Semimartingales and their Statistical Inference

Rating:
Summary: Rao's Semimartingales and their statistical inference
Review: This book and the journal articles on which it is based pioneer a new branch of statistics. The reader who is unfamiliar with semimartingales can think of them as a generalization of supermartingales, where the latter are roughly a sequence of variables whose means increase and such that each variable is bigger than its conditional mean (a conditional mean is the mean of one thing or variable with another thing or variable fixed). Intuitively, when you use semimartingales to approximate or model something in the real world, you are approximating its mean by a sequence of increasing or decreasing means or alternately both, keeping track of where and when the means increase or decrease inside the mathematics. This allows you to make fewer assumptions than usual about how the things that you are modeling behave, since roughly you only have to keep track of the means. This has an advantage over computer techniques now in common use such as linear and polynomial regression. This book shows that you can actually make statistical estimates for the unknown quantities in your model for large samples ("asymptotically"). The reader is cautioned that a somewhat parallel but interesting theory exists with conditional means replaced by logic-based probability (LBP) means, which is to say that division is replaced by subtraction and adding 1 to the result. The latter allows study of very rare events since it is defined when events have probability zero, and also other events of importance. See some of my reviews of other mathematics books or my articles abstracted on the internet at the Institute for Logic of the University of Vienna for LBP methods.

Rating:
Summary: Rao's Semimartingales and their statistical inference
Review: This book and the journal articles on which it is based pioneer a new branch of statistics. The reader who is unfamiliar with semimartingales can think of them as a generalization of supermartingales, where the latter are roughly a sequence of variables whose means increase and such that each variable is bigger than its conditional mean (a conditional mean is the mean of one thing or variable with another thing or variable fixed). Intuitively, when you use semimartingales to approximate or model something in the real world, you are approximating its mean by a sequence of increasing or decreasing means or alternately both, keeping track of where and when the means increase or decrease inside the mathematics. This allows you to make fewer assumptions than usual about how the things that you are modeling behave, since roughly you only have to keep track of the means. This has an advantage over computer techniques now in common use such as linear and polynomial regression. This book shows that you can actually make statistical estimates for the unknown quantities in your model for large samples ("asymptotically"). The reader is cautioned that a somewhat parallel but interesting theory exists with conditional means replaced by logic-based probability (LBP) means, which is to say that division is replaced by subtraction and adding 1 to the result. The latter allows study of very rare events since it is defined when events have probability zero, and also other events of importance. See some of my reviews of other mathematics books or my articles abstracted on the internet at the Institute for Logic of the University of Vienna for LBP methods.