risk function in an almost homogeneous environment, Theory Probab. Math. Statist. 73, pp. 71–79.
Kass R. E. and Wasserman L. (1996) The Selection of Prior Distributions by Formal Rules, Journal of the American Statistical Association, 91, 1343-1370.
Kendall M. and Stuart A. (1966) Distribution Theory, Vol. 1, Charles Griffit & Company Limited, London.
Kendall M. and Stuart A. (1970) The Advanced Theory of Statistics. Vol. 2. - Charles Griffin & Company Limited, London.
Klockars A. J. and Sax G. (1986) Multiple Comparison. Newbury Park, CA: Sage.
Kiefer J. (1977) Conditional confidence statement and confidence estimations (with discussion). J. Amer. Statist. Assoc., 72(360): 789-808.
Korolev, V. V. (2008). A System for Measuring Mechanical Stresses in a Vessel Hull. Russian Journal of Nondestructive Testing. 44, 1, 45–53.
Kotz S., Balakrishnan N. and Johnson N. L. (2000) Continuous Multivariate Distributions. Models and Applications, Vol. 1, 2nd Edition, John Wiley & Sons Ltd, New York.
Kullback, S. (1978) Informatin Theory and Statistics. Wiley & Sons, Inc.
Kuzmin S. Z. (1974) The basis of digital processing of radio-location information, Sovetskoe Radio, Moscow.
Lee P.M. (1989) Bayesian Statistics: An Introduction. Edward Arnold, London.
Lehmann, E. L. (1950). Some principles of the theory of the theory of testing hypotheses. The Annals of Mathematical Statistics, 20(1), 1-26.
Lehmann, E. L. (1957a). A theory of some multiple decision problems I. The Annals of Mathematical Statistics,

28, 1-25.
Lehmann, E. L. (1957b). A theory of some multiple decision problems II. The Annals of Mathematical Statistics, 28, 547-572.
Lehmann E. L. (1993) The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two? American Statistical Association Journal, Theory and Methods, 88(424): 1242-1249.
Lehmann E. L. (1997) Testing Statistical Hypotheses, 2nd ed. New York: Springer.
Lepage, R. (1979). Smoothing Wald’s test. Technical Report, Department of Statistics, Michigan State University, East Lansing.
Leventhal, L. & Huynh, C. (1996). Directional decisions for two-tailed tests: Power, error rates, and sample size. Psychological Methods, 1, 278-292.
Liang T. (1999) Monotone empirical Bayes tests for a discrete normal distribution. Statist. Probab. Lett. 44 3 241-249.
Lindley D.V. (1961) The Use of Prior Probability Distribution in Statistical Inference and Decisions. Proceedings of the 4th Berkley Symposium on Mathematical Statistics and Probability, Vol. 1, pp. 453-468.
Lindley D.V. (1990) The 1988 Wald Memorial Lectures: The present position of Bayesian Statistics. Statist. Sci. 5, 44-89.
Lindley, D.V. & Phillips, L.D. (1976). Inference for a Bernoulli process (a Bayesian view). Amer. Statist. 30, 112-119.
Lopez-Granados F., Jurado-Exposito M., Atenciano S. and others. (2002) Spatial variability of agricultural soil parameters in southern Spain. Plant and Soil, 246, 97-105.
Marden J. I. (2000) Hypothesis Testing: From p Values to Bayes Factors, Journal of the American Statistical Association, Vol. 95, No. 452.
McDaniel, S. and Hemedinger, Ch. (2007) SAS for Dummies, John Wiley & Sons.
Meng, C. and Dempster, A. (1987). A Bayesian Approach to the Multiplicity Problem for Significance Testing with Binomial Data, Biometrics, 43, 301-311.
Meyer, R.K. and Krueger, D.D. (2004) A Minitab Guide to Statistics, Prentice-Hall Publishing.
Millard, S.P. (1987) Proof of Safety as Proof of Hazard, Biometrics, 43, 719-725.
Miller R. G. (1966) Simultaneous Statistical Inference, New York: Wiley.
Moreno E. and Cano J.A. (1989) Testing a point null hypothesis: Asymptotic robust Bayesian analysis with respect to the priors given on a sub-sigma field. Int. Statist. Rev. 57 221-232.
Moreno E. and Giron F. J. (2006) On the Frequentist and Bayesian Approaches to Hypothesis Testing, SORT 30 (1) January-June; 3-28.
Nadarajah S. and Mitov K. (2003) Product Moments of Multivariate Random Vectors, Communications in Statistics. Theory and Methods, Vol. 32, No. 1, pp. 47-60.
Nath S.N. (1968) On Product Moments from a Finite Universe, Journal of the American Statistical Association, Vol. 63, No. 322, pp. 535-541.
Nath S.N. (1969) More results on Product Moments from a Finite Universe, Journal of the American Statistical Association, Vol. 64, No. 327, pp. 864-869.
Neyman J. and Pearson E. (1928) On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference. Part I, Biometrica, 20A: 175-240.
Neyman J. and Pearson E. (1933) On the Problem of the Most Efficient Tests of Statistical Hypotheses, Philos. Trans. Roy. Soc., Ser. A, 231: 289-337.
O’Brien, P.C. (1984) Procedures for comparing samples with multiple endpoints. Biometrics, 40, 1079–1087.
Pastushko O. N. and Nevludov I. Sh. (2012) Analysis of Quality Indicators of Biometrical Systems of Authentication of Users. Problems of Telecommunication, #4 (9), http://pt.journal.kh.ua.
Pocock, S.J., Geller, N.L., Tsiatis, A.A. (1987) The analysis of multiple endpoints in clinical trials. Biometrics, 43, 487–498.
Potapov, A.I., Vinogradov, A.G., Goritskyi, I.A. and Pertsov, E.E. (1975). About decision-making of presence of objects at group measurements. Questions of radio-electronics, Series OT, 6, 69-76.
Primak A. V., Kafarov V. V. and Kachiashvili K. J. (1991) System analysis of air and water quality control. Naukova Dumka, Kiev.
Rao C. R. (2006) Linear Statistical Inference and Its Application, 2nd ed. New York: Wiley.
Ray S. N. (1965). Bounds on the maximum sample size of a Bayes sequential procedure. Ann. Math. Statist. 36, 3, 859-878.
Robinson G. K. (1979a) Conditional properties of statistical procedures. Ann. Statist., 7, 742-755.
Robinson G. K. (1979b) Conditional properties of statistical procedures for location and scale families. Ann. Statist. 7 756-771.
Rom D. M. (1990) A Sequentially Rejective Test Procedure Based on a Modified Bonferroni Inequality, Biometrika, 77: 663-665.
Rosenthal R. and Rubin D. B. (1983) Psychological Bulletin, 94(3): 540-541.
Sage A. P. and Melse J. L. (1972) Estimation Theory with Application to Communication and Control. New York: McGraw-Hill.
Samaniego F.J. and Vestrup E. (1999) On improving standard estimators via linear empirical Bayes methods. Statist. Probab. Lett. 44 3 309-318.
Savin N. E. (1980) The Bonferroni and Scheffe Multiple Comparison Procedures, Review of Economic Studies, 47:255-213.
Savin N. E. (1984) Multiple Hypothesis Testing, Trinity College, Cambridge, Chapter 14 of Handbook of Econometrics, Volume II, Edited by Z. Griliches and M.D. Intriligator, Elsevier Science Publishers.
Schaarfsma W., Tobloom J. and Van der Menlen B. (1989) Discussing truth or falsity by computing a q-value. Statistical Data Analysis and Inference, Ed. Y. Dodge, pp. 85-100. Amsterdam: North-Holland.
Schervish M. J. (1989) A general method for comparing probability assessors. Ann. Statist., 17, 1856-1879.
Seber G. A. F. (1977) Linear Regression Analysis. New York: Wiley.
Shaffer J. P. (1986) Modified Sequentially Rejective Multiple Procedures. Journal of the American Statistical Association, 81(395): 826-831.
Shaffer J. P. (1995) Multiple hypothesis testing. Annu. Rev. Psychol., 46: 561-84.
Shaffer, J. P. (2002). Multiplicity, directional (Type III) errors, and the null hypothesis. Psychological Methods, 7(3), 356-369.
Shellard G. D. (1952) Estimating the product of several random variables, Journal of the American Statistical Association, Vol. 47, pp. 216-221.
Shi J. and Wan F. (1999) Diagnostics for empirical Bayes models. J. Systems Sci. Math. Sci. 12 2 104-114.
Shiryaev A.N. (2008) Optimal Stopping Rules. Springer-Verlag, Berlin.
Siegmund D. (1985) Sequential Analysis. Springer Series in Statistics. Springer-Verlag, NY.
Sloan I. H. and Joe S. (1994) Lattice Methods for Multiple Integration, Clarendon Press, Oxford.
Sobol I. M. (1973) Numerical Methods of Monte-Carlo. Nauka, Moscow.
Stoica P. and Viberg, M. (1996) Maximum likelihood parameter and rank estimation in reduced-rank multivariate linear regression. IEEE Transaction Signal Processing, 44, 3069-3078.
Storey, J. D. (2003). The positive false discovery rate: a Bayesian interpretation and the q-value. The Annals of Statistics, 31(6), 2013–2035.
Stuart A., Ord J. K. and Arnols S. (1994) Kendall’s Advanced Theory of Statistics. Distribution Theory, Vol. 1, 6th Edition, Oxford University Press Inc., New York.
Stuart A., Ord, J. K. and Arnols, S. (1999) Kendall’s Advanced Theory of Statistics. Classical Inference and the Linear Model. Vol. 2A, Six edition. Oxford University Press Inc., New York.
Süli E. and Mayers D. F. (2003) An Introduction to Numerical Analysis, University of Oxford: Cambridge University Press.
Szego G. (1959) Orthogonal Polynomials, American Mathematical Society, New York.
Tartakovsky A., Nikiforov I. and Basseville M. (2015) Sequential Analysis. Hypothesis Testing and Chalengepoint Detection. Taylor & Francis Group, New York.
Tartakovsky A. G., Li X. R., Yaralov G. (2003) Sequential detection of targets in multichannel systems. IEEE Transactions on Information Theory 49 (2), 425–445.
Tartakovsky A. G., Veeravalli, V. V. (2004) Change-point detection in multichannel and distributed systems with applications. In: Mukhopadhyay, N., Datta, S., Chattopadhyay, S. (Eds.), Applications of Sequential Methodologies, Marcel Dekker, Inc., New York, pp. 339–370.
Tierney L. and Kadane J. B. (1986) Accurate approximations for posterior moments and marginal densities,” Journal of The American Statistical Association, Vol. 81, pp. 82-86.
Thompson P. M. (1989). Admissibility of p-values. Ph.D. dissertation, Dept. Statistics, University Illinois-Champaign.
Thompson S. (2011) On the Distribution of Type II Errors in Hypothesis Testing. Applied Mathematics, Vol. 2, No. 2, pp. 189-195.
Toothaker L. E. (1991) Multiple Comparisons for Researchers, NewBury Park, CA: Sage.
Turin I. N. and Kulaichev A. (1999) The methods and facilities of data analysis in Windows environment. STADIA 6.0. Moscow: Informatics and komputers.
Utkin L. V. (2007) Risk Analysis under Partial Prior Information and Nonmonotone Utility Functions, International Journal Information Technology & Decision Making, 6, 4, 625-647.
Wald A. (1947a) Sequential analysis. New-York: Wiley.
Wald A. (1947b) Foundations of a General Theory of Sequential Decision Functions. Econometrica, 15: 279-313.
Wald A. and Wolfowitz J. (1948) Optimum Character of the Sequential Probability Ratio Test. Ann. Math. Statist. 19, 326-339.
Wald A. (1950) Statistical Decision Functions. Wiley, New York.
Weerahandi S. and Zidek J. V. (1981). Multi-Bayesian Statistical Decision theory, J. R. Statist. Soc. A, 144, 1, 85-93.
Westfall P. H. (1997) A Bayesian perspective on the Bonferroni adjustment. Biometrika, 84 2 419-427.
Westfall P. H., Johnson W. O. and Utts J. M. (1997) A Bayesian Perspective on the Bonferroni Adjustment. Biometrica, 84(2): 419-427.
Westfall P. H. and Krishen A. (2001)Optimally weighted, 'xed sequence and gatekeeper multiple testing procedures. Journal of Statistical Planning and Inference, 99: 25–40.
Westfall P. H. and Young S. S. (1993) Resampling-based Multiple Testing, New York: Wiley.
Wijsman R. A. (1967) Cross-Sections of Orbits and Their Application to Densities of Maximal Invariants, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1, 389-400.
Wilks S. S. (1962) Mathematical Statistics, Wiley, New York.
Wolpert R. L. (1996) Testing simple hypotheses. Data Analysis and Information Systems. Ed. H.H. Bock & W. Polasek,

7,pp.289–297. Heidelberg: Springer.
Zacks S. (1971) The Theory of Statistical Inferences, Wiley, New York.

Keywords: Statistical Hypotheses Testing; Constrained Bayesian Methods; The Parallel Methods; The Sequential Methods; Simple, Composite, Multiple and Directional Hypotheses Testing Methods

The book will be useful for undergraduates and postgraduates in the field of mathematics, mathematical statistics, applied statistics and application of statistical methods in research; Researchers in the areas of hypotheses testing and estimation theory who develop new methods as well as apply these methods to the solution of problems in different spheres of knowledge. The new approach gives to the young generation modern tools for novel, considerable achievements for solving theoretical and applied problems of mathematical statistics. Thus the book is very useful and necessary exactly for the students (coming generation) as it gives them new methods and opportunities for research.

You have not viewed any product yet.