Upper Record Values from a Univariate Normal Distribution

Maximum likelihood estimation of the two parameters of a univariate normal distribution when the observations are upper record values.

Usage

rec.normal(lmean = "identitylink", lsd = "loglink",
          imean = NULL, isd = NULL, imethod = 1, zero = NULL)

Arguments

lmean, lsd

Link functions applied to the mean and sd parameters. See Links for more choices.

imean, isd

Numeric. Optional initial values for the mean and sd. The default value NULL means they are computed internally, with the help of imethod.

imethod

Integer, either 1 or 2 or 3. Initial method, three algorithms are implemented. Choose the another value if convergence fails, or use imean and/or isd.

zero

Can be an integer vector, containing the value 1 or 2. If so, the mean or standard deviation respectively are modelled as an intercept only. Usually, setting zero = 2 will be used, if used at all. The default value NULL means both linear/additive predictors are modelled as functions of the explanatory variables. See CommonVGAMffArguments for more information.

Details

The response must be a vector or one-column matrix with strictly increasing values.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

References

Arnold, B. C. and Balakrishnan, N. and Nagaraja, H. N. (1998). Records, New York: John Wiley & Sons.

Author

T. W. Yee

Note

This family function tries to solve a difficult problem, and the larger the data set the better. Convergence failure can commonly occur, and convergence may be very slow, so set maxit = 200, trace = TRUE, say. Inputting good initial values are advised.

This family function uses the BFGS quasi-Newton update formula for the working weight matrices. Consequently the estimated variance-covariance matrix may be inaccurate or simply wrong! The standard errors must be therefore treated with caution; these are computed in functions such as vcov() and summary().

Examples

nn <- 10000; mymean <- 100
# First value is reference value or trivial record
Rdata <- data.frame(rawy = c(mymean, rnorm(nn, mymean, exp(3))))
# Keep only observations that are records:
rdata <- data.frame(y = unique(cummax(with(Rdata, rawy))))

fit <- vglm(y ~ 1, rec.normal, rdata, trace = TRUE, maxit = 200)
#> Iteration 1: loglikelihood = -30.664191
#> 12 weight matrices not updated out of 13 
#> Iteration 2: loglikelihood = -26.76061
#> 11 weight matrices not updated out of 13 
#> Iteration 3: loglikelihood = -26.743649
#> 12 weight matrices not updated out of 13 
#> Iteration 4: loglikelihood = -26.738771
#> 12 weight matrices not updated out of 13 
#> Iteration 5: loglikelihood = -26.738448
#> 12 weight matrices not updated out of 13 
#> Iteration 6: loglikelihood = -26.738398
#> 12 weight matrices not updated out of 13 
#> Iteration 7: loglikelihood = -26.738365
#> 12 weight matrices not updated out of 13 
#> Iteration 8: loglikelihood = -26.738332
#> 12 weight matrices not updated out of 13 
#> Iteration 9: loglikelihood = -26.7383
#> 12 weight matrices not updated out of 13 
#> Iteration 10: loglikelihood = -26.738267
#> Iteration 11: loglikelihood = -26.635721
#> Iteration 12: loglikelihood = -26.633118
#> 1 weight matrices not updated out of 13 
#> Iteration 13: loglikelihood = -26.632205
#> 1 weight matrices not updated out of 13 
#> Iteration 14: loglikelihood = -26.631345
#> Iteration 15: loglikelihood = -26.605746
#> Iteration 16: loglikelihood = -26.605587
#> 3 weight matrices not updated out of 13 
#> Iteration 17: loglikelihood = -26.605436
#> 12 weight matrices not updated out of 13 
#> Iteration 18: loglikelihood = -26.605367
#> 1 weight matrices not updated out of 13 
#> Iteration 19: loglikelihood = -26.605291
#> 1 weight matrices not updated out of 13 
#> Iteration 20: loglikelihood = -26.605279
#> 1 weight matrices not updated out of 13 
#> Iteration 21: loglikelihood = -26.605277
#> 1 weight matrices not updated out of 13 
#> Iteration 22: loglikelihood = -26.605276
coef(fit, matrix = TRUE)
#>                 mean loglink(sd)
#> (Intercept) 92.58273    2.888791
Coef(fit)
#>       mu       sd 
#> 92.58273 17.97157 
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = rec.normal, data = rdata, trace = TRUE, 
#>     maxit = 200)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1  92.5827    11.9673   7.736 1.02e-14 ***
#> (Intercept):2   2.8888     0.1794  16.101  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: mean, loglink(sd)
#> 
#> Log-likelihood: -26.6053 on 24 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 22 
#>