Calculates the probability density for the double Poisson distribution (V.M. Cave).

### Options

`PRINT` = string tokens |
Controls printed output (`probability` , `summary` ); default `prob` |
---|---|

`PLOT` = string token |
Whether to plot the k terms used to approximate the normalizing constant by the `kpartialsum` method (`yes` , `no` ); default `no` |

`METHOD` = string token |
How to approximate the normalizing constant (`kpartialsum` , `edgeworth` ); default `kpar` |

`LOCATION` = scalar or variate |
Location parameter; no default, must be set |

`SHAPE` = scalar or variate |
Shape parameter; default 1 |

`MAXCYCLE` = scalar or variate |
Limits the number of terms, k, used to approximate the normalizing constant by the `kpartialsum` method; default `MAX` (`1000, 2*LOCATION` ) |

`TOLERANCE` = scalar |
Convergence criterion used when approximating the normalizing constant by the `kpartialsum` method; default `1E-12` |

### Parameters

`DATA` = scalar or variate |
Non-negative integer values for which the double Poisson probabilities are to be calculated |
---|---|

`DECIMALS` = scalars |
Number of decimal places for printing; default `*` |

`PROBABILITY` = variate |
Saves the probabilities |

### Description

PRDOUBLEPOISSON calculates the probability density for the two-parameter double Poisson distribution. The double Poisson probability density is given by

*P(X=x) = c(μ,θ) θ ^{1/2}* e

*( e*

^{-θμ}*x*

^{-x}*/ x!) (e*

^{x}*μ / x*)

^{θx }for

*μ*> 0,

*θ*> 0, x = 0, 1, 2 …

where *c(μ,θ)* is the normalizing constant, *μ* is the location parameter, and *θ* is the shape parameter. The double Poisson distribution is over-dispersed when *θ* > 1, under-dispersed when 0< θ < 1, and is identical to the Poisson distribution when *θ* =1.

The non-negative integers, for which the double Poisson probabilities are to be calculated, are supplied by the `DATA`

parameter.

The location parameter, *μ*, must be specified using the `LOCATION`

option. The shape parameter, *θ*, can be set using the `SHAPE`

option; default 1. For both the `LOCATION`

and `SHAPE`

options, either a single value (scalar or variate of length 1) or a variate containing the same number of values as `DATA`

may be supplied.

The `METHOD`

option specifies the method used to approximate the normalizing constant. The default (`METHOD=kpartialsum`

) is to use the more accurate and reliable *k*-th partial sum method proposed by Zou et al. (2013). This method involves summing the first *k* terms of an infinite sum (see the *Method* section). The number of terms, *k*, is determined by the `TOLERANCE`

and `MAXCYCLE`

options. The `TOLERANCE`

option can supply a scalar to specify the tolerance for convergence of the infinite sum; default `1E-12`

. The `MAXCYCLE`

option places a limit on *k*, where the default is the maximum of 1000 and twice the value of the location parameter. If the infinite sum fails to converge within `k = MAXCYCLE`

, the probability density is not calculated and a warning is given. `MAXCYCLE`

may supply either a single value (scalar or variate of length 1) or a variate containing the same number of values as `DATA`

. (However, if both `LOCATION`

and `SHAPE`

supply single values, only the first value of `MAXCYCLE`

is used.) The `PLOT`

option allows you to request a plot of the *k* terms used to approximate the normalizing constant. By default no plot is produced.

Although the *k-*th partial sum method converges very quickly when the location parameter is small, convergence for large values of the location parameter requires a large value for *k*. The closed-form Edgeworth series method of Efron (1986) may then provide an alternative way of approximating the normalizing constant (`METHOD=edgeworth`

). However, the Edgeworth series approximation is highly unreliable for small values of the location parameter, i.e. values less than about 10.

Printed output is controlled by the `PRINT`

option, with settings:

`probability ` (the default) |
prints the probability density, and |

`summary` |
prints a description and a table containing; the data value, the location and shape parameters, the approximation of the normalizing constant, k (if `METHOD=kpartialsum` ), and the probability density. |

The `DECIMALS`

parameter allows you to set the number of decimal places to appear in the printed output.

The `PROBABILITY`

parameter can save the probability densities, in a variate.

Options: `PRINT`

, `PLOT`

, `METHOD`

, `LOCATION`

, `SHAPE`

, `MAXCYCLE`

, `TOLERANCE`

.

Parameters: `DATA`

, `DECIMALS`

, `PROBABILITY`

.

### Method

The normalizing constant for the double Poisson distribution, *c(μ,θ)*, is given by an infinite sum. `PRDOUBLEPOISSON`

offers two methods for approximating the constant: the *k*-th partial sum method of Zou et al. (2013), i.e. `METHOD=kpartialsum`

, and the Edgeworth series method of Efron (1986), i.e. `METHOD=edgeworth`

.

The *k*-th partial sum method uses the sum of the first *k* terms of the infinite sum. The number of terms is determined by the `TOLERANCE`

option, which specifies the convergence criterion. The infinite sum is assumed to have converged when

f_{μ,θ} (*X = k-1*) > *f _{μ,θ} (X = k)*

and

*f _{μ,θ} (X = k)* <

`TOLERANCE`

.If the infinite sum fails to converge within `k = MAXCYCL`

E, the probability density is not calculated and a warning is given.

The Edgeworth series method of Efron (1986) provides a closed-form approximation to the infinite sum.

1 / c(*μ,θ*) = ∑_{x = 0…∞} ^{f}_{μ, θ}*(x)*

The *k*-th partial sum method is more accurate and more reliable than the Edgeworth series approximation. In particular, the Edgeworth series method is highly unreliable when the location parameter, *θ*, is small (i.e. less than about 10), and may even produce negative values.

### Action with `RESTRICT`

The `DATA`

, `LOCATION`

, `SHAPE`

and `MAXCYCLE`

variates can be restricted.

### References

Efron, B. (1986). Double exponential families and their use in generalized linear regression. *Journal of the American Statistical Association*, **81**, 709-721.

Zou, Y., Geedipally, S.R. & Lord, D. (2013). Evaluating the double Poisson generalized linear model. *Accident Analysis & Prevention*, **59**, 497-505.

### See also

Commands for: Basic and nonparametric statistics.

### Example

CAPTION 'PRDOUBLEPOISSON example'; STYLE=meta PRDOUBLEPOISSON [PRINT=summary; PLOT=yes; LOCATION=10; SHAPE=0.5] !(0...24) PRDOUBLEPOISSON [PRINT=summary; METHOD=edgeworth; LOCATION=10; SHAPE=0.5] !(0...24) PRDOUBLEPOISSON [PRINT=summary; PLOT=yes; LOCATION=!(1...10); SHAPE=0.5] !(10(2)) PRDOUBLEPOISSON [PRINT=summary; PLOT=yes; LOCATION=10; SHAPE=!(1...10)/10] !(10(2))