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Previously suggested methods for constructing confidence bands for cumulative distribution functions have been based on the classical Kolmogorov-Smirnov test for an empirical distribution function. This paper gives a method based on maximum likelihood estimation of the parameters. The method is described for a general continuous distribution. Detailed results are given for a location-scale parameter model, which includes the normal and extreme-value distributions as special cases. Results are also given for the related lognormal and Weibull distributions. The formulas derived for these distributions give a band with exact confidence coefftcient. A chi-squared approximation, which avoids the use of special tables, is also described. An example is used to compare the resulting bands with those obtained by previously published methods.

In 1907, the Bergen Institute of Marine Research started regular sampling of scales and lengths from landings of mature Norwegian spring-spawning herring. The actual age of each fish when caught was recorded, and from the early 1920s also the age at which it spawned for the first time. The present analyses concern biological samples secured during the fishing seasons 1940–1964. Herring in this stock do not all reach maturity at the same age. A small proportion of any one year class matures at 3 years. The majority matures from the age of 4–7 years, and a small proportion of some year classes at 8 and even 9 years of age. Subsequent age composition and growth of each maturation cohort were followed throughout mature life after spawning for the first time. The maximum age was found to increase with age at maturation, rising to an asymptote of about 22 years. The von Bertalanffy parameter L∞ shows an increasing trend with age at maturation, while K decreases. There is no strict length threshold at maturation and the curve joining the length at which each maturation cohort reaches maturity is less steep than the growth curve itself over the range of maturation ages. The data suggest that fish in this stock spawn, on average, eight times during a period of their life history in which the mortality rate is independent of age. After these eight spawnings, at an age referred to in this paper as the hinge age, the mortality rate increases sharply. Thus, the adult life is divided into two phases, called here pre-senescent and senescent. The total mortality rates in the pre-senescent phase are relatively stable for all maturation cohorts 3–9, but there is some evidence of a trend towards higher mortality rates during the senescent phase for the youngest maturing fish. This trend is caused mainly by a reduced natural mortality in the fish that mature when older. These findings have interesting demographic implications. Additional mortality due to fishing will change the relative contribution of young and old maturation cohorts in the senescent phase, thus making it appear that natural mortality is dependent on the intensity of fishing. Consequently, for stock assessment, analysis on a cohort basis seems advisable.

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