Package 'zipfextR'

Calculates initial values for the parameters of the models.

Description

The selection of appropiate initial values to compute the maximum likelihood estimations reduces the number of iterations which in turn, reduces the computation time. The initial values proposed by this function are computed using the first two empirical frequencies.

Usage

getInitialValues(data, model = "zipf")

Arguments

data	Matrix of count data.
model	Specify the model that requests the initial values (default='zipf').

Details

The argument data is a two column matrix with the first column containing the observations and the second column containing their frequencies. The argument model refers to the selected model of those implemented in the package. The possible values are: zipf, moezipf, zipfpe, zipfpss or its zero truncated version zt_zipfpss. By default, the selected model is the Zipf one.

For the MOEZipf, the Zipf-PE and the zero truncated Zipf-PSS models that contain the Zipf model as a particular case, the $\beta$ value will correspond to the one of the Zipf model (i.e. $\beta = 1$ for the MOEZipf, $\beta = 0$ for the Zipf-PE and $\lambda = 0$ for the zero truncated Zipf-PSS model) and the initial value for $\alpha$ is set to be equal to:

$\alpha_0 = log_2 \big (\frac{f_r(1)}{f_r(2)} \big),$

where $f_r(1)$ and $f_r(2)$ are the empirical relative frequencies of one and two. This value is obtained equating the two empirical probabilities to their theoritical ones.

For the case of the Zipf-PSS the proposed initial values are obtained equating the empirical probability of zero to the theoretical one which gives:

$\lambda_0 = -log(f_r(0)),$

where $f_r(0)$ is the empirical relative frequency of zero. The initial value of $\alpha$ is obtained equating the ratio of the theoretical probabilities at zero and one to the empirical ones. This gives place to:

$\alpha_0 = \zeta^{-1}(\lambda_0 * f_r(0)/f_r(1)),$

where $f_r(0)$ and $f_r(1)$ are the empirical relative frequencies associated to the values 0 and 1 respectively. The inverse of the Riemman Zeta function is obtained using the optim routine.

Value

Returns the initial values of the parameters for a given distribution.

References

Güney, Y., Tuaç, Y., & Arslan, O. (2016). Marshall–Olkin distribution: parameter estimation and application to cancer data. Journal of Applied Statistics, 1-13.

Examples

data <- rmoezipf(100, 2.5, 1.3)
data <- as.data.frame(table(data))
data[,1] <- as.numeric(levels(data[,1])[data[,1]])
initials <- getInitialValues(data, model='zipf')

---

The Marshal-Olkin Extended Zipf Distribution (MOEZipf).

Description

Probability mass function, cumulative distribution function, quantile function and random number generation for the MOEZipf distribution with parameters $\alpha$ and $\beta$. The support of the MOEZipf distribution are the strictly positive integer numbers large or equal than one.

Usage

dmoezipf(x, alpha, beta, log = FALSE)

pmoezipf(q, alpha, beta, log.p = FALSE, lower.tail = TRUE)

qmoezipf(p, alpha, beta, log.p = FALSE, lower.tail = TRUE)

rmoezipf(n, alpha, beta)

Arguments

x, q	Vector of positive integer values.
alpha	Value of the $\alpha$ parameter ($\alpha > 1$ ).
beta	Value of the $\beta$ parameter ($\beta > 0$ ).
log, log.p	Logical; if TRUE, probabilities p are given as log(p).
lower.tail	Logical; if TRUE (default), probabilities are $P[X \leq x]$, otherwise, $P[X > x]$.
p	Vector of probabilities.
n	Number of random values to return.

Details

The probability mass function at a positive integer value $x$ of the MOEZipf distribution with parameters $\alpha$ and $\beta$ is computed as follows:

$p(x | \alpha, \beta) = \frac{x^{-\alpha} \beta \zeta(\alpha) }{[\zeta(\alpha) - \bar{\beta} \zeta (\alpha, x)] [\zeta (\alpha) - \bar{\beta} \zeta (\alpha, x + 1)]},\, x = 1,2,...,\, \alpha > 1, \beta > 0,$

where $\zeta(\alpha)$ is the Riemann-zeta function at $\alpha$, $\zeta(\alpha, x)$ is the Hurtwitz zeta function with arguments $\alpha$ and x, and $\bar{\beta} = 1 - \beta$.

The cumulative distribution function, at a given positive integer value $x$, is computed as $F(x) = 1 - S(x)$, where the survival function $S(x)$ is equal to:

$S(x) = \frac{\beta\, \zeta(\alpha, x + 1)}{\zeta(\alpha) - \bar{\beta}\,\zeta(\alpha, x + 1)},\, x = 1, 2, ..$

The quantile of the MOEZipf$(\alpha, \beta)$ distribution of a given probability value p is equal to the quantile of the Zipf$(\alpha)$ distribution at the value:

$p\prime = \frac{p\,\beta}{1 + p\,(\beta - 1)}$

The quantiles of the Zipf$(\alpha)$ distribution are computed by means of the tolerance package.

To generate random data from a MOEZipf one applies the quantile function over n values randomly generated from an Uniform distribution in the interval (0, 1).

Value

dmoezipf gives the probability mass function, pmoezipf gives the cumulative distribution function, qmoezipf gives the quantile function, and rmoezipf generates random values from a MOEZipf distribution.

References

Casellas, A. (2013) La distribució Zipf Estesa segons la transformació Marshall-Olkin. Universitat Politécnica de Catalunya.

Devroye L. (1986) Non-Uniform Random Variate Generation. Springer, New York, NY.

Duarte-López, A., Prat-Pérez, A., & Pérez-Casany, M. (2015). Using the Marshall-Olkin Extended Zipf Distribution in Graph Generation. European Conference on Parallel Processing, pp. 493-502, Springer International Publishing.

Pérez-Casany, M. and Casellas, A. (2013) Marshall-Olkin Extended Zipf Distribution. arXiv preprint arXiv:1304.4540.

Young, D. S. (2010). Tolerance: an R package for estimating tolerance intervals. Journal of Statistical Software, 36(5), 1-39.

Examples

dmoezipf(1:10, 2.5, 1.3)
pmoezipf(1:10, 2.5, 1.3)
qmoezipf(0.56, 2.5, 1.3)
rmoezipf(10, 2.5, 1.3)

---

MOEZipf parameters estimation.

Description

For a given sample of strictly positive integer numbers, usually of the type of ranking data or frequencies of frequencies data, estimates the parameters of the MOEZipf distribution by means of the maximum likelihood method. The input data should be provided as a frequency matrix.

Usage

moezipfFit(data, init_alpha = NULL, init_beta = NULL, level = 0.95,
  ...)

## S3 method for class 'moezipfR'
residuals(object, ...)

## S3 method for class 'moezipfR'
fitted(object, ...)

## S3 method for class 'moezipfR'
coef(object, ...)

## S3 method for class 'moezipfR'
plot(x, ...)

## S3 method for class 'moezipfR'
print(x, ...)

## S3 method for class 'moezipfR'
summary(object, ...)

## S3 method for class 'moezipfR'
logLik(object, ...)

## S3 method for class 'moezipfR'
AIC(object, ...)

## S3 method for class 'moezipfR'
BIC(object, ...)

Arguments

data	Matrix of count data in form of a table of frequencies.
init_alpha	Initial value of $\alpha$ parameter ($\alpha > 1$).
init_beta	Initial value of $\beta$ parameter ($\beta > 0$).
level	Confidence level used to calculate the confidence intervals (default 0.95).
...	Further arguments to the generic functions. The extra arguments are passing to the optim function.
object	An object from class "moezipfR" (output of moezipfFit function).
x	An object from class "moezipfR" (output of moezipfFit function).

Details

The argument data is a two column matrix with the first column containing the observations and the second column containing their frequencies.

The log-likelihood function is equal to:

$l(\alpha, \beta; x) = -\alpha \sum_{i = 1} ^m f_{a}(x_{i}) log(x_{i}) + N (log(\beta) + \log(\zeta(\alpha)))$

$- \sum_{i = 1} ^m f_a(x_i) log[(\zeta(\alpha) - \bar{\beta}\zeta(\alpha, x_i)(\zeta(\alpha) - \bar{\beta}\zeta(\alpha, x_i + 1)))],$

where $f_{a}(x_i)$ is the absolute frequency of $x_i$, $m$ is the number of different values in the sample and $N$ is the sample size, i.e. $N = \sum_{i = 1} ^m x_i f_a(x_i)$.

By default the initial values of the parameters are computed using the function getInitialValues.

The function optim is used to estimate the parameters.

Value

Returns a moezipfR object composed by the maximum likelihood parameter estimations jointly with their standard deviation and confidence intervals. It also contains the value of the log-likelihood at the maximum likelihood estimator.

See Also

getInitialValues.

Examples

data <- rmoezipf(100, 2.5, 1.3)
data <- as.data.frame(table(data))
data[,1] <- as.numeric(as.character(data[,1]))
data[,2] <- as.numeric(as.character(data[,2]))
initValues <- getInitialValues(data, model='moezipf')
obj <- moezipfFit(data, init_alpha = initValues$init_alpha, init_beta = initValues$init_beta)

---

Expected value.

Description

Computes the expected value of the MOEZipf distribution for given values of parameters $\alpha$ and $\beta$.

Usage

moezipfMean(alpha, beta, tolerance = 10^(-4))

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 2$).
beta	Value of the $\beta$ parameter ($\beta > 0$).
tolerance	Tolerance used in the calculations (default = $10^{-4}$).

Details

The mean of the distribution only exists for $\alpha$ strictly greater than 2. It is computed by calculating the partial sums of the serie, and stopping when two consecutive partial sums differ less than the tolerance value. The value of the last partial sum is returned.

Value

A positive real value corresponding to the mean value of the distribution.

Examples

moezipfMean(2.5, 1.3)
moezipfMean(2.5, 1.3, 10^(-3))

---

Distribution Moments.

Description

General function to compute the k-th moment of the MOEZipf distribution for any integer value $k \geq 1$, when it exists. The k-th moment exists if and only if $\alpha > k + 1$. For k = 1, this function returns the same value as the moezipfMean function.

Usage

moezipfMoments(k, alpha, beta, tolerance = 10^(-4))

Arguments

k	Order of the moment to compute.
alpha	Value of the $\alpha$ parameter ($\alpha > k + 1$).
beta	Value of the $\beta$ parameter ($\beta > 0$).
tolerance	Tolerance used in the calculations (default = $10^{-4}$).

Details

The k-th moment is computed by calculating the partial sums of the serie, and stopping when two consecutive partial sums differ less than the tolerance value. The value of the last partial sum is returned.

Value

A positive real value corresponding to the k-th moment of the distribution.

Examples

moezipfMoments(3, 4.5, 1.3)
moezipfMoments(3, 4.5, 1.3,  1*10^(-3))

---

Variance of the MOEZipf distribution.

Description

Computes the variance of the MOEZipf distribution for given values of $\alpha$ and $\beta$.

Usage

moezipfVariance(alpha, beta, tolerance = 10^(-4))

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 3$).
beta	Value of the $\beta$ parameter ($\beta > 0$).
tolerance	Tolerance used in the calculations. (default = $10^{-4}$)

Details

The variance of the distribution only exists for $\alpha$ strictly greater than 3.

Value

A positive real value corresponding to the variance of the distribution.

See Also

moezipfMoments, moezipfMean.

Examples

moezipfVariance(3.5, 1.3)

---

The Zero Inflated Zipf-Poisson Stop Sum Distribution (ZI Zipf-PSS).

Description

Probability mass function for the zero inflated Zipf-PSS distribution with parameters $\alpha$, $\lambda$ and $w$. The support of thezero inflated Zipf-PSS distribution are the positive integer numbers including the zero value.

Usage

d_zi_zipfpss(x, alpha, lambda, w, log = FALSE)

Arguments

x	Vector of positive integer values.
alpha	Value of the $\alpha$ parameter ($\alpha > 1$ ).
lambda	Value of the $\lambda$ parameter ($\lambda > 0$ ).
w	Value of the $w$ parameter (0 < $w < 1$ ).
log	Logical; if TRUE, probabilities p are given as log(p).

Details

The support of the $\lambda$ parameter increases when the distribution is truncated at zero being $\lambda \geq 0$. It has been proved that when $\lambda = 0$ one has the degenerated version of the distribution at one.

References

Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 22-26.

Sundt, B., & Jewell, W. S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 27-39.

---

Zero Inflated Zipf-PSS parameters estimation.

Description

For a given sample of strictly positive integer numbers, usually of the type of ranking data or frequencies of frequencies data, estimates the parameters of the zero inflated Zipf-PSS distribution by means of the maximum likelihood method. The input data should be provided as a frequency matrix.

Usage

zi_zipfpssFit(data, init_alpha = 1.5, init_lambda = 1.5,
  init_w = 0.1, level = 0.95, ...)

## S3 method for class 'zi_zipfpssR'
residuals(object, ...)

## S3 method for class 'zi_zipfpssR'
fitted(object, ...)

## S3 method for class 'zi_zipfpssR'
coef(object, ...)

## S3 method for class 'zi_zipfpssR'
plot(x, ...)

## S3 method for class 'zi_zipfpssR'
print(x, ...)

## S3 method for class 'zi_zipfpssR'
summary(object, ...)

## S3 method for class 'zi_zipfpssR'
logLik(object, ...)

## S3 method for class 'zi_zipfpssR'
AIC(object, ...)

## S3 method for class 'zi_zipfpssR'
BIC(object, ...)

Arguments

data	Matrix of count data in form of table of frequencies.
init_alpha	Initial value of $\alpha$ parameter ($\alpha > 1$).
init_lambda	Initial value of $\lambda$ parameter ($\lambda > 0$).
init_w	Initial value of $w$ parameter ($0 < w < 1$).
level	Confidence level used to calculate the confidence intervals (default 0.95).
...	Further arguments to the generic functions. The extra arguments are passing to the optim function.
object	An object from class "zpssR" (output of zipfpssFit function).
x	An object from class "zpssR" (output of zipfpssFit function).

Details

The argument data is a two column matrix with the first column containing the observations and the second column containing their frequencies.

References

Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 22-26.

Sundt, B., & Jewell, W. S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 27-39.

See Also

getInitialValues.

Examples

data <- rzipfpss(100, 2.5, 1.3)
data <- as.data.frame(table(data))
data[,1] <- as.numeric(as.character(data[,1]))
data[,2] <- as.numeric(as.character(data[,2]))
obj <- zipfpssFit(data, init_alpha = 1.5, init_lambda = 1.5)

---

The Zipf-Poisson Extreme Distribution (Zipf-PE).

Description

Probability mass function, cumulative distribution function, quantile function and random number generation for the Zipf-PE distribution with parameters $\alpha$ and $\beta$. The support of the Zipf-PE distribution are the strictly positive integer numbers large or equal than one.

Usage

dzipfpe(x, alpha, beta, log = FALSE)

pzipfpe(q, alpha, beta, log.p = FALSE, lower.tail = TRUE)

qzipfpe(p, alpha, beta, log.p = FALSE, lower.tail = TRUE)

rzipfpe(n, alpha, beta)

Arguments

x, q	Vector of positive integer values.
alpha	Value of the $\alpha$ parameter ($\alpha > 1$ ).
beta	Value of the $\beta$ parameter ($\beta\in (-\infty, +\infty)$ ).
log, log.p	Logical; if TRUE, probabilities p are given as log(p).
lower.tail	Logical; if TRUE (default), probabilities are $P[X \leq x]$, otherwise, $P[X > x]$.
p	Vector of probabilities.
n	Number of random values to return.

Details

The probability mass function of the Zipf-PE distribution with parameters $\alpha$ and $\beta$ at a positive integer value $x$ is computed as follows:

$p(x | \alpha, \beta) = \frac{e^{\beta (1 - \frac{\zeta(\alpha, x)}{\zeta(\alpha)})} (e^{\beta \frac{x^{-\alpha}}{\zeta(\alpha)}} - 1)} {e^{\beta} - 1},\, x= 1,2,...,\, \alpha > 1,\, -\infty < \beta < +\infty,$

where $\zeta(\alpha)$ is the Riemann-zeta function at $\alpha$, and $\zeta(\alpha, x)$ is the Hurtwitz zeta function with arguments $\alpha$ and x.

The cumulative distribution function at a given positive integer value $x$, $F(x)$, is equal to:

$F(x) = \frac{e^{\beta (1 - \frac{\zeta(\alpha, x + 1)}{\zeta(\alpha)})} - 1}{e^{\beta} -1}$

The quantile of the Zipf-PE$(\alpha, \beta)$ distribution of a given probability value p is equal to the quantile of the Zipf$(\alpha)$ distribution at the value:

$p\prime = \frac{log(p\, (e^{\beta} - 1) + 1)}{\beta}$

The quantiles of the Zipf$(\alpha)$ distribution are computed by means of the tolerance package.

To generate random data from a Zipf-PE one applies the quantile function over n values randomly generated from an Uniform distribution in the interval (0, 1).

Value

dzipfpe gives the probability mass function, pzipfpe gives the cumulative function, qzipfpe gives the quantile function, and rzipfpe generates random values from a Zipf-PE distribution.

References

Young, D. S. (2010). Tolerance: an R package for estimating tolerance intervals. Journal of Statistical Software, 36(5), 1-39.

Examples

dzipfpe(1:10, 2.5, -1.5)
pzipfpe(1:10, 2.5, -1.5)
qzipfpe(0.56, 2.5, 1.3)
rzipfpe(10, 2.5, 1.3)

---

Zipf-PE parameters estimation.

Description

For a given sample of strictly positive integer values, usually of the type of ranking data or frequencies of frequencies data, estimates the parameters of the Zipf-PE distribution by means of the maximum likelihood method. The input data should be provided as a frequency matrix.

Usage

zipfpeFit(data, init_alpha = NULL, init_beta = NULL, level = 0.95,
  ...)

## S3 method for class 'zipfpeR'
residuals(object, ...)

## S3 method for class 'zipfpeR'
fitted(object, ...)

## S3 method for class 'zipfpeR'
coef(object, ...)

## S3 method for class 'zipfpeR'
plot(x, ...)

## S3 method for class 'zipfpeR'
print(x, ...)

## S3 method for class 'zipfpeR'
summary(object, ...)

## S3 method for class 'zipfpeR'
logLik(object, ...)

## S3 method for class 'zipfpeR'
AIC(object, ...)

## S3 method for class 'zipfpeR'
BIC(object, ...)

Arguments

data	Matrix of count data in form of table of frequencies.
init_alpha	Initial value of $\alpha$ parameter ($\alpha > 1$).
init_beta	Initial value of $\beta$ parameter ($\beta \in (-\infty, +\infty)$).
level	Confidence level used to calculate the confidence intervals (default 0.95).
...	Further arguments to the generic functions.The extra arguments are passing to the optim function.
object	An object from class "zpeR" (output of zipfpeFit function).
x	An object from class "zpeR" (output of zipfpeFit function).

Details

The argument data is a two column matrix with the first column containing the observations and the second column containing their frequencies.

The log-likelihood function is equal to:

$l(\alpha, \beta; x) = \beta\, (N - \zeta(\alpha)^{-1}\, \sum_{i = 1} ^m f_{a}(x_{i})\, \zeta(\alpha, x_i)) + \sum_{i = 1} ^m f_{a}(x_{i})\, log \left( \frac{e^{\frac{\beta\, x_{i}^{-\alpha}}{\zeta(\alpha)}} - 1}{e^{\beta} - 1} \right),$

where $f_{a}(x_i)$ is the absolute frequency of $x_i$, $m$ is the number of different values in the sample and $N$ is the sample size, i.e. $N = \sum_{i = 1} ^m x_i f_a(x_i)$.

By default the initial values of the parameters are computed using the function getInitialValues.

The function optim is used to estimate the parameters.

Value

Returns an object composed by the maximum likelihood parameter estimations jointly with their standard deviation and confidence intervals. It also contains the value of the log-likelihood at the maximum likelihood estimator.

See Also

getInitialValues.

Examples

data <- rzipfpe(100, 2.5, 1.3)
data <- as.data.frame(table(data))
data[,1] <- as.numeric(as.character(data[,1]))
data[,2] <- as.numeric(as.character(data[,2]))
initValues <- getInitialValues(data, model='zipfpe')
obj <- zipfpeFit(data, init_alpha = initValues$init_alpha, init_beta = initValues$init_beta)

---

Expected value of the Zipf-PE distribution.

Description

Computes the expected value of the Zipf-PE distribution for given values of parameters $\alpha$ and $\beta$.

Usage

zipfpeMean(alpha, beta, tolerance = 10^(-4))

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 2$).
beta	Value of the $\beta$ parameter ($\beta \in (-\infty, +\infty)$).
tolerance	Tolerance used in the calculations (default = $10^{-4}$).

Details

The mean of the distribution only exists for $\alpha$ strictly greater than 2. It is computed by calculating the partial sums of the serie, and stopping when two consecutive partial sums differ less than the tolerance value. The value of the last partial sum is returned.

Value

A positive real value corresponding to the mean value of the Zipf-PE distribution.

Examples

zipfpeMean(2.5, 1.3)
zipfpeMean(2.5, 1.3, 10^(-3))

---

Distribution Moments.

Description

General function to compute the k-th moment of the Zipf-PE distribution for any integer value $k \geq 1$, when it exists. The k-th moment exists if and only if $\alpha > k + 1$. For k = 1, this function returns the same value as the zipfpeMean function.

Usage

zipfpeMoments(k, alpha, beta, tolerance = 10^(-4))

Arguments

k	Order of the moment to compute.
alpha	Value of the $\alpha$ parameter ($\alpha > k + 1$).
beta	Value of the $\beta$ parameter ($\beta \in (-\infty, +\infty)$).
tolerance	Tolerance used in the calculations (default = $10^{-4}$).

Details

The k-th moment of the Zipf-PE distribution is finite for $\alpha$ values strictly greater than $k + 1$. It is computed by calculating the partial sums of the serie, and stopping when two consecutive partial sums differ less than the tolerance value. The value of the last partial sum is returned.

Value

A positive real value corresponding to the k-th moment of the distribution.

Examples

zipfpeMoments(3, 4.5, 1.3)
zipfpeMoments(3, 4.5, 1.3,  1*10^(-3))

---

Variance of the Zipf-PE distribution.

Description

Computes the variance of the Zipf-PE distribution for given values of $\alpha$ and $\beta$.

Usage

zipfpeVariance(alpha, beta, tolerance = 10^(-4))

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 3$).
beta	Value of the $\beta$ parameter ($\beta \in (-\infty, +\infty)$).
tolerance	Tolerance used in the calculations. (default = $10^{-4}$)

Details

The variance of the distribution only exists for $\alpha$ strictly greater than 3.

Value

A positive real value corresponding to the variance of the distribution.

See Also

zipfpeMoments, zipfpeMean.

Examples

zipfpeVariance(3.5, 1.3)

---

The Zipf-Polylog Distribution (Zipf-Polylog).

Description

Probability mass function of the Zipf-Polylog distribution with parameters $\alpha$ and $\beta$. The support of the Zipf-Polylog distribution are the strictly positive integer numbers large or equal than one.

Usage

dzipfpolylog(x, alpha, beta, log = FALSE, nSum = 1000)

pzipfpolylog(x, alpha, beta, log.p = FALSE, lower.tail = TRUE,
  nSum = 1000)

qzipfpolylog(p, alpha, beta, log.p = FALSE, lower.tail = TRUE,
  nSum = 1000)

rzipfpolylog(n, alpha, beta, nSum = 1000)

Arguments

x	Vector of positive integer values.
alpha	Value of the $\alpha$ parameter ($\alpha > 1$ ).
beta	Value of the $\beta$ parameter ($\beta > 0$ ).
log, log.p	Logical; if TRUE, probabilities p are given as log(p).
nSum	The number of terms used for computing the Polylogarithm function (Default = 1000).
lower.tail	Logical; if TRUE (default), probabilities are $P[X \leq x]$, otherwise, $P[X > x]$.
p	Vector of probabilities.
n	Number of random values to return.

Details

The probability mass function at a positive integer value $x$ of the Zipf-Polylog distribution with parameters $\alpha$ and $\beta$ is computed as follows:

Value

dzipfpolylog gives the probability mass function

Examples

dzipfpolylog(1:10, 1.61, 0.98)
pzipfpolylog(1:10, 1.61, 0.98)
qzipfpolylog(0.8, 1.61, 0.98)

---

ZipfPolylog parameters estimation.

Description

For a given sample of strictly positive integer numbers, usually of the type of ranking data or frequencies of frequencies data, estimates the parameters of the ZipfPolylog distribution by means of the maximum likelihood method. The input data should be provided as a frequency matrix.

Usage

zipfPolylogFit(data, init_alpha, init_beta, level = 0.95, ...)

## S3 method for class 'zipfPolyR'
residuals(object, ...)

## S3 method for class 'zipfPolyR'
fitted(object, ...)

## S3 method for class 'zipfPolyR'
coef(object, ...)

## S3 method for class 'zipfPolyR'
plot(x, ...)

## S3 method for class 'zipfPolyR'
print(x, ...)

## S3 method for class 'zipfPolyR'
summary(object, ...)

## S3 method for class 'zipfPolyR'
logLik(object, ...)

## S3 method for class 'zipfPolyR'
AIC(object, ...)

## S3 method for class 'zipfPolyR'
BIC(object, ...)

Arguments

data	Matrix of count data in form of a table of frequencies.
init_alpha	Initial value of $\alpha$ parameter ($\alpha > 1$).
init_beta	Initial value of $\beta$ parameter ($\beta > 0$).
level	Confidence level used to calculate the confidence intervals (default 0.95).
...	Further arguments to the generic functions. The extra arguments are passing to the optim function.
object	An object from class "zipfPolyR" (output of zipfPolylogFit function).
x	An object from class "zipfPolyR" (output of zipfPolylogFit function).

Details

The argument data is a two column matrix with the first column containing the observations and the second column containing their frequencies.

The log-likelihood function is equal to:

The function optim is used to estimate the parameters.

Value

Returns a zipfPolyR object composed by the maximum likelihood parameter estimations jointly with their standard deviation and confidence intervals. It also contains the value of the log-likelihood at the maximum likelihood estimator.

---

Expected value of the ZipfPolylog distribution.

Description

Computes the expected value of the ZipfPolylog distribution for given values of parameters $\alpha$ and $\beta$.

Usage

zipfpolylogMean(alpha, beta, tolerance = 10^(-4))

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 2$).
beta	Value of the $\beta$ parameter ($\beta \in (-\infty, +\infty)$).
tolerance	Tolerance used in the calculations (default = $10^{-4}$).

Value

A positive real value corresponding to the mean value of the ZipfPolylog distribution.

Examples

zipfpolylogMean(0.5, 0.8)
zipfpolylogMean(2.5, 0.8, 10^(-3))

---

Moments of the Zipf-Polylog Distribution.

Description

General function to compute the k-th moment of the ZipfPolylog distribution for any integer value $k \geq 1$, when it exists. #' For k = 1, this function returns the same value as the zipfpoylogMean function.

Usage

zipfpolylogMoments(k, alpha, beta, tolerance = 10^(-4), nSum = 1000)

Arguments

k	Order of the moment to compute.
alpha	Value of the $\alpha$ parameter ($\alpha > k + 1$).
beta	Value of the $\beta$ parameter ($\beta \in (-\infty, +\infty)$).
tolerance	Tolerance used in the calculations (default = $10^{-4}$).
nSum	The number of terms used for computing the Polylogarithm function (default = 1000).

Details

The k-th moment of the Zipf-Polylog distribution is always finite, but, for $\alpha >1$ and $\beta = 0$ the k-th moment is only finite for all $\alpha > k + 1$. It is computed by calculating the partial sums of the serie, and stopping when two consecutive partial sums differ less than the tolerance value. The value of the last partial sum is returned.

Value

A positive real value corresponding to the k-th moment of the distribution.

Examples

zipfpolylogMoments(1, 0.2, 0.90)
zipfpolylogMoments(3, 4.5, 0.90,  1*10^(-3))

---

Variance of the ZipfPolylog distribution.

Description

Computes the variance of the ZipfPolylog distribution for given values of $\alpha$ and $\beta$.

Usage

zipfpoylogVariance(alpha, beta, tolerance = 10^(-4))

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 3$).
beta	Value of the $\beta$ parameter ($\beta \in (-\infty, +\infty)$).
tolerance	Tolerance used in the calculations. (default = $10^{-4}$)

Details

The variance of the distribution only exists for $\alpha$ strictly greater than 3.

Value

A positive real value corresponding to the variance of the distribution.

See Also

zipfpolylogMoments, zipfpoylogMean.

Examples

zipfpoylogVariance(0.5, 0.75)

---

The Zipf-Poisson Stop Sum Distribution (Zipf-PSS).

Description

Probability mass function, cumulative distribution function, quantile function and random number generation for the Zipf-PSS distribution with parameters $\alpha$ and $\lambda$. The support of the Zipf-PSS distribution are the positive integer numbers including the zero value. In order to work with its zero-truncated version the parameter isTruncated should be equal to True.

Usage

dzipfpss(x, alpha, lambda, log = FALSE, isTruncated = FALSE)

pzipfpss(q, alpha, lambda, log.p = FALSE, lower.tail = TRUE,
  isTruncated = FALSE)

rzipfpss(n, alpha, lambda, log.p = FALSE, lower.tail = TRUE,
  isTruncated = FALSE)

qzipfpss(p, alpha, lambda, log.p = FALSE, lower.tail = TRUE,
  isTruncated = FALSE)

Arguments

x, q	Vector of positive integer values.
alpha	Value of the $\alpha$ parameter ($\alpha > 1$ ).
lambda	Value of the $\lambda$ parameter ($\lambda > 0$ ).
log, log.p	Logical; if TRUE, probabilities p are given as log(p).
isTruncated	Logical; if TRUE, the zero truncated version of the distribution is returned.
lower.tail	Logical; if TRUE (default), probabilities are $P[X \leq x]$, otherwise, $P[X > x]$.
n	Number of random values to return.
p	Vector of probabilities.

Details

The support of the $\lambda$ parameter increases when the distribution is truncated at zero being $\lambda \geq 0$. It has been proved that when $\lambda = 0$ one has the degenerated version of the distribution at one.

References

Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 22-26.

Sundt, B., & Jewell, W. S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 27-39.

---

Zipf-PSS parameters estimation.

Description

For a given sample of strictly positive integer numbers, usually of the type of ranking data or frequencies of frequencies data, estimates the parameters of the Zipf-PSS distribution by means of the maximum likelihood method. The input data should be provided as a frequency matrix.

Usage

zipfpssFit(data, init_alpha = NULL, init_lambda = NULL, level = 0.95,
  isTruncated = FALSE, ...)

## S3 method for class 'zipfpssR'
residuals(object, isTruncated = FALSE, ...)

## S3 method for class 'zipfpssR'
fitted(object, isTruncated = FALSE, ...)

## S3 method for class 'zipfpssR'
coef(object, ...)

## S3 method for class 'zipfpssR'
plot(x, isTruncated = FALSE, ...)

## S3 method for class 'zipfpssR'
print(x, ...)

## S3 method for class 'zipfpssR'
summary(object, isTruncated = FALSE, ...)

## S3 method for class 'zipfpssR'
logLik(object, ...)

## S3 method for class 'zipfpssR'
AIC(object, ...)

## S3 method for class 'zipfpssR'
BIC(object, ...)

Arguments

data	Matrix of count data in form of table of frequencies.
init_alpha	Initial value of $\alpha$ parameter ($\alpha > 1$).
init_lambda	Initial value of $\lambda$ parameter ($\lambda > 0$).
level	Confidence level used to calculate the confidence intervals (default 0.95).
isTruncated	Logical; if TRUE, the truncated version of the distribution is returned.(default = FALSE)
...	Further arguments to the generic functions. The extra arguments are passing to the optim function.
object	An object from class "zpssR" (output of zipfpssFit function).
x	An object from class "zpssR" (output of zipfpssFit function).

Details

The argument data is a two column matrix with the first column containing the observations and the second column containing their frequencies.

The log-likelihood function is equal to:

$l(\alpha, \lambda, x) = \sum_{i =1} ^{m} f_a(x_i)\, log(P(Y = x_i)),$

where $m$ is the number of different values in the sample, being $f_{a}(x_i)$ is the absolute frequency of $x_i$.The probabilities are calculated applying the Panjer recursion. By default the initial values of the parameters are computed using the function getInitialValues. The function optim is used to estimate the parameters.

Value

Returns a zpssR object composed by the maximum likelihood parameter estimations jointly with their standard deviation and confidence intervals and the value of the log-likelihood at the maximum likelihood estimator.

References

Panjer, H. H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 22-26.

Sundt, B., & Jewell, W. S. (1981). Further results on recursive evaluation of compound distributions. ASTIN Bulletin: The Journal of the IAA, 12(1), 27-39.

See Also

getInitialValues.

Examples

data <- rzipfpss(100, 2.5, 1.3)
data <- as.data.frame(table(data))
data[,1] <- as.numeric(as.character(data[,1]))
data[,2] <- as.numeric(as.character(data[,2]))
initValues <- getInitialValues(data, model='zipfpss')
obj <- zipfpssFit(data, init_alpha = initValues$init_alpha, init_lambda = initValues$init_lambda)

---

Expected value of the Zipf-PSS distribution.

Description

Computes the expected value of the Zipf-PSS distribution for given values of parameters $\alpha$ and $\lambda$.

Usage

zipfpssMean(alpha, lambda, isTruncated = FALSE)

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 2$).
lambda	Value of the $\lambda$ parameter ($\lambda > 0$).
isTruncated	Logical; if TRUE Use the zero-truncated version of the distribution to calculate the expected value (default = FALSE).

Details

The expected value of the Zipf-PSS distribution only exists for $\alpha$ values strictly greater than 2. The value is obtained from the law of total expectation that says that:

$E[Y] = E[N]\, E[X],$

where E[X] is the mean value of the Zipf distribution and E[N] is the expected value of a Poisson one. From where one has that:

$E[Y] = \lambda\, \frac{\zeta(\alpha - 1)}{\zeta(\alpha)}$

Particularlly, if one is working with the zero-truncated version of the Zipf-PSS distribution. This values is computed as:

$E[Y^{ZT}] = \frac{\lambda\, \zeta(\alpha - 1)}{\zeta(\alpha)\, (1 - e^{-\lambda})}$

Value

A positive real value corresponding to the mean value of the distribution.

References

Sarabia Alegría, J. M., Gómez Déniz, E. M. I. L. I. O., & Vázquez Polo, F. (2007). Estadística actuarial: teoría y aplicaciones. Pearson Prentice Hall.

Examples

zipfpssMean(2.5, 1.3)
zipfpssMean(2.5, 1.3, TRUE)

---

Distribution Moments.

Description

General function to compute the k-th moment of the Zipf-PSS distribution for any integer value $k \geq 1$, when it exists. The k-th moment exists if and only if $\alpha > k + 1$.

Usage

zipfpssMoments(k, alpha, lambda, isTruncated = FALSE,
  tolerance = 10^(-4))

Arguments

k	Order of the moment to compute.
alpha	Value of the $\alpha$ parameter ($\alpha > k + 1$).
lambda	Value of the $\lambda$ parameter ($\lambda > 0$).
isTruncated	Logical; if TRUE, the truncated version of the distribution is returned.
tolerance	Tolerance used in the calculations (default = $10^{-4}$).

Details

The k-th moment of the Zipf-PSS distribution is finite for $\alpha$ values strictly greater than $k + 1$. It is computed by calculating the partial sums of the serie, and stopping when two consecutive partial sums differ less than the tolerance value. The value of the last partial sum is returned.

Value

A positive real value corresponding to the k-th moment of the distribution.

Examples

zipfpssMoments(1, 2.5, 2.3)
zipfpssMoments(1, 2.5, 2.3, TRUE)

---

Variance of the Zipf-PSS distribution.

Description

Computes the variance of the Zipf-PSS distribution for given values of parameters $\alpha$ and $\lambda$.

Usage

zipfpssVariance(alpha, lambda, isTruncated = FALSE)

Arguments

alpha	Value of the $\alpha$ parameter ($\alpha > 3$).
lambda	Value of the $\lambda$ parameter ($\lambda > 0$).
isTruncated	Logical; if TRUE Use the zero-truncated version of the distribution to calculate the expected value (default = FALSE).

Details

The variance of the Zipf-PSS distribution only exists for $\alpha$ values strictly greater than 3. The value is obtained from the law of total variance that says that:

$Var[Y] = E[N]\, Var[X] + E[X]^2 \, Var[N],$

where X follows a Zipf distribution with parameter $\alpha$, and N follows a Poisson distribution with parameter $\lambda$. From where one has that:

$Var[Y] = \lambda\, \frac{\zeta(\alpha - 2)}{\zeta(\alpha)}$

Particularlly, if one is working with the zero-truncated version of the Zipf-PSS distribution. This values is computed as:

$Var[Y^{ZT}] = \frac{\lambda\, \zeta(\alpha)\, \zeta(\alpha - 2)\, (1 - e^{-\lambda}) - \lambda^2 \, \zeta(\alpha - 1)^2 \, e^{-\lambda}}{\zeta(\alpha)^2 \, (1 - e^{-\lambda})^2}$

Value

A positive real value corresponding to the variance of the distribution.

References

Sarabia Alegría, JM. and Gómez Déniz, E. and Vázquez Polo, F. Estadística actuarial: teoría y aplicaciones. Pearson Prentice Hall.

Examples

zipfpssVariance(4.5, 2.3)
zipfpssVariance(4.5, 2.3, TRUE)

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