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Probability Density Function | The general formula for the probability density function of the double exponential distribution is where is the location parameter and is the scale parameter. The case where = 0 and = 1 is called the standard double exponential distribution. The equation for the standard double exponential distribution is Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the double exponential probability density function. | ||||||||||||||||
Cumulative Distribution Function | The formula for the cumulative distribution function of the double exponential distribution is The following is the plot of the double exponential cumulative distribution function. | ||||||||||||||||
Percent Point Function | The formula for the percent point function of the double exponential distribution is The following is the plot of the double exponential percent point function. | ||||||||||||||||
Hazard Function | The formula for the hazard function of the double exponential distribution is The following is the plot of the double exponential hazard function. | ||||||||||||||||
Cumulative Hazard Function | The formula for the cumulative hazard function of the double exponential distribution is The following is the plot of the double exponential cumulative hazard function. | ||||||||||||||||
Survival Function | The double exponential survival function can be computed from the cumulative distribution function of the double exponential distribution.The following is the plot of the double exponential survival function. | ||||||||||||||||
Inverse Survival Function | The formula for the inverse survival function of the double exponential distribution is The following is the plot of the double exponential inverse survival function. | ||||||||||||||||
Common Statistics |
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Parameter Estimation | The maximum likelihood estimators of the location and scale parameters of the double exponential distribution are where is the sample median. | ||||||||||||||||
Software | Some general purpose statistical software programs support at least some of the probability functions for the double exponential distribution. |
Runs Test for Detecting Non-randomness Purpose: Detect Non-Randomness The runs test ( Bradley, 1968 ) can be used to decide if a data set is from a random process. A run is defined as a series of increasing values or a series of decreasing values. The number of increasing, or decreasing, values is the length of the run. In a random data set, the probability that the ( I +1)th value is larger or smaller than the I th value follows a binomial distribution , which forms the basis of the runs test. Typical Analysis and Test Statistics The first step in the runs test is to count the number of runs in the data sequence. There are several ways to define runs in the literature, however, in all cases the formulation must produce a dichotomous sequence of values. For example, a series of 20 coin tosses might produce the f
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