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An inequality for Mill's ratio for the type III population

  • G. Baikunth Nath (a1)

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Mills(7), Gordon(5), Birnbaum(1), and the author(6) have studied the ratio of the area of the standardized normal curve from x to ∞ and the ordinate at x. Des Raj (4) established the monotonic character of, and obtained lower and upper bounds for this ratio for the standardized type III population. This ratio, as shown by Cohen (2) and Des Raj (3), has to be calculated for several values of x when solving approximately the equations involved in the problem of estimating the para-meters of type III populations from truncated samples. When the areas and ordinates are small, either this ratio cannot be obtained from existing tables prepared by Salvosa(8) or that very few significant digits are available for its calculation. The object of this note is to obtain an upper bound which could satisfactorily locate this ratio over the range x > −1.

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(1)Birnbaum, Z. W.An inequality for Mill's Ratio. Ann. Math. Statist. 13 (1942), 245246.
(2)Cohen, A. C.Estimating parameters of Pearson Type III populations from truncated samples. J. Amer. Statist. Assoc. 45 (1950), 411423.
(3)Des, Raj. Estimation of the parameters of Type III populations from truncated samples. J. Amer. Statist. Assoc. 48 (1953), 336349.
(4)Des, Raj. On Mill's Ratio for the Type III populations. Ann. Math. Statist. 24 (1953), 309312.
(5)Gordon, R. D.Values of Mill's Ratio of area to bounding ordinates of the normal probability integral for large values of the argument. Ann. Math. Statist. 12 (1941), 364366.
(6)Gupta, Baikunth N.On Mill's Ratio. Proc. Cambridge Philos. Soc. 67 (1970), 363364.
(7)Mills, J. P.Table of the ratio: area to bounding ordinate, for any portion of the normal curve. Biometrika 18 (1926), 395400.
(8)Salvosa, L. R.Tables of Pearson's Type III function. Ann. Math. Statist. 1 (1930), appended.

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