Generally–Altered, –Inflated and –Truncated Zeta Distribution

Description

Density, distribution function, quantile function and random generation for the generally–altered, –inflated and –truncated zeta distribution. Both parametric and nonparametric variants are supported; these are based on finite mixtures of the parent with itself and the multinomial logit model (MLM) respectively. Altogether it can be abbreviated as GAAIIT–Zeta(shape.p)–Zeta(shape.a)–MLM–Zeta(shape.i)–MLM, and it is also known as the GAIT-Zeta PNP combo.

Usage

dgaitzeta(x, shape.p, alter.mix = NULL, alter.mlm = NULL,
          inflate.mix = NULL, inflate.mlm = NULL, truncate = NULL,
          max.support = Inf, pobs.mix.a = 0, pobs.mlm.a = 0,
          pstr.mix.i = 0, pstr.mlm.i = 0, shape.a = shape.p,
          shape.i = shape.p, byrow.arg = FALSE, deflation = FALSE,
          log.arg = FALSE)
pgaitzeta(q, shape.p, alter.mix = NULL, alter.mlm = NULL,
          inflate.mix = NULL, inflate.mlm = NULL, truncate = NULL,
          max.support = Inf, pobs.mix.a = 0, pobs.mlm.a = 0,
          pstr.mix.i = 0, pstr.mlm.i = 0, shape.a = shape.p,
          shape.i = shape.p, byrow.arg = FALSE)
qgaitzeta(p, shape.p, alter.mix = NULL, alter.mlm = NULL,
          inflate.mix = NULL, inflate.mlm = NULL, truncate = NULL,
          max.support = Inf, pobs.mix.a = 0, pobs.mlm.a = 0,
          pstr.mix.i = 0, pstr.mlm.i = 0, shape.a = shape.p,
          shape.i = shape.p, byrow.arg = FALSE)
rgaitzeta(n, shape.p, alter.mix = NULL, alter.mlm = NULL,
          inflate.mix = NULL, inflate.mlm = NULL, truncate = NULL,
          max.support = Inf, pobs.mix.a = 0, pobs.mlm.a = 0,
          pstr.mix.i = 0, pstr.mlm.i = 0, shape.a = shape.p,
          shape.i = shape.p, byrow.arg = FALSE)

Arguments

Details

These functions allow any combination of 3 operator types: truncation, alteration and inflation. See Gaitpois for generic information. These functions do what Oazeta, Oizeta, Otzeta collectively did plus much more.

In the notation of Yee and Ma (2020) these functions allow for the special cases: (i) GAIT–Zeta(shape.p)–Zeta(shape.a, alter.mix, pobs.mix.a)–Zeta(shape.i, inflate.mix, pstr.mix.i); (ii) GAIT–Zeta(shape.p)–MLM(alter.mlm, pobs.mlm.a)–MLM(inflate.mlm, pstr.mlm.i). Model (i) is totally parametric while model (ii) is the most nonparametric possible.

Value

dgaitzeta gives the density, pgaitzeta gives the distribution function, qgaitzeta gives the quantile function, and rgaitzeta generates random deviates. The default values of the arguments correspond to ordinary dzeta, pzeta, qzeta, rzeta respectively.

Note

See Gaitpois for general information also relevant to this parent distribution.

Author(s)

T. W. Yee.

See Also

Gaitpois, multinomial, gaitzeta.mix, Oazeta, Oizeta, Otzeta.

Examples

ivec <- c(2, 10); avec <- ivec + 4; shape <- 0.95; xgrid <- 0:29
tvec <- 15; max.support <- 25; pobs.a <- 0.10; pstr.i <- 0.15
(ddd <- dgaitzeta(xgrid, shape, truncate = tvec,
   max.support = max.support, pobs.mix.a = pobs.a,
   alter.mix = avec, pstr.mix.i = pstr.i, inflate.mix = ivec))
## Not run: plot(xgrid, ddd, type = "n", ylab = "Probability", xlab = "x",
              main = "GAIT PMF---Zeta Parent")
mylwd <- 0.5
abline(v = avec, col = 'green', lwd = mylwd)
abline(v = ivec, col = 'red', lwd = mylwd)
abline(v = tvec, col = 'tan', lwd = mylwd)
abline(v = max.support, col = 'magenta', lwd = mylwd)
abline(h = c(pobs.a, pstr.i, 0:1), col = 'gray', lty = "dashed")
lines(xgrid, dzeta(xgrid, shape), col = 'gray', lty = "dashed")  # f_{\pi}
lines(xgrid, ddd, type = "h", col = "blue", lwd = 3)  # GAIT PMF
points(xgrid[ddd == 0], ddd[ddd == 0], pch = 16)  
## End(Not run)