Some probability density functions and their characteristic functions

Author:
Frank McNolty

Journal:
Math. Comp. **27** (1973), 495-504

MSC:
Primary 65C99; Secondary 60E05

MathSciNet review:
0329193

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Abstract: This paper presents, without derivation, several generalized density functions together with their characteristic functions. The densities are expressed variously in terms of special functions such as: , the modified Bessel function of the first kind of order *v*; , the modified Bessel function of the second kind of order *v*; , the confluent hypergeometric function; , the hypergeometric function; , Whittaker's function; , a generalized hypergeometric function (type I);

**[1]**B. C. Bhattacharyya,*The use of McKay’s Bessel function curves for graduating frequency distributions*, Sankhyā**6**(1942), 175–182. MR**0009272****[2]**R. C. Bose, "On the exact distribution and moment-coefficients of the -statistic,"*Sankhyā*, v. 2, 1936, pp. 143-154.**[3]**S. S. Bose, "On a Bessel function population,"*Sankhyā*, v. 3, 1938, pp. 253-261.**[4]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Tables of Integral Transforms*, Vol. 1, McGraw-Hill, New York and London, 1954. MR**15**, 868.**[5]**J. O. Irwin, "On the frequency distribution of the means of samples,"*Biometrika*, v. 19, 1927, pp. 225-239.**[6]**K. V. Krishna Sastry,*On a Bessel function of the second kind and Wilks’ 𝑍-distribution*, Proc. Indian Acad. Sci., Sect. A.**28**(1948), 532–536. MR**0027995****[7]**R. G. Laha,*On some properties of the Bessel function distributions*, Bull. Calcutta Math. Soc.**46**(1954), 59–72. MR**0063617****[8]**R. D. Lord,*The use of the Hankel transform in statistics. I. General theory and examples*, Biometrika**41**(1954), 44–55. MR**0061791****[9]**R. D. Lord,*The use of the Hankel transform in statistics. II. Methods of computation*, Biometrika**41**(1954), 344–350. MR**0065090****[10]**R. D. Lord,*The distribution of distance in a hypersphere*, Ann. Math. Statistics**25**(1954), 794–798. MR**0065048****[11]**Yudell L. Luke,*Integrals of Bessel functions*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR**0141801****[12]**Yudell L. Luke,*The Spectral Functions and Their Approximations*, Vol. 1, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR**39**#3039.**[13]**Yudell L. Luke,*The Special Functions and Their Approximations*. Vol. 2, Math. in Sci. and Engineering, vol. 53, Academic Press, New York, 1969. MR**40**#2909.**[14]**A. T. McKay, "A Bessel function distribution,"*Biometrika*, v. 24, 1931, pp. 39-44.**[15]**Frank McNolty,*A contour-integral derivation of the non-central chi-square distribution*, Ann. Math. Statist.**33**(1962), 796–800. MR**0137195****[16]**Frank McNolty,*Applications of Bessel function distributions*, Sankhyā Ser. B**29**(1967), 235–248. MR**0228088****[17]**Frank McNolty, "A note on radial distributions,"*Operations Res.*, v. 16, 1968a, pp. 211-216.**[18]**Frank McNolty, "Expected coverage for targets of nonuniform density,"*Operations Res.*, v. 16, 1968b, pp. 1027-1040.**[19]**Frank McNolty,*Quadratic form distributions associated with special functions*, Sankhyā Ser. B**34**(1972), 21–26. MR**0326923****[20]**Frank McNolty and Jack Tomsky,*Some properties of special-function, bivariate distributions*, Sankhyā Ser. B**34**(1972), 251–264. MR**0336873****[21]**Frank McNolty, "Random vectors with non-uniform phase distributions." (Submitted.)**[22]**Frank McNolty,*Reliability density functions when the failure rate is randomly distributed*, Sankhyā Ser. A**26**(1964), 287–292. MR**0187266****[23]**Frank McNolty, R. Clow & E. Hansen, "Some matched filter configurations for infrared systems,"*IEEE Trans. Aerospace and Electronic Systems*, v. AES-8, 1972, pp. 428-438.**[24]**Frank McNolty, R. Clow & E. Hansen, "Some properties of the output of integrator in an infrared system,"*IEEE Trans. Aerospace and Electronic Systems*, v. AES-8, 1972, pp. 552-558.**[25]**Karl Pearson, "Further applications in statistics of the Bessel function,"*Biometrika*, v. 24, 1932, pp. 293-350.**[26]**Raj Des,*On a generalised Bessel function population*, Ganita**3**(1953), 111—115. MR**0053444****[27]**Norman L. Johnson and Samuel Kotz,*Distributions in statistics. Continuous univariate distributions. 1.*, Houghton Mifflin Co., Boston, Mass., 1970. MR**0270475****[28]**Norman L. Johnson and Samuel Kotz,*Distributions in statistics. Continuous univariate distributions. 2.*, Houghton Mifflin Co., Boston, Mass., 1970. MR**0270476**

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DOI:
https://doi.org/10.1090/S0025-5718-1973-0329193-3

Article copyright:
© Copyright 1973
American Mathematical Society