Application of algebraic properties of statistical models to deriving distributions of statistics

Sapozhnikov, P. N.

doi:10.1007/bf02362493

Cited by 4 publications

(7 citation statements)

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“…We also present a new method for the estimation of a density function based on the determination of the distribution of a sufficient statistic.This paper extends the results of the research reported at the XV Seminar on Stability Problems for Stochastic Models in Perm which were published in [1,2]. Some new results were discussed at the International Conference in honor of N. G. Chebotarev [3].…”

mentioning

confidence: 60%

“…Moreover, due to its algebraic structure, each shift family possesses some additional properties which rather stringently determine the choice of the coefficients c~(g) and the function h(t) as is described in the following statement proved in [2]. into (7) with regard to the definition of an(t) we obtain another form of the p.d.f.…”

Section: Definition 3 the Family (1) Is Called Regular If The Suppormentioning

confidence: 97%

“…Mathematical grounds of this approach with respect to a special case of exponential shift families were considered in [2]. These grounds are essentially based on the investigation of the structure of the families under consideration and are of separate interest being of the type of characterization results.…”

Section: H'(t) -Q(w0(g) + V~(g)t~)h'(w0(g) + V~(g)t)ly~(g)i Q(tg) Fmentioning

confidence: 99%

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Completeness conditions for sufficient statistics in shift families and unification of estimation procedures

Sapozhnikov

1997

J Math Sci

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We show that the consideration of algebraic properties of exponential shift families allows one to simplify and unify the procedures of determination of the distribution of a sufficient statistic. We also present a new method for the estimation of a density function based on the determination of the distribution of a sufficient statistic.This paper extends the results of the research reported at the XV Seminar on Stability Problems for Stochastic Models in Perm which were published in [1,2]. Some new results were discussed at the International Conference in honor of N. G. Chebotarev [3]. It was shown in those papers that by the consideration of algebraic properties of exponential shift families one can considerably simplify and unify the procedures of determination of the distribtition of a sufficient statistic in these families as well as solutions of estimation problems. In this paper, an account of these results is presented and one new method for the estimation of a density function is described which does not require any additional statistical procedures besides those of determining the distribution of a sufficient statistic.Let X be a homogeneous space of a coherent Lee group G satisfying the second countability axiom, and let /~ be a quasiinvariant measure with the modular function A on the q-field Bx of Borel sets of A'. Throughout the paper, we shall assume that G acts on ,t' by left shifts and write g-x,, instead of (g. x 1,..., g-z,,) to emphasize that g is a translation of the vector x,~. Definition 1. A family of probability density functions (p.d.f's)on (X, Bx,p) is called a shift family with the generating p.d.f, f(x) and the family ~ = g~0 with ~0 being a random element having f(z) as its p.d.f, will be called a shift family.As a parametric space of a shift family one can take the group G itself, but this is not the best way to introduce parametrization since the correspondence between the elements g E G and the densities f(. I g) = f(g-t')A(g) is not one-to-one. To improve the situation, it is better to consider as a parametric space the set of left co-sets of G with respect to a stationary subgroup of the generating density K = {g: f(z I g) = f(z) for all x E X} (see [5]). This space and its element will be denoted by O and 0, respectively. It is clear that f(. ]gx) = f(" 192) for all gx, g2 E 0, where 0 is a fixed element of e = G/K while the random elements gx~0 and g2~0 may be different. On the contrary, from the analytical point of view it is much more convenient to operate with elements of the group G, so in what follows we shall use the notation f(z I g), g E G, remembering where necessary that f(z ] g) is one and the same for all g E 0, 0 E O. For example, if T,,(Xn) = Tn(X1,...,Xn) is a sufficient statistic for the parameter g E G, then the same statistic is also sufficient for the parameter 0 E O. Moreover, the converse is also true by virtue of the Halmos-Savage characterization theorem (see [6]).Denote by q(t, 9) the main factor in the characterization criterion, namely, letWe shall a...

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confidence: 60%

Section: Definition 3 the Family (1) Is Called Regular If The Suppormentioning

confidence: 97%

Section: H'(t) -Q(w0(g) + V~(g)t~)h'(w0(g) + V~(g)t)ly~(g)i Q(tg) Fmentioning

confidence: 99%

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Completeness conditions for sufficient statistics in shift families and unification of estimation procedures

Sapozhnikov

1997

J Math Sci

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“…Comparing the proposed formula to the formula for the maximal invariant a~" l (T,~ (Xn))x~,, density function. we can see that H, .... (u~,n) = r Now the theorem follows from general ideas concerning relations between the unbiased estimators with uniformly minimum variance and maximal invariant density functions given in [10] or [3,4].…”

Section: Qk(to =~O'~(to -Tt-~:(uk~)))p((r~:(to--tt_~:(m:t))) ' Ifukmentioning

confidence: 98%

“…In our previous paper [4], it was shown that the structure elements of an exponential shift family satisfy rather strong restrictions. Namely, cr(g) = c r (e)V(g) + at(g), where a(g) satisfies the following set of equations:…”

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On optimal estimators of shift population density functions and conditions of their consistency

Sapozhnikov

1997

J Math Sci

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Let ~or(x) = (~ol (z), ~o2(z) ..... ~o~(x)) be a linear independent (with constant) system of functions on a homogeneous space X of a coherent Lie group G satisfying the second countability axiom. We shall assume that the components of ~o(x) are functions of C(t)-class and the linear space generated by the functions 1, ~o(z) is invariant with respect to the left shift operators lb(x) ::r ~b(g-tz), g G G. If V(g) is a matrix of the group G representation in the above-mentioned base, then it has the block form , where e is the unit of the group G, will be called the generating density of the considered family and the population ~ = g~0, where the random element ~0 has f(x) as its density function, will be called the exponential shift family. In our previous paper [4], it was shown that the structure elements of an exponential shift family satisfy rather strong restrictions. Namely, cr(g) = c r (e)V(g) + at(g), where a(g) satisfies the following set of equations:r(g~g~) = ar(g~)+ar(g2)V(g~) ' a(e) = 0for arbitrary elements gl, g2 E G. If a(g) is given, then h(x) is uniquely determined from the set of equations

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Sapozhnikov¹

2001

Journal of Mathematical Sciences

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Application of algebraic properties of statistical models to deriving distributions of statistics

Cited by 4 publications

References 11 publications

Completeness conditions for sufficient statistics in shift families and unification of estimation procedures

Completeness conditions for sufficient statistics in shift families and unification of estimation procedures

On optimal estimators of shift population density functions and conditions of their consistency

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