Maximum Likelihood 161 where 6 is a continuous function that decreases from 1 to 0 on [0,1].If it tends to zero rapidly enough as e-1,the peaks of the triangles will distract the m.l.e.from its appointed rounds.In Example 1,$2,the m.l.e.led a precarious existence.Here everything is compact and continuous and all of Wald's conditions,except one,are satisfied.To convert the example into one that satisfies Cramer's conditions,for ee(0,1),Ferguson replaces the triangles by Beta densities. The above example relies heavily on the fact that ratios of the type f(x,0)/f(x,00)are unbounded functions of 6.One can also make up examples where the ratios stay bounded and m.l.e.still misbehaves. A possible example is as follows.For each integer m>1 divide the interval (0,1]by binary division,getting 2"intervals of the form (2m,(0+1)2m](0=0,1,..,2m-1). For each such division there are 2m 2m-1 ways of selecting 2-of the intervals.Make a selection s.On the selected ones,letm be equal to 1.On the remaining ones let .m be equal to(-1). This gives a certain countable family of functions. Now for given m and for the selection s let psm be the measure whose density with respect to Lebesgue measure on(0,1]is 1+(1-e-m)中.m In this case the ratio of densities is always between and 2.The measures are all distinct from one another. Application of a maximum likelihood technique would lead us to estimate m by + (This is essentially equivalent to another example of Bahadur. 5 An Example from Biostatistics The following is intended to show that even for 'straight'exponential families one can sometimes do better than the m.l.e. The example has a long history,which we shall not recount.It occurs from the evaluation of dose responses in biostatistics. Suppose that a chemical can be injected to rats at various doses y,y2,...,y:.For a particular dose,one just observes whether or not there is a response.There is then for each y a certain probability of response.Biostatisticians,being complicated people,prefer to work out not with the dose y but with its logarithm x=log y. We shall then let p(x)be the probability of response if the animal is given the log dose t. Some people,including Sir Ronald,felt that the relation xp(x)would be well described by a cumulative normal distribution,in standard form 1 p-V2m_e护d业 I do not know why.Some other people felt that the probability p has a derivative p' about proportional to p except that for p close to unity (large dose)the poor animal is saturated so that the curve has a ceiling at 1.Maximum Likelihood where 6 is a continuous function that decreases from 1 to 0 on [0, 1]. If it tends to zero rapidly enough as 0- 1, the peaks of the triangles will distract the m.l.e. from its appointed rounds. In Example 1, ? 2, the m.l.e. led a precarious existence. Here everything is compact and continuous and all of Wald's conditions, except one, are satisfied. To convert the example into one that satisfies Cramer's conditions, for 0 E (0, 1), Ferguson replaces the triangles by Beta densities. The above example relies heavily on the fact that ratios of the type f(x, 0)/f(x, 00) are unbounded functions of 0. One can also make up examples where the ratios stay bounded and m.l.e. still misbehaves. A possible example is as follows. For each integer m > 1 divide the interval (0, 1] by binary division, getting 2m intervals of the form (j2-, (j + 1)2-m] (j = , 1,...,2m 1). For each such division there are (2m) 2M-1 ways of selecting 2m-1 of the intervals. Make a selection s. On the selected ones, let )s,m be equal to 1. On the remaining ones let qPs,m be equal to (-1). This gives a certain countable family of functions. Now for given m and for the selection s let Ps,m be the measure whose density with respect to Lebesgue measure on (0, 1] is 1 + (1 - e-m)s,m. In this case the ratio of densities is always between I and 2. The measures are all distinct from one another. Application of a maximum likelihood technique would lead us to estimate m by +00. (This is essentially equivalent to another example of Bahadur.) 5 An Example from Biostatistics The following is intended to show that even for 'straight' exponential families one can sometimes do better than the m.l.e. The example has a long history, which we shall not recount. It occurs from the evaluation of dose responses in biostatistics. Suppose that a chemical can be injected to rats at various doses yl, Y2, . . , yi > 0. For a particular dose, one just observes whether or not there is a response. There is then for each y a certain probability of response. Biostatisticians, being complicated people, prefer to work out not with the dose y but with its logarithm x = log y. We shall then let p(x) be the probability of response if the animal is given the log dose x. Some people, including Sir Ronald, felt that the relation x->p(x) would be well described by a cumulative normal distribution, in standard form p() =/(2r) e- dt I do not know why. Some other people felt that the probability p has a derivative p' about proportional to p except that for p close to unity (large dose) the poor animal is saturated so that the curve has a ceiling at 1. 161