Weighted Distributions and Size-Biased Sampling with Applications to Wildlife Populations and Human Families

Abstract

When an investlgator records an observation by nature according to a certain stochastic model the recorded observation will not have the original distribution unless everJ observation is given an equal chance of being recorded. A number of papers have appeared during the last ten years implicitly using the concepts of weighted and size-biased sampling distributions. In this paper we examine some general models leading to weighted distributiosls with weight functions not necessarily bounded by unity. The examples include. probability sampling in sasnple surveys additive damage models visibility bias dependent on the nature of data collection and two-stage sampling. Several important distributions and their size-biased fornls are recorded. A few theorems are given on the inequalities between the mean values of two weighted distributions. The results are applied to the analysis of data relating to human populations and wildlife management. For human populations the following is raised and discussed. Let us ascertain from each snale student in a class the nusHber of brothers including himself and sisters he has and denote by k the number of students and by B and S the total numbers of brothers and sisters. What would be the approximate values of B/(B + S) the ratio of brothers to the total number of children asld (B + S)/k the average number of children per family? It is shown that B/(B + S) will be an overestimate of the proportion of boys among the children per family in the general population which is about halB and similarly (B + S)/k is biased upwards as an estimate of the average number of children per family in the general population. Some suggestiosls are oJ%ered for the estimation of these population parameters. Lastly for the purpose of estimating wildlife population density certain results are formulated within the framework of quadrat sasnpling involving visibility bias. fw(x) = w(x) f(x) (1.1)

On Size-Biased Weighted Transmuted Weibull Distribution

“…Especially the weibull distribution and its generalizations in the literature attract the most of the researchers due to its wide range applications. The Weibull distribution includes the exponential and the Rayleigh distributions as sub models, the usefulness and applications of parametric distributions including Weibull, Rayleigh are seen in various areas including reliability, renewal theory, and branching processes which can be seen in papers by many authors such as in { [16], [17], [25]}. Different generalizations of the Weibull distribution are common in the literature as in { [4], [5], [21], [22], [28], [38]} and another generalization of the weibull distribution using the concept of weighted distributions is available as in { [6], [8], [19], [24], [30], [34], [36], [37]}.…”

Vol. 16, No. 2 (Full Issue)

“…Weighted distributions are employed mainly in research associated with reliability, bio-medicine, meta-analysis, econometrics, survival analysis, renewal processes, physics, ecology and branching processes which are found in Patil and Rao (1978), Gupta and Kirmani (1990), Gupta and Keating (1985), Oluyede (1999), Patil and Ord (1976) and Zelen and Feinleib (1969). A weighted form of Rayleigh distribution has been published by Reshi et al (2014).…”

“…Being first introduced by Rayleigh (1880), this statistical model was originally derived in connection with a problem in acoustics. More details on the Rayleigh distribution can be found in Johnson et al (1994) and references therein.The Rayleigh distribution has the following probability density function (pdf) and the cumulative distribution function (cdf), respectively,Weighted distributions are employed mainly in research associated with reliability, bio-medicine, meta-analysis, econometrics, survival analysis, renewal processes, physics, ecology and branching processes which are found in Patil and Rao (1978), Gupta and Kirmani (1990), Gupta and Keating (1985), Oluyede (1999), Patil andOrd (1976) and Zelen and Feinleib (1969). A weighted form of Rayleigh distribution has been published by Reshi et al (2014).…”

Double Bias: Estimation of Causal Effects from Length-Biased Samples in the Presence of Confounding

“…Recently, Bergeron et al [3], Shen et al [4], Qin and Shen [5] and Ning et al [6] studied analysis of covariates under biased sampling. Studies on lengthbiased sampling can be traced as far back as Wicksell [7], Fisher [8], Neyman [9], Cox and Lewis [1], Zelen and Feinlein [2] and Patil and Rao [10]. An updated review of the subject can be found in Asgharian et al [11].…”