1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

Suppose that we have adopted the particular model, i. e. declaredthat the magnitudes (1.2) are identically distributed random variables.Will this be the sole necessary assumption? No, since we badly need toknow how large can be the error of the approximate equality (1.6). Forexample, if ˆλ=2, can the real value of λ be 4? In other words, weshould be able to calculate the variance of (1.5). It is equal tonˆ 1var λ = [ var µ + cov(µ µ )].∑ ∑2i i jn i= 1i≠jFor the Poisson lawvarµ i = Eµ i = λand, as a rough estimate, it is possible to assume varµ i = ˆλ µ, = but wecan not say anything abut the covariations. The available data areusually far from adequate for estimating it. Nothing is left than tosuppose that the variables (1.2) are independent, i. e. to consider thatthe covariations vanish.We thus arrive at a model of independent identically distributedrandom variables, that is, to a sample. The reader will probably agreethat our considerations, if not logically prove that only a model of asample is useful, are still sufficiently convincing in showing that it isdifficult to tear away from the sphere of ideas leading to the model of asample. It is therefore very popular and researchers are trying to workwith it provided that its falsity is not proven.The chapters of mathematical statistics devoted to samples areundoubtedly in its best and the most developed part. However, themodel of sample is sufficiently (and even too) often wrong. We sawthat if the machinery is aging, the observations (1.2) do not compose asample. The same is true when a preliminary period is involved, whenthe work begins by eliminating defects after which the number offailures drops. Other causes violating the identity of the distribution ofthe variables (1.2) also exist.Their independence can also be violated. For example, if a failurewill lead to a capital repair with the replacement of many depreciatedalthough still workable machine parts, a negative correlation betweenµ i and the depreciated and worn-out µ j will appear. If, however, thewear and tear of a machine part intensifies the depreciation of the otherparts and no replacements are made, the appeared correlation will bepositive.When introducing models differing from a model of a sample, weshould evidently specify their distinction by a small number ofparameters determinable either theoretically or by available statisticaldata. Complicated models, as stated above, are absolutely useless. It ispractically possible to allow for either deviations from the identity ofdistributions given by determinate functions or from independenceprovided that identity is preserved. We will now consider such models.It should be borne in mind that both these models and the model ofsample are sufficiently tentative. True, if a model is proper, our51