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Bivariate Normal Distribution and Heuristic-Algorithm of BIVNOR for Generating Biquantile Pairs

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Decision Processes by Using Bivariate Normal Quantile Pairs

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Abstract

As it is the key chapter on software development, it begins with a characterization of bivariate normal distribution by Ludeman (Random processes, filtering, estimation and detection, Wiley India, New Delhi, 2010). That is followed by a brief presentation of the various properties of bivariate normal distribution and its applications by Essenwagner (Applied statistics in atmospheric science, part A. Frequencies and curve fitting, Elsevier, New York, 1976). Thereafter, Owen’s (1956) computational scheme for numerical integration of the bivariate normal integral is presented stepwise. The envisaged algorithm is to make iterative use of this scheme. It is very well realized that unlike univariate normal, which has a unique quantile value for a given probability level, its bivariate extension would have multiple (or infinite) quantile pairs for the same probability level. Apart from this, there arose other problems in their generation, which were sorted out and strategies to meet such problems were listed and used in perfecting the same algorithm. This was followed by finding the role of equi-quantile value BIGH for each probability level and correlation value as the initial point of such iterative scheme for the entire computational horizon of four hundred grids. In fact, the approach adopted is entirely innovative and of great economic and other consequences. Such action resulted in expansion of decision alternatives for the given or estimated correlation value without any change in probability (risk) level. Methods of forming simultaneous (joint) confidence intervals have emanated by using such biquantile pairs, having a definite edge over the existing Bonferroni’s joint confidence interval. Multiplicity of biquantile pairs offers a scope even for multiple joint confidence intervals for the same confidence probability. That could yield larger number of decision alternatives and hence the scope of choice amongst them by the decision maker on some criterion function. Those who are interested only in application of quantile pairs may skip Sect. 4.1, but they also should read its concluding part.

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Notes

  1. 1.

    Also discussed in Sect. 3.1 (in Chap. 3).

  2. 2.

    For further theoretical background, Ludeman’s text, or that of Papoulis (1965), should be consulted.

  3. 3.

    The impact of these shall be seen in examples in Chap. 7.

  4. 4.

    Further, their illustrations through varied examples are given of Chap. 7.

  5. 5.

    The role of Galton (1886, 1908) and that of Pearson (1900) in the development and application of correlation and regression at early stages has already been dealt with in Chap. 2.

  6. 6.

    His exposition of bivariate elliptical distribution, an outcome of non-zero correlation between x 1 and x 2, where x 1 (zonal wind speed data) and x 2 (meridional wind speed data), shall be discussed in Sect. 7.5 of Chap. 7.

  7. 7.

    This was needed to meet moderately high precision requirements to be explained in Sect. 4.10(8) and Chap. 5 subsequently.

  8. 8.

    It was not considered necessary to compare the results so obtained with those of others because results obtained met all the software reliability test criteria as presented in Chap. 5.

  9. 9.

    As discussed in Chap. 3.

  10. 10.

    Some of which are discussed in Chap. 7.

  11. 11.

    As mentioned in Sect. 3.5 in Chap. 3.

  12. 12.

    Some of which have been illustrated through numerous examples in Chap. 7.

  13. 13.

    Economic advantages of such properties without any change in risk level are perceptible in examples given in Sect. 7.7(A)–(F), and even in other sections of Chap. 7.

  14. 14.

    Thus the problem of the size of the two-way table having been determined, its 21 columns for the range of the correlation values between +0.95 and −0.95 are required to be spread in three sheets of Table 8.1 of Chap. 8 (seven correlation values in descending order of magnitude in a sheet), whereas its rows are required to accommodate all the 19 probability levels between 0.99 and 0.50, common for all the three sheets of the said Table 8.1 with intervals as decided above. Thus, there could be 21 columns for differing correlation values and 19 rows for changing probability levels having 21 × 19 = 399 grids for which iso-probable quantile (biquantile) pair tables are required to be generated.

  15. 15.

    Such a table of 399 grids, providing suitable data structure, is placed in Table 8.1 of Chap. 8. Its three-dimensional graph obtained by the use of MATLAB 7 with some changes in data to meet its programming requirements will be presented on a separate sheet following this page to illustrate its features.

  16. 16.

    It has been placed at the said Table 8.1 of Chap. 8 along with the corresponding SMHL values just below SMH in the same cell.

  17. 17.

    It is discussed in Sect. 3.5 in Chap. 3.

  18. 18.

    Refer to the illustration given in Sect. 7.4 in Chap. 7.

  19. 19.

    However, a negative correlation has been used in more than one example given in Sects. 7.1, 7.3 and 7.9 in Chap. 7, from which its uses and implications can easily be understood. The earliest version of the software BIVNOR, the software for generating tables of bivariate normal quantile pairs was presented by Das (1993) at the 46th Annual Conference of I.S.A.S. held at Bhubaneshwar in February 1993.

  20. 20.

    Numerical examples taken from Johnson and Wichern (op. cit.) comparing all the above methods with joint confidence intervals based on biquantile pairs have been presented in Sect. 7.2 of Chap. 7).

  21. 21.

    Please refer to http://www.biostat.wustl.edu/archives/html/s-news/2003-2msg00004.html. This monographic text and especially the tables generated and placed in Chap. 8, which are Tables 8.1 and 8.2 are the answers to David’s questions. It is also advised to read sections of Chap. 7 and descriptions on Tables in Chap. 5.

  22. 22.

    Refer to “quantile-bivariate-normal distribution” posted on Tuesday 20 December 2005, 02:37:30 GMT.

  23. 23.

    However, the abstract is released to http://atlas-conferences do not throw any light on the role of correlation values on the estimates of quantiles. Hopefully, they could have done so. Though the said work appears to be nearer to the present one, it throws no light on the actual generation of quantile pairs for a given probability level and known or estimated correlation values as has been obtained in Chap. 8 of the book.

  24. 24.

    Also presented in Table 8.1 of Chap. 8.

  25. 25.

    This is reported in Chap. 5 with examples in Chap. 9.

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  1. Department of Statistics, Birsa Agricultural University, Ranchi, Jharkhand, India

    N. C. Das

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Das, N.C. (2015). Bivariate Normal Distribution and Heuristic-Algorithm of BIVNOR for Generating Biquantile Pairs. In: Decision Processes by Using Bivariate Normal Quantile Pairs. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2364-1_4

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