In probability theory, an exponentially modified Gaussian distribution describes the sum of independent normal and exponential random variables. An exGaussian random variableZ may be expressed as, where X and Y are independent, X is Gaussian with meanμ and variance σ2, and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component. It may also be regarded as a weighted function of a shifted exponential with the weight being a function of the normal distribution.
An alternative but equivalent form of the EMG distribution is used for description of peak shape in chromatography. This is as follows where This function cannot be calculated for some values of parameters because of arithmetic overflow. Alternative, but equivalent form of writing the function was proposed by Delley: where is a scaled complementary error function In the case of this formula arithmetic overflow is also possible, region of overflow is different from the first formula, except for very small τ. For small τ it is reasonable to use asymptotic form of the second formula: Decision on formula usage is made on the basis of the parameter : Mode is calculated using derivative of formula 2; inverse of scaled complementary error function erfcxinv is used for calculation. The apex is always located on the original Gaussian.
Parameter estimation
There are three parameters: the mean of the normal distribution, the standard deviation of the normal distribution and the exponential decay parameter. The shape K = τ / σ is also sometimes used to characterise the distribution. Depending on the values of the parameters, the distribution may vary in shape from almost normal to almost exponential. The parameters of the distribution can be estimated from the sample data with the method of moments as follows: where m is the sample mean, s is the sample standard deviation, and γ1 is the skewness. Solving these for the parameters gives:
Recommendations
Ratcliff has suggested that there be at least 100 data points in the sample before the parameter estimates should be regarded as reliable. Vincent averaging may be used with smaller samples, as this procedure only modestly distorts the shape of the distribution. These point estimates may be used as initial values that can be refined with more powerful methods, including maximum likelihood.
Confidence intervals
There are currently no published tables available for significance testing with this distribution. The distribution can be simulated by forming the sum of two random variables one drawn from a normal distribution and the other from an exponential.
Skew
The value of the nonparametric skew of this distribution lies between 0 and 0.31. The lower limit is approached when the normal component dominates, and the upper when the exponential component dominates.
This family of distributions is a special or limiting case of the normal-exponential-gamma distribution. This can also be seen as a three-parameter generalization of a normal distribution to add skew; another distribution like that is the skew normal distribution, which has thinner tails. The distribution is a compound probability distribution in which the mean of a normal distribution varies randomly as a shifted exponential distribution. A Gaussian minus exponential distribution has been suggested for modelling option prices. If such a random variable Y has parameters μ, σ, λ, then its negative -Y has an exponentially modified Gaussian distribution with parameters -μ, σ, λ, and thus Y has mean and variance.
