Zero bias transform

Last updated October 30, 2019

The zero-bias transform is a transform from one probability distribution to another. The transform arises in applications of Stein's method in probability and statistics.

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable $X$ is used to denote the outcome of a coin toss, then the probability distribution of $X$ would take the value 0.5 for $X = heads$ , and 0.5 for $X = tails$ . Examples of random phenomena can include the results of an experiment or survey.

Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972, to obtain a bound between the distribution of a sum of $-dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform) metric and hence to prove not only a central limit theorem, but also bounds on the rates of convergence for the given metric.$

Formal definition

The zero bias transform may be applied to both discrete and continuous random variables. The zero bias transform of a density function $f (t)$ , defined for all real numbers $t \geq 0$ , is the function $g (s)$ , defined by

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as $π$ (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

g(s)=\int _{s}^{\infty }tf(t)1(t>s)\,dt

where s and t are real numbers and f(t) is the density or mass function of the random variable T.^[1]

An equivalent but alternative approach is to deduce the nature of the transformed random variable by evaluating the expected value

\operatorname {E} (TH(T))=\sigma ^{2}E(h(T^{z}))

where the right-side superscript denotes a zero biased random variable whereas the left hand side expectation represents the original random variable. An example from each approach is given in the examples section beneath.

If the random variable is discrete the integral becomes a sum from positive infinity to s. The zero bias transform is taken for a mean zero, variance 1 random variable which may require a location-scale transform to the random variable.

Applications

The zero bias transformation arises in applications where a normal approximation is desired. Similar to Stein's method the zero bias transform is often applied to sums of random variables with each summand having finite variance an mean zero.

The zero bias transform has been applied to CDO tranche pricing.^[2]

Examples

1. Consider a Bernoulli(p) random variable B with Pr(B = 0) = 1 − p. The zero bias transform of T = (B − p) is:

{\begin{aligned}\operatorname {E} (TH(T))&=-p(1-p)H(-p)+(1-p)pH(1-p)\\&=p(1-p)[H(1-p)-H(-p)]\\&=p(1-p)\int _{-p}^{1-p}h(s)\,ds\end{aligned}}

where h is the derivative of H. From there it follows that the random variable S is a continuous uniform random variable on the support (−p, 1 − p). This example shows how the zero bias transform smooths a discrete distribution into a continuous distribution.

2. Consider the continuous uniform on the support $(-{\sqrt {3}},{\sqrt {3}})$ .

\int _{s}^{\sqrt {3}}t1(t>s)f(t)\,dt=\int _{s}^{\sqrt {3}}{\frac {t}{2{\sqrt {3}}}}\,dt={\frac {\sqrt {3}}{4}}-{\frac {s^{2}}{{\sqrt {3}}\,4}}{\text{ where }}-{\sqrt {3}}<s<{\sqrt {3}}

This example shows that the zero bias transform takes continuous symmetric distributions and makes them unimodular.

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References

↑ Goldstein, Larry; Reinert, Gesine (1997), "Stein's Method and the Zero Bias Transformation with Application to Simple Random Sampling" (PDF), The Annals of Applied Probability, 7 (4): 935–952
↑ Karoui, N. El; Jiao, Y. (2009). "Stein's method and zero bias transformation for CDO tranche pricing". Finance and Stochastics. 13 (2): 151–180. doi:10.1007/s00780-008-0084-6.

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[1] Goldstein, Larry; Reinert, Gesine (1997), "Stein's Method and the Zero Bias Transformation with Application to Simple Random Sampling" (PDF), The Annals of Applied Probability, 7 (4): 935–952

[2] Karoui, N. El; Jiao, Y. (2009). "Stein's method and zero bias transformation for CDO tranche pricing". Finance and Stochastics. 13 (2): 151–180. doi:10.1007/s00780-008-0084-6.