Chi robust tests in some parametric models(Praca wpiynqla, do Redakcji 16.10.1990)

ROCZNIKI POtSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I IE MATEMATYKA STOSOWANA XXXV (1992)

L

M

, P

M

Wroclaw

Chi robust tests in some parametric models

( Praca wpiynqla, do Redakcji 16.10.1990)

S u m m a ry . For the exponential-scale and Pareto-shape testing problems the unbiased tests based on order statistics are considered. Under the violations defined by hazards and of the contamination type the tests with the most robust (stable) power functions axe explicitly constructed.

L. Introduction. Robustness of the test concerns the insensitivity of its characteristic (size, power, risk) to small deviations of the actual situation from the idealized theoretical model. In this paper we are interested in the approach to the robustness given by Zielinski ([II], [12]). This means, the purpose is to investigate the performance of the test characteristic when passing from the theoretical model to the supermodel, i.e. to the extended model described by the violation. Then the robustness of the test is identified with the stability of its characteristic under a given violation. Size and power robustness problems have been studied among others in [4], [7], [15] and [8], [9], [II], [14] respectively. In the paper the robustness of the test concerns the behaviour of the power function identified with its graph on the plane, and is measured by tire area of the region in which the power function can oscillate under a given violation. We confine ourselves to the exponential and Pareto testing models. The above models are widely used in practical problems (see e,g. [5], [6], [IT]).

2„ Notation. Let 2L be a random sample of size n, n > 2, from tire distribution function (d.I.) F with density / . Denote by X\,n < ... < X mn tire order statistics of A. Define Acun = sup{£ : F (x) = 0}, if it is finite.

The corresponding random variables Di:n = (n — i + l)(A i.n — A';_kn),.2 =

lj, ..., n, are called normalized spacings and the dT. of ^ ciiDi:n is denoted by Fa. The inverse and failure rate functions are defined by F~1(x) = inf{y : F(y) > x} and rF(x) = f(x)/ (l — F{x)) respectively. The notation F\j, T 7 means F\(x) = F(\x), F7(z) = F(zr7), x > 0, Moreover, p denotes the Lebesque measure on R2. If rp is nonincreasing then we write F G DFR.

3L Exponential model. Let I be a sample of size n, n > 2, from the exponential cLf. K\(x) = 1 - exp(-Aar), x > O^with an unknown scale parameter A > 0.. We consider tire problem of testing the null hypethesis H q : A > Ao (Ao > 0) against the alternative hypothesis Hi: A < Ao by using the unbiased tests based on normalized spacings. Given a G (0,1), each considered test y?a(X ), a = (a-i,..., an) > Q has the critical region of the form {£ ^ =1 «iA :n > ca,a}, where sup EKx(pa(2Q = a, and EF(paf2Q denotes the expectation of y>a(X ) if the sample comes from ic.

Suppose that due to measurement errors the observations are slightly disturbed and come from an unknown d.f. F, where F belongs to w(K\) -

“some neighbourhood” of AV Then

(1) Ra = : A > 0,?/ G ( inf Epipa(2Q, sup Epipa(X_))},

F€ir{hx) F £ n (K x)

if it is finite, describes the dispersion of the power function (risk) under the violation 7r and gives us a measure of robustness of the test y v

In the set of life d.f.’s F (i.e. F (0) = 0), consider the violation of the hazard type, Le.

KG_,H{Kxr) = i F : rGx < r F <rH^}, w-here G / if, rG < rK < rH (see [2}), and of Tukey type, i.e.

ird(K\) = {(1 - e)K\ + £rKcx : d <jc < 1}, where 0 < < 1, A > 0.

Given 7r G { k g , h -> ^1}? we find the test y?^ such that R± = min{R& : af> 0}.

We prove first some auxiliary lemmas. Let F and G be the life d.R’s.

L em -MA X. If F or G G D'FR and rG < rp then rGG~l < rFF ~l . Tire proof follows directly from the definition of DFR class.

L e m m a 2. If rcG ^1 < rpF W'1 and a > 0 then Ga < A*

P r o o f. By assumption, G~1F(x) — x is nondecreasing. Let I be a sam- ple from F and a > 0. Obviously,

n

^ ^ a>i{n i T l)(AG;-n ^

2 — 1

On robust tests in some parametric models 89

< Y , - » + l)(G _1F(A'ijn) - G -'F fX i_ 1<n)).

i = i

From the faet that G -1 F(Xj)., i = 1 ,..., n, are i.i.d. random variables from 6*, tire proof is complete.

L e m m a 3. Let F = (1 - e]K + sK c, G = (1 - e)K + sK d. If O < s < 1,

d <_e < 1 then i ' g G~1 < rp F ~l .

P ro o f. By assumption, dG~l < cJF~l . Consequently for 0 < re < 1 rpF~1(x) — =(1 - .r)“ 1£'((l — d)(l - K(dG~x(x)))

- (1 - c)(l - # (c F _1(:r)))) > 0,‘

whieh completes the proof.

Let 7r G { k g , h , TTrf}, £ g X be finite and G,LT G DFR. We introduce the following notation. Define p; = EaDi-.n — EfiDi-.m i = l,...,.n , where in the case 7r = 7F^ we put H — K .,G = (l — e)K sKd. Note that under tire definition of 7r, pi > 0 for each i. Let S.n be the permutation group on { l , .. . ,n). Given a = (a*,. ..,a n), r G 5n, let a(r) = (aT(i),.. .,a T(n}), fti^n < ... < amn be n ordered coordinates of a and N{i) be the rank of Pi in (Pn:n, • •-,Pi:n)- If Pi = Pj then N(i) < N(j) for i < j. Let n° = {^N(l t:n , • • •, 0,N(nFn ) € ( L, 0, . . ., 0).

T h e o r e m 1. If 0 < a < 1, a > 0, r G Sn then R^o < R^T).

P r o o f. Let a G (0,1) be a given significance level. By the definition of the test pa we obtain Ca,a — [/C^]—1 (1 — o)/Ao, and hence Eppa(X_) = l - 4 ( [ ^ a ] _1( l - « ) / V ) .

By using Lemmas 1, 2, if 7r = it g . h or Lemmas 2, 3, if 7r = 7r^, (1) gives

n

(2) R& = A0 a,-/?,-/[A'o]_ 1 (1 7 o ).

Z=1

Given r G we have

n n

^ ^ 0/N(iy.nP'i — ^ ^ E-a_ Ea(T)i

i = 1 i=l

which completes the proof.

T h e o r e m 2. Let 0 < o < 1 and a > 0 be such that an:n/akin <

/or some L. Then far the vector b with coordinates

(S') hi — ai>n for i ^ k^n, 6j — q f —*-+-Axn/ n—*-i-1-:n /or i — L,n,

where q = (ak:npn- k + + an:nlh:n)/{ak-.n + amn) it holds R#> < R^o.

P roof. Let a £ (O ^l , 1 ). Without loss of generality we assume that «^ > 0 for i = 1 ,..., 7%. Let b he defined by (3). Then

0'k:niPn — k + l:n T O ninPlin

^kPn — k+V.n T 1 ^/bk T 1 ^{/ b n} — 1 ^{/ ttk\n} T 1 ^{I Omn} and b'k < (Ikin'

From the fact that the function K a{t), t > d, is strictly Schur-concave in ( 1/ ai , . . 1 /an) (see e.g. [3]) we have [/t'£]-1('l — cr) < [/tT]-1 (l — a), and the result follows by (2) and Theorem 1.

C o r o l l a r y . Let 0 < a < 1 ,Pi ^ Pj fo r some i^j- Then for the vector

b SUch that b\ — ( p i m T Pn:n^)/^Pn:nt bn — ^{(p i;n} T P m n )/ ‘^Pl'.ni b{ — 1 for i 1,71, it holds Rbo < R( ¹ fi -

This means, the test <pbo is more robust than the uniformly most powerful test.

T ^{he o r e m} 3. Let to = exp(—3/2), tn = exp( —(.n + 1)) for n > 2. Then there exists t £ (tn, 1] such that for every 0 < a < t and all q > 0 Reo < Ra-

P roof. It is easily seen that we may restrict our attention to the vectors

n _

a > 0 with Y ttj- = 1. Based on the results due to Bock et al. [3], Kgj^t), t >

n *— l n

(— logtn) Y aii is Schur-concave in a > 0. Consequently, if a > 0, Y ai = 1

i=l i=l

then K{t) < K a{i) for l > — logfn, and hence [iifa]-1(l — a) < I(~ 1(X — o ) for a < tn. On the other hand pPn < Y aiPi- n

By (2) the proof is complete. i—1

If G = K ( // = if, as in 7r^) then by the stochastic monotonicity of the normalized spacings (see [I]) one can obtain that pi is decreasing (increasing) in i. Consequently, the most robust is the test based on Dn.n (D\.n).

4. Pareto model. Let X be a sample of size n, n > 2, from the Pareto d.f. P%{x) — 1 — a,*"7, x > 1, with an unknown shape parameter 7 > 0.

To verify the hypothesis H q : 7 > 70 (70 > 0) versus H[\ 7 < 70, the unbiased o-level tests 0a(2Q> « = (« i, • • •, an) > 0

with the critical regions { Y ai(n - i + l)\og(XUn/Xi_1:n) > da,a}, where sup EP^ifa{2Q = a-, are n

considered.

In the set of d.f.’s F such that F(l) = 0, consider the violations

7T g , h ( P7) = { F : rGl <.rF < rHl], G ± //, rG < rP < rH,

w*d(P*) = {(1 - e)P 7 + eP ^ : d < c < 1}, 0 < .£, d < 1, 7 > 0.

On robust tests in some parametric models ⁹¹ Let 7r £ Then the robustness of tire test ipa under tire violation

% is measured by the quantity

R* = /*{( 7, y) : 7 > 0,2/ e ( inf £ f W 2Q, sup PF^a(A))}- Fe%(P^)

Assume that E g log A' is finite, rnjrp, t q J t p are decreasing functions and write & = (n - i + l)(EGrlog(Ai:n/Ai-i;.n) - EH\og(Xi:nJ X i-i:n)), i = 1-,..., n., where for the case ir = ired we put H = P and G = (1 — e)P + eP d.

Since the Pareto d.f. is closely connected with the exponential d.f. through the relation K(t) = P(exp(f)), / > Q, it follows that all results of the previous section remain valid if pi and Ra are replaced by pi and respectively.

Moreover, if G = P (H = P, as in 7r^) then the most robust is the test based on log(A^n/AX-i-rn), (nlogX hn).

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