Nonparametric roughness penalties for probability densities

ONE of the most fundamental problems in statistics is the estimation of a probability density function from a sample, the smoothing of a histogram being the usual non-parametric method. This method requires a large sample and even so it is difficult to decide whether “bumps” are genuinely in the population. A method is presented here that should help to overcome this difficulty.

[1]  H. Jeffreys,et al.  Theory of probability , 1896 .

[2]  Arthur Eddington.  The mathematical theory of relativity , 1923 .

[3]  Erwin Schrödinger,et al.  Über das Verhältnis der Heisenberg‐Born‐Jordanschen Quantenmechanik zu der meinem , 1926 .

[4]  M. Plancherel,et al.  Sur les valeurs asymptotiques des polynomes d'Hermite $$H_n (x) = ( - I)^n e^{\frac{{x^2 }}{2}} \frac{{d^n }}{{dx^n }}\left( {e^{ - \frac{{x^2 }}{2}} } \right),$$ , 1929 .

[5]  H. Weyl.  The Theory Of Groups And Quantum Mechanics , 1931 .

[6]  N. Wiener.  The Fourier Integral: and certain of its Applications , 1933, Nature.

[7]  G. N. Watson,et al.  Notes on Generating Functions of Polynomials: (2) Hermite Polynomials , 1933 .

[8]  A. Erdélyi.  Über eine erzeugende Funktion von Produkten Hermitescher Polynome , 1939 .

[9]  D. Jackson,et al.  fourier series and orthogonal polynomials , 1943, The Mathematical Gazette.

[10]  Henry Margenau,et al.  The mathematics of physics and chemistry , 1943 .

[11]  H. Jeffreys.  An invariant form for the prior probability in estimation problems , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[12]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[13]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[14]  Erwin Schrödinger,et al.  Space-time structure , 1951 .

[15]  L. J. Savage,et al.  Probability and the weighing of evidence , 1951 .

[16]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[17]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[18]  P. Whittle.  On the Smoothing of Probability Density Functions , 1958 .

[19]  A. Messiah.  Quantum Mechanics , 1961 .

[20]  Granino A. Korn,et al.  Mathematical handbook for scientists and engineers , 1961 .

[21]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[22]  M. Kendall,et al.  The advanced theory of statistics , 1945 .

[23]  I. Good.  Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables , 1963 .

[24]  Irving John Good,et al.  The Estimation of Probabilities: An Essay on Modern Bayesian Methods , 1965 .

[25]  I. Good.  A Bayesian Significance Test for Multinomial Distributions , 1967 .

[26]  I. Good.  Corroboration, Explanation, Evolving Probability, Simplicity and a Sharpened Razor , 1968, The British Journal for the Philosophy of Science.

[27]  C. I. Bliss.  Statistics in biology. Vol. 2. , 1968 .

[28]  James M. Dickey,et al.  Estimation of disease probabilities conditioned on symptom variables , 1968 .

[29]  I. Good,et al.  Utility of a Distribution , 1968, Nature.

[30]  Leslie E Ballentine,et al.  The statistical interpretation of quantum mechanics , 1970 .

[31]  David G. Kendall,et al.  Spline Transformations: Three New Diagnostic Aids for the Statistical Data‐Analyst , 1971 .