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The classical analysis is based on random samples. However, in many situations the observations are recorded according to a nonnegative function of observations. In this case the mechanism of sampling is called weighted sampling. The usual statistical methods based on a weighted sample may be not valid and have to be adjusted. In this paper adjusted methods under some particular weight functions for normal distribution are studied and a new distribution called double normal distribution, is introduced as a weighted normal distribution.

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  One of the important problems that bring up in genetic fields is determining of loci of special gene in order to gene mapping and generating more effective drugs in medicine. Genetic linkage analysis is one important stage in this way. Haseman-Elston method is a quantitative statistical method that is used by biostatisticians and geneticists for genetic linkage analysis. The original Haseman-Elston method is presented in the year 1972 and ever after many investigators recommended some suggestions to make better it. In this article, we introduce the Haseman-Elston regression method and its extensions through 1972 to 2009. and finally we show performance of these methods in a practical example.

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In weighted sampling as a generalization of random sampling, every observation, y, is recorded with probably proportional to a non-negative function of y. In this paper, the normal regression model is investigated under the weighted sampling for a common weight function. Parameters of the model are estimated for known and unknown weight parameters. Using simulation, efficiency of estimators is studied when they have not closed forms. As an application, the data of number of visited  patients by specialist doctors in Social Security Organization of Ahvaz in Iran (SSOAI) are analyzed.

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The standard Beta distribution is a suitable distribution for modeling the data that include proportions. In many situations which the data of proportions include a considerable number of zeros and ones, the inflated beta distributions are more appropriate. When probabilities of recording such observations are proportional to a nonnegative weight function, the recorded observations distributed as a weighted inflated Beta. This article focuses on the size biased inflated Beta distribution as a special case of weighted inflated Beta distribution with the power weight function. Some properties of this distribution is studied and its parameters are estimated using maximum likelihood and method of moments approaches. The estimators are compared via a simulation study. Finally, the real mortality data set is fitted for this model.

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‎Usually in survey sampling when the sensitive questions are asked directly‎, ‎the respondents do not provide true answers‎. ‎The randomized response techniques have been introduced to protect the privacy responses‎. ‎In this article we focus on Simons randomized response technique for qualitative variables‎. ‎Using the combination of the two different‎ ‎Simmons’ models‎, ‎a new combined randomized response technique is introduced to increase protection of privacy‎. ‎Using simulation in R package‎, ‎efficiency of the proposed model is compared to the Simmons’ and Alavi and Tajodini's (1394) models‎. ‎Finally‎, ‎the proposed model has been employed for estimating the proportion of student cheating in Shahid Chamran University‎.

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In many sample surveys, the variables of interest, such as student cheating in a university are sensitive in nature. In such situations, the interviewees respond to direct questions untruthful, or refuse to answer. The various indirect methods such as randomized response technique and item count technique are introduced to collect sensitive information. In this paper a new item count is proposed, then its randomized version called randomized item count model is introduced. Using this model an unbiased estimator for the sensitive proportion of the population is obtained. The variance of the estimator and an estimate for its variance are obtained. A criterion for comparing efficiency and privacy is introduced simultaneously. Using simulation, the proposed model is evaluated and its efficiency and privacy are compared with the Simons’ technique. Based on this criterion, it is shown that the proposed method is better than the Simons method. The proportion of student cheating in the Shahid Chamran University of Ahvaz is estimated using the proposed model.

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Often, in sample surveys, respondents refused to answer some questions of a sensitive nature. Randomized response methods are designed not to reveal respondent confidentiality. In this article, a new quantitative randomized response method is introduced, and by conducting a series of simulation studies, we show that the proposed method is preferable to the cumulative and multiplicative methods. By using unbiased predictors, we estimate the covariance between two sensitive variables. In an experimental study using the proposed method, the average number of cheating and the average daily cigarette consumption of the Shahid Chamran University of Ahvaz students are estimated along with their variance, and an estimate for the covariance between them is provided.