Statistical Inference Based On Rounded Data

Most traditional statistical analyses treat numerical data as if they were real (infinite-number-of-decimal-places) observations. It is a practical problem that data on hand are sometimes obtained using crude gaging. We might call such data "rounded data" since they are really obtainly by measuring to the nearest multiple of precision unit. The difference of between rouding data and true data value is called as rounding error. To explain this, let rounding precision unit equal to 1. When rounding phenomenon occurs, if the true value is 0.6, then rounded value is 1 and corresponding rounding error 1 — 0.6 = 0.4. Similarly, if the true value is 5.4, then rounded value 5 and corresponding rounding error 5 - 5.4 = -0.4. We can see that The absolute value of rounding error is smaller than (or equals to )a half of width of precision unit. For simplification, we choose rounding off unit δ = 1 except especial case.Rounded data arise in a number of ways. It is so popular that nowhere observations are not rounded, unless the model is discrete. The rounding of data may be due to the precision of the measuring instruments in experiments or that of the recording/storage mechanism, say the computer precision, or due to the unnecessary need of high accuracy, say, the blood pressure on a patient is not needed to be measured to decimals. In some cases, the rounding errors may be very large, especially in social survey, e.g. the GDP of a large country may be rounded at tens of million dollars, the annual income may be roundedat thousands of dollars. In some cases, the rounding error may be small, e.g. in high technology, the rounding errors may be as small as millimeters, or even 10~8 meters. No matter the precision is high or low, no observations are not rounded.Now, exploratory developments on rounding error are still involved in almost each and every applied region. Up to date issues have Chung(2005), Kodek and Krisper (2005), Mastrolilli and Hutter (2006) ,Panigrahi and Le-Ngoc(2005) and Salazar-Gnzalez (2006). modern computing science also considers effect of rounding of data in algorithm research region, (see Toring (1948), Carr (1959), Doerr (2006) and Devillers (2006)). some pioneer statisticians may have noticed that the rounding errors may cause serious effects to statistical inferences. Early discussions and bibliographies are referred to Shep-pard(1898) and Fisher (1936). Whereas, as using standard statistical method does statistical inference in classical statistical analysis, rounding error of data is often ignored by people. In fact, rounding error of data can result in a serious effect to accuracy of the statistical inference. If there is not effective method to deal with this case, then the conclusion we obtained may be incorrect or contrary ? For example, true sample X\, ? ? ?, Xn follow N(fj,,