Estimation Of Location Parameter On The Skew Normal Setting With Known Coefficient Of Variation

It is well known that the confidence intervals for the normal means can be constructed in two cases;(a) parameter variance is known(b) parameter variance is unknown. However, in fact, there are several situations in area of agricultural, biological, environmental and physical sciences that the coefficient of variation can be obtained. Hence, the confidence intervals for the normal means with known coefficient of variation have been studied recently. Nevertheless, in most real life cases, many data sets often do not follow the normal distribution, many follow the skew-normal distribution. This is the reason why more and more people start to study skew-normal distribution. In practice, skew-normal is an extension for normal and normal can be seen a special case of skew-normal. Skew-normal distribution has one more parameter which is called skewness. Skewness is a measure of the asymmetry of the probability distribution of a random variable. Its value can be positive or negative, or even undefined. Positive skew means that the tail on the right side of the probability density function is longer than the left one. Conversely, negative skew implies that the tail on the left side is longer than the right one. Zero value indicates that the tails of both sides are equal, which is the case of symmetric. In this paper, we do the following researches:1. Confidence intervals for the normal meanWe propose five kinds of confidence intervals for the normal mean with known coefficient of variation. The first one is using the pivotal statistic. Then, with known coefficient of variation, we construct confidence intervals by using the least mean squares estimator, the best unbiased estimator and the maximum likelihood estimator.2. Confidence intervals for the skew-normal location parameterWe derive three kinds of confidence intervals for the skew-normal location parameter with known coefficient of variation. Similar as the normal case, at first we use a pivotal statistic. After that, with known coefficient of variation, we construct confidence intervals by using the least mean squares estimator and the best unbiased estimator. At last, we show the way to minimize the confidence intervals.3. Monte-Carlo SimulationIn the paper, we use Monte-Carlo simulation to access the performance of the confidence intervals based on their coverage probabilities and average lengths.