154 L.LE CAM some other principle instead,although we give a few guidelines in 6.For other views see the discussion of the paper by Berkson (1980). This paper is adapted from lectures given at the University of Maryland,College Park, in the Fall of 1975.We are greatly indebted to Professor Grace L.Yang for the invitation to give the lectures and for the permission to reproduce them. 2 A Few Old Examples Let X,X2,...,X be independent identically distributed observations with values in some space X,A).Suppose that there is a o-finite measure A on A and that the distribution Pe of X;has a density f(x,0)with respect to u.The parameter 0 takes its values in some set e. For n observations x1,x2,...,xn the maximum likelihood estimate is any value 6 such that IIf(0)=sup IIf(0). j=1 ej=1 Note that such a 6 need not exist,and that,when it does,it usually depends on what version of the densities f(x,6)was selected.A function (x1,...,x)(x1,...,x) selecting a value 6 for each n-tuple (x1,...,x)may or may not be measurable. However all of this is not too depressing.Let us consider some examples. Example 1.(This may be due to Kiefer and Wolfowitz or to whoever first looked at mixtures of Normal distributions.)Let a be the number=10-10.Let =(u,o), u(-0,+)o>0.Let fi(x,0)be the density defined with respect to Lebesgue measureλon the line by a,o-a高e卿{-+vp{, Then,for (x1,...,x)one can take u=x1 and note that supΠfk;h,o)=o. 0=1 If o=0 was allowed one could claim that=(x,0)is maximum likelihood. Example 2.The above Example 1 is obviously contaminated and not fit to drink.Now a variable X is called log normal if there are numbers (a,b,c)such that X=c+ear+b with a Y which is N(0,1).Let 0=(a,b,c)in R3.The density of X can be taken zero for xsc and for x>c,and is equal to ,o)-v2np{-aloge-a)-br() 1 A sample (x1,...,x)from this density will almost surely have no ties and a unique minimum z minxi. The only values to consider are those for which c<z.Fix a value of b,say b =0.Take aL. LE CAM some other principle instead, although we give a few guidelines in ? 6. For other views see the discussion of the paper by Berkson (1980). This paper is adapted from lectures given at the University of Maryland, College Park, in the Fall of 1975. We are greatly indebted to Professor Grace L. Yang for the invitation to give the lectures and for the permission to reproduce them. 2 A Few Old Examples Let X1, X2, ... , X, be independent identically distributed observations with values in some space {X,A}. Suppose that there is a a-finite measure A on A and that the distribution P0 of Xj has a density f(x, 0) with respect to M. The parameter 0 takes its values in some set 0. For n observations x,l, x,.. ., xn the maximum likelihood estimate is any value 0 such that n n f (x0) sup f(x,e 0). j=1 0eO j= Note that such a 0 need not exist, and that, when it does, it usually depends on what version of the densities f(x, 0) was selected. A function (xl,..., x,n) 0((x,.. ., x,) selecting a value 0 for each n-tuple (xl,..., x,) may or may not be measurable. However all of this is not too depressing. Let us consider some examples. Example 1. (This may be due to Kiefer and Wolfowitz or to whoever first looked at mixtures of Normal distributions.) Let ca be the number c = 10-1017. Let 0= (,u, a), M e (-00, +oo), a>0. Let fl(x, 0) be the density defined with respect to Lebesgue measure A on the line by - 2p{( -^ 1 (X{-7)2} fi(x, 0) = (2r) exp -2 (x - P)2 + a(2r) exp {- (a2 Then, for (xl, ..., xn) one can take p = xl and note that n sup fi(x,;p, o)= o. a j=l If a = 0 was allowed one could claim that 0 = (xl, 0) is maximum likelihood. Example 2. The above Example 1 is obviously contaminated and not fit to drink. Now a variable X is called log normal if there are numbers (a, b, c) such that X = c + eaY+b with a Y which is N(0, 1). Let 0 = (a, b, c) in R3. The density of X can be taken zero for x < c and for x > c, and is equal to 2(X, ) = (2) exp 2 [log (x - c) - b]2} - (-x ). A sample (x1, .. ., Xn) from this density will almost surely have no ties and a unique minimum z = min xj. The only values to consider are those for which c < z. Fix a value of b, say b = 0. Take a 154