# Drift and the Risk-Free Rate

Authors: Anda Gadidov; M. C. Spruill
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

It is proven, under a set of assumptions differing from the usual ones in the
unboundedness of the time interval, that, in an economy in equilibrium consisting of
a risk-free cash account and an equity whose price process is a geometric Brownian
motion on[0,∞), the drift rate must be close to the risk-free rate; if the drift rate
μ and the risk-free rate r are constants, then r=μ and the price process is the
same under both empirical and risk neutral measures. Contributing in some degree
perhaps to interest in this mathematical curiosity is the fact, based on empirical
data taken at various times over an assortment of equities and relatively short
durations, that no tests of the hypothesis of equality are rejected.


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## Body

## 1. Introduction

In the Black-Scholes model of a market with a single equity, its priceSt is a geometric Brownian motion (GBM) satisfying for time t≥0 the stochastic differential equation(1.1)dSt=μStdt+σStdBt,
where the volatility σ, the drift rate μ, and the rate r for the risk-free security are all constants. The stochastic process Bt, 0≤t is a standard Brownian motion. In the formulation of Harrison and Kreps [1] the process is on t∈[0,T], T<∞, and is defined on the probability space (Ω,ℱT,PT), where the filtration ℱt=σ(Bs,0≤s≤t), t≥0, is that generated by Bt. They show in this case that under equilibrium pricing for their securities market model, allowing only simple trading strategies, there is a measure PT* equivalent to PT and prices can be expressed as expectations with respect to PT*. Furthermore, under PT*, e-rtSt is a martingale on [0,T] with respect to ℱt. There are three free parameters in the model, μ, σ, and r. It is shown here that, if the equity's prices are given by (1.1) on [0,∞) and again only simple trading strategies are allowed on finite but arbitrary sets of nonrandom times, then there is an equivalent martingale measure and pricing with respect to it in the same manner represents a viable pricing system in the sense of Kreps [2] if and only if r=μ. In this case, there are really only two free parameters.Besides the results found in Lemma4.5 relating to the Black-Scholes model, results of a somewhat more general nature in which r, μ, and σ depend upon time deterministically, can be found in Lemma 4.1.The arguments given here are for an economy consisting of a single equity and a cash account. To the extent, therefore, that such models are pertinent to actual equities prices, an empirical investigation ofμ=r for real market data is of interest. Assuming that the model is true for our empirical data consisting of daily closes of some selected equities, the hypothesis that r=μ is tested and in no case is the hypothesis of equality rejected by these optimal tests.The organization of the paper is as follows. First the terms, definitions, and basic results of Harrison and Kreps [1] are recalled in the context of our model on [0,∞). Then connections are made between a presumed empirical GBM process with drift μ and the martingale arising from viability. It is not assumed but shown that the price process must also be a GBM under the equivalent measure; its drift is r and its volatility agrees with that of the empirical GBM. It is shown also that the arguments used in [1] to obtain this result on [0,T] cannot generally be used here. Next, the main result on μ=r is presented; namely, that under equilibrium the drift of the empirical GBM must be the risk-free rate. If the price process is a GBM under the empirical measure, then a consequence of viability is that it is also a GBM under an equivalent (risk-neutral) measure. Finally, the development and results of our hypothesis tests appear in Tables 1 and 2.Table 1
P values for testing r=μ. None are significant at 0.05.
SymbolDatesUMPULRTEffect sizeAAPL1-5-2001 to 12-6-20010.4160.4135.2×10-2APD1-5-2001 to 12-6-20010.5200.5173.8×10-2C1-1-1983 to 1-1-20000.0590.0593.1×10-2CVS8-30-2001 to 11-30-20010.4200.3979.9×10-2DAL11-03-1994 to 02-02-19950.9140.913-8.3×10-3F11-03-1994 to 02-02-19950.2580.226-1.5×10-1IBM3-29-1984 to 07-31-19840.6180.6114.8×10-2K11-12-1999 to 02-10-20000.9960.9951.5×10-1LMT1-5-2001 to 12-6-20010.2370.2775.1×10-2PG11-12-1999 to 02-10-20000.4570.4459.2×10-2Table 2
P values for testing r=μ, March 1, 2007, to August 30, 2007.
SymbolUMPULRTEffect sizeAAPL0.0860.0681.6×10-1APD0.2720.2611.0×10-1C0.5960.593-4.7×10-2CVS0.2570.2441.0×10-1F0.9530.953-5.3×10-3GE0.5280.5255.6×10-2GM0.9390.939-6.8×10-3IBM0.1210.1031.5×10-1K0.3890.3827.8×10-2LMT0.9740.9742.9×10-3PG0.9110.9111.0×10-2Proofs of technical details most pertinent to the main ideas appear in the body of the paper; proofs of more tangential ones have been placed in the appendix.


## 2. Viability, the Extension Property, and Equivalent Martingale Measures

The notation and assumptions are those of [1] except that here there is an infinite rather than a finite horizon. Thus, there is a linear space X of functions x:Ω→R which are random variables defined on a probability space (Ω,ℱ,P). The points ω∈Ω represent states of the world; the points x∈X represent bundles of goods in some abstract economy. A subspace M⊂X represents the space of bundles that can be constructed out of marketed bundles of goods. There is a bounded linear functional π defined on M, with π(m) representing the market price of m∈M and a collection A of agents represented by complete transitive binary relations ≽ on the space X. The pair (M,π) is viable (as a model of economic equilibrium) if there is an order ≽∈A of the above specifications and an m*∈M such that π(m*)≤0 and m*≽m for all m∈M such that π(m)≤0. Letting Ψ denote the collection of positive bounded linear functionals on X, Kreps [2] proves the following lemma.Lemma 2.1.
The price system(M,π) is viable with respect to X if and only if there is a ψ∈Ψ such that ψ∣M=π.Here the securities market model of Harrison and Kreps [1] is extended to [0,∞) and involves, as there, a risk-free security with rate 0 and a security whose price at time t under the state of the world ω∈Ω is Zt(ω), where the second-order stochastic process Zt is measurable with respect to a filtration ℱt⊂ℱ∞=σ(∪s≥0ℱs)=ℱ. Only simple trades are allowed. Simple trading strategies are denoted by θ and implicit in θ's description is a finite set of nonrandom trading times 0≤t1<t2<⋯<tk. The 2-vector of functions θ(t)=(θR(t),θE(t)) has elements which indicate the units held in the risk-free asset and the equity so that the value of the portfolio at time t is θ(t)·V(t), the ordinary inner product of the vector θ with V(t,ω)=(1,Zt(ω)). The function θ is ℱt-adapted, and, for each ω,θ(t) is constant on ti-1≤t<ti. As in [1] simple trades involve finite arbitrary collections of nonrandom points of time at which trades occur but, in contrast, here there is no fixed “consumption time.” Instead, if the trading strategy has its last trading time at tk as above, then at the last time all is placed in the risk-free cash account so that, at times t≥tk,θ(t)=(θ(tk)·V(tk),0). The subspace M is the linear span of the random variables θ·V, where θ is a simple trading strategy. It is implicit that EP[∥θ(tj)∥2]<∞ for each j=1,2,…,k, an assumption made throughout. A simple trading strategy θ is self-financing if θ(tj-1)·V(tj)=θ(tj)·V(tj) for each j=1,2,…,k. If a security market model (M,π) is viable and if θ is self-financing, then θ(0)·V(0)=π(m) for m=limt→∞θ(t)·V(t)=θ(tk)·V(tk). The existence of an equivalent martingale measure is asserted in Lemma 2.2 and its proof can be carried out as by Harrison and Kreps.Lemma 2.2.
If(M,π) is viable with respect to X=L2(Ω,ℱ,P), then there is an equivalent measure P* with dP*/dP∈X and under this measure Zt is a martingale with respect to ℱt.The risk-free security of concern here has instantaneous rater(t) at time t, the price process St solves (3.1), and the corresponding trades relative to the process of real interest here, Ṽt=(e∫0tr(s)ds,St), can be obtained by taking Zt=e-∫0tr(s)dsSt and Vt=e-∫0tr(s)dsṼt (see [1, Section 7]). Under a viable pricing system, it follows that e-∫0tr(s)dsSt is a martingale with respect to ℱt. Thus, pricing for a final transaction time T is given by π(m)=EP*[e-∫0Tr(s)dsm].


## 3. Price Process underP and P*

It is assumed that underP the price process St(ω), t≥0 solves the SDE(3.1)dSt=μ(t)Stdt+σ(t)StdBt
analogous to (1.1) but with deterministically varying r(·), μ(·), and σ(·) subject to the following assumptions:  (A1)
the functionsμ(·) and r(·) are continuous on [0,∞) and σ(·) is absolutely continuous with a derivative bounded on compact intervals,(A2)
for some0<σL<σU<∞ and all s≥0, σL≤σ(s)≤σU holds,(A3)
the risk premiumρ(s)=σ-1(s)(μ(s)-r(s)) is uniformly bounded: for some ρU<∞ and all s>0, |ρ(s)|≤ρU,(A4)
the risk free rate is uniformly bounded: for some0<rU<∞ and all s≥0, |r(s)|≤rU.The solutionS to (3.1) at time T given its value at time 0≤t<T can be written explicitly as(3.2)ST=Stexp⁡{∫tTμ(s)ds-12∫tTσ2(s)ds+∫tTσ(s)dBs}.Under the measureP* the continuous discounted value process (3.3)Xt=Stexp⁡⁡{-∫0tr(s)ds}
is a martingale, so by Theorems 4.2 (Chapter 3) of Karatzas and Shreve [3] and IX.5.3 of Doob [4], for example, provided the quadratic variation process 〈X〉t(ω) is an absolutely continuous function of t for P*-almost every ω with a nonzero derivative, there exists Wt a Wiener process under P* and a measurable ℱt-adapted process Φ(t,ω) such that d(e-∫0tr(s)dsSt(ω))=Φ(t,ω)dWt(ω). Moreover 〈X〉t(ω)=∫0tΦ2(s,ω)ds. It also follows by Lemma 3.1 that under the equivalence of P and P* one has Φ2(s,ω)=Xs2(ω)σ2(s).Lemma 3.1.
Let conditions (A) be satisfied, and letSt satisfy (3.1), where Bt is a standard Brownian motion under P. Suppose that Xt=e-∫0tr(s)dsSt is a martingale under the equivalent measure P*. Then, under the measure P*, 〈X〉t(ω)=∫0tXs2(ω)σ2(s)ds. See proof in the Appendix.By Ito’s formula,d(e-∫0tr(s)dsSt)=-r(t)e-∫0tr(s)dsStdt+e-∫0tr(s)dsdSt, so(3.4)dSt=r(t)Stdt+e∫0tr(s)dsΦ(t,St)dWt.
It follows from Lemma 3.1 that, under the equivalence of P and P*, one has (3.5)Φ(t,ω)=e-∫0tr(s)dsSt(ω)σ(t)
and hence that(3.6)dSt=r(t)Stdt+σ(t)StdWt.
Thus, if T>t, then under P*(3.7)ST=Stexp⁡⁡{∫tTr(s)ds-12∫tTσ2(s)ds+∫tTσ(s)dWs}.It has been shown that under the equivalent martingale measureP* the price process satisfies on [0,∞) an SDE with a standard Brownian motion Wt, t≥0, the same volatility as the empirical one, and a drift coincident with the risk-free rate. How does this result on [0,∞) relate to the results of [1] on [0,T], 0<T<∞?Harrison and Kreps show that there is a probability measurePT* equivalent to PT under which e-∫0tr(s)dsSt, t≤T is a martingale, where PT is P restricted to ℱT. Karatzas and Shreve [3, Section 3.5A] (see also [5, Section 1.7, Proposition 7.4]), show that there is a probability measure P̃* on (Ω,ℱ∞) with the property that for every T>0 the measure restricted to ℱT is PT*. They point out, however, that generally P̃* need not be equivalent to P. That is, in fact the case if ∫0∞ρ2(s)ds=∞ as can be seen from Proposition 3.2 (see also the remark preceding Example 7.6, Chapter 1, of [5]). In that case, not only is our measure P* not obtained from the arguments of Harrison and Kreps, it is singular with respect to one that is. The scope of our main result below would be more limited if the finiteness of the integral were assumed.Proposition 3.2.
If the measureP̃* satisfies ζ=dP̃*/dP∈L2(Ω,ℱ,P) and, for each t>0, P̃*|ℱt=Pt* then P̃*⊥P* if ∫0∞ρ2(s)ds=∞. See proof in the Appendix.


## 4. Relationship betweenμ and r

Letα>0 be arbitrary. Suppose that there is an essentially disjoint collection 𝒞α={Ij}j≥1 of subintervals of the real line satisfying (4.1)⋃j≥1Ij⊂{s≥0:|μ(s)-r(s)|≥α}.
Under the assumption that the functions μ and r are continuous observe that, unless μ(s)=r(s) for all s>0, for α>0 sufficiently small, the collections 𝒞α are nonempty. Denote the Lebesgue measure of an interval I by m(I) and fix α>0 for which 𝒞α is nonempty. The equities price process under the actual probability measure P is given in (3.2) and if the pricing system is viable then under P* the same process is given by (3.7). Lemmas 4.2, 4.3, and 4.4 are used in the proof of Lemma 4.1, the key to our main result.Lemma 4.1.
Under the assumptions (A) andP[S0=s0]=1 for some s0>0, if (M,π) is a viable pricing system for M, the class of marketable claims under simple trading strategies, then for every γ>0, the set of indices j for which m(Ij)≥γ is at most finite.Proof.
Suppose that(M,π) is viable and the claim is not true. Then, for some γ>0, there is an infinite collection of such intervals [aj,bj]⊂Iij, j=1,2,… with bj-aj≥γ. Writing
(4.2)|μ(s)-r(s)|=(μ(s)-r(s))+-(μ(s)-r(s))-,
assume without loss of generality that thereon |μ(s)-r(s)|=(μ(s)-r(s))+≥α. Then, also without loss of generality, one can assume that there are points dj<uj, where aij≤dj<uj≤bij, and ∫djuj(μ(s)-r(s))ds=β/2=c, j=1,2,…, where β=γα.
Consider a sequence of trading strategiesθn. Under the simple strategy θn, buy (nSd1)-1e∫0u1r(s)ds shares of the equity at time d1 to spend n-1e∫0u1r(s)ds units. At this time also sell n-1e∫0u1r(s)ds of the risk free security to spend -n-1e∫0u1r(s)ds at time d1. Cash flow at time d1 is then 0. At time t=u1, sell the equity to spend -e∫0u1r(s)dsSu1/nSd1 and redeem the bond to spend n-1e∫0u1r(s)dse∫d1u1r(s)ds. “Cash” on hand at this stage is
(4.3)e∫0u1r(s)dsn(Su1Sd1-e∫d1u1r(s)ds).
Invest it in the risk-free security. At times d2,d3,…,dn, repeat this, buying at time dj, (nSdj)-1e∫0ujr(s)ds shares of the equity and selling n-1e∫0ujr(s)ds of the risk-free security to spend a total of 0. At time uj, sell the shares to obtain -e∫0ujr(s)dsSuj/nSdj and redeem the bond to spend n-1e∫0ujr(s)dse∫dju1r(s)ds and invest the “cash” in the risk-free security. At time un, there will be an amount
(4.4)Cn=n-1∑j=1ne∫ujunr(s)ds×e∫0ujr(s)ds(SujSdj-e∫djujr(s)ds).
The price of this will be
(4.5)π(Cn)=EP*[e-∫0unr(s)dsCn]=EP*[1n∑j=1n(SujSdj-e∫djujr(s)ds)].
Under the geometric Brownian motion model (3.1), the term inside the expectation is n-1∑j=1nYj, where
(4.6)Yj=e∫djuj(μ(s)-σ2(s)/2)ds+∫djujσ(s)dBs-e∫djujr(s)ds,
an average of independent, but not identically distributed, random variables. By Lemma 4.2, there is a subsequence nk and an L>0 such that
(4.7)1nk∑j=1nke∫djujr(s)ds⟶L
and, under P,
(4.8)Jk=Vnk=1nk∑i=1nkYi⟶pL(ec-1).
Since a sequence converging in probability has a subsequence which converges almost surely, there is a subsequence k′ such that Jk′→a.s.PL(ec-1). By equivalence of P and P*, that convergence is also almost surely P*. By Lemma 4.4, the sequence Jk is uniformly integrable under P*, so expectations converge (see [6, Theorem 5.4]) and one concludes that EP*[Jk′]→L(ec-1). On the other hand, by Lemma 4.3, EP*[(nk)-1∑i=1nkYi]=0 for every k. It follows that c=0, a contradiction.Lemma 4.2.
Under conditions (A), there is a subsequencenk and L>0 such that
(4.9)1nk∑j=1nke∫djujr(s)ds⟶L,1nk∑j=1nkYj⟶pL(ec-1)
(see proof in the Appendix).Lemma 4.3.
EP*[Yj]=0 (see proof in the Appendix).Lemma 4.4.
Under conditions (A) the sequencen-1∑j=1nYj is uniformly integrable under P* (see proof in the Appendix).Lemma 4.5.
Ifμ(·) and r(·) are constant and if the model is viable, then r=μ.Proof.
Suppose that|r-μ|=d>0. Then, for 0<α<d, and any γ>0, essentially disjoint intervals I in 𝒞α can be chosen in such a way that there is an infinite collection satisfying m(I)≥γ. By Lemma 4.1 this violates the viability of the model.
The next most interesting case is when(4.10)μ(s)-r(s)=ρσ(s),
where ρ, the possibly varying risk premium, is assumed constant.Lemma 4.6.
Under conditions (A), if the pricing system(M,π) is viable and if (4.10) holds for all s≥0, then ρ=0.Proof.
Assume thatρ≠0, and let 0<α<|ρ|σL. Then, for any γ>0 essentially disjoint intervals in 𝒞α can be chosen in such a way that there is an infinite collection, contradicting Lemma 4.1 unless ρ=0.
Returning to the market expressed in terms ofV(t), define the functional ψ on X=L2(Ω,ℱ,P), where P=Pμ is the empirical measure under which St satisfies (3.1), by ψ(x)=∫x(ω)dP(ω). Our interest in the following theorem centers on the pricing system defined on M by π(m)=∫m(ω)dPr(ω), and Pr is the measure corresponding to the process solving (3.6).Theorem 4.7.
For the Black-Scholes model on[0,∞) (r=0, μ, and σ are constants), the pricing system (M,π) is viable with respect to X if and only if μ=0.Proof.
By Lemma4.5 it is known that, if the system (M,π) is viable then r=μ. It suffices to show that, if r=μ, then the pricing system given by π is viable. According to Lemma 2.1, it suffices to show, as it plainly does here, that ψ extends π.


## 5. Empirical Considerations

For these considerations to have relevance to real data, equities prices should be adequately modeled as solutions to SDE (3.1). For roughly a century, models in agreement with (3.1) [7–10] have appeared in the literature, and we will assume this model here. Statistical tests of r=μ, assuming the Black-Scholes model over a suitably brief time span, are then employed on a small set of data. The results found in Tables 1 and 2 are consistent with the truth of the hypothesis of equality.


### 5.1. UMPU Test

Assuming the model (3.1) with μ and r constant, some tests of μ=r are developed here based on readily available daily log return data, Rt=ln(St+1/St), where Rt are i.i.d. N(μ-ξ2/2,ξ2), and applied to different underlying equities at various historical times.LetXi=Ri-r. Setting η=E[Xi] and ξ2=Var(Xi), the hypothesis pair H0′:μ=r versus Ha′:μ≠r becomes H0:η+ξ2/2=0 versus Ha:η+ξ2/2≠0, which will be tested based on observing X1,…,Xn i.i.d. N(η,ξ2). Writing (5.1)f(x∣η,ξ2)=1(2πξ2)n/2exp⁡⁡{-12ξ2∑i=1n(xi-η)2},
one has that (∑Xi2,∑Xi) are joint sufficient statistics for (-1/2ξ2,η/ξ2)=(θ1,θ2). It can be seen, based upon the theory of tests of a single parameter from an exponential family (see [11]), that a uniformly most powerful unbiased (UMPU) test ϕα of size α exists for testing the hypothesis of interest here,(5.2)H0:θ2=-12versusHa:θ2≠-12.
Furthermore, denoting by Pα(θ) the power function of the test ϕα, one has the following result.Lemma 5.1.
The test which rejectsH0 if |τn|>zα/2, where
(5.3)τn=n(X¯n/Sn2+1/2)1/Sn2+3/2+Sn2/4,
satisfies for all θ in H0(5.4)lim⁡n→∞⁡P[|τn|>zα/2∣θ]=Pα(θ).
See proof in the Appendix.The results of testing hypothesis (5.2) for various equities at varying times are found in Table 1 and for a fixed set of times in Table 2. The risk-free rates were determined from the US treasury for the corresponding time spans at each initial time and, in the latter case, reveal that none of the drift rates differ significantly (at α=0.05) from r=2.0113×10-4, the daily rate for 26-week treasury bills during that stretch. In the tables, the entry Effect Size is η̂/σ̂+σ̂/2, a rough estimate of μ-r in terms of the volatility.It is perhaps surprising that there were no significances especially in the case of C in Table1 and AAPL in Table 2, the former because of the long time span and the approximation assuming a fixed risk-free rate in a world in which it is constantly changing, and the latter because on June 29, 2007 Apple introduced its first iPhone a major milestone in the company's rising fortunes. The former is in line with the sample size and observed effect size while the latter is consistent with a fundamental change.


### 5.2. Likelihood Ratio Tests

Likelihood ratio tests present an alternative possibility. They are known to be optimal (see, e.g., [12]) in large samples. According to Wilks’ theorem, the null hypothesis should be rejected when -2lnλ(x) is too large. Setting(5.5)w*=2[1+∑j=1nxj2n-1],
one has the following.Lemma 5.2.
The likelihood ratio test of (5.2) of size α rejects H0 if the test statistic
(5.6)τ=n(ln⁡⁡w*-1)+1w*∑j=1n(xj+w*2)2-nln⁡(∑j=1n(xj-x¯)2n)
exceeds χα2. See proof in the Appendix.P values are found in Tables 1 and 2 both for the UMPU test and for the LRT and one observes that they are quite close and, again, there are no significances at α=0.05.


## 5.1. UMPU Test

Assuming the model (3.1) with μ and r constant, some tests of μ=r are developed here based on readily available daily log return data, Rt=ln(St+1/St), where Rt are i.i.d. N(μ-ξ2/2,ξ2), and applied to different underlying equities at various historical times.LetXi=Ri-r. Setting η=E[Xi] and ξ2=Var(Xi), the hypothesis pair H0′:μ=r versus Ha′:μ≠r becomes H0:η+ξ2/2=0 versus Ha:η+ξ2/2≠0, which will be tested based on observing X1,…,Xn i.i.d. N(η,ξ2). Writing (5.1)f(x∣η,ξ2)=1(2πξ2)n/2exp⁡⁡{-12ξ2∑i=1n(xi-η)2},
one has that (∑Xi2,∑Xi) are joint sufficient statistics for (-1/2ξ2,η/ξ2)=(θ1,θ2). It can be seen, based upon the theory of tests of a single parameter from an exponential family (see [11]), that a uniformly most powerful unbiased (UMPU) test ϕα of size α exists for testing the hypothesis of interest here,(5.2)H0:θ2=-12versusHa:θ2≠-12.
Furthermore, denoting by Pα(θ) the power function of the test ϕα, one has the following result.Lemma 5.1.
The test which rejectsH0 if |τn|>zα/2, where
(5.3)τn=n(X¯n/Sn2+1/2)1/Sn2+3/2+Sn2/4,
satisfies for all θ in H0(5.4)lim⁡n→∞⁡P[|τn|>zα/2∣θ]=Pα(θ).
See proof in the Appendix.The results of testing hypothesis (5.2) for various equities at varying times are found in Table 1 and for a fixed set of times in Table 2. The risk-free rates were determined from the US treasury for the corresponding time spans at each initial time and, in the latter case, reveal that none of the drift rates differ significantly (at α=0.05) from r=2.0113×10-4, the daily rate for 26-week treasury bills during that stretch. In the tables, the entry Effect Size is η̂/σ̂+σ̂/2, a rough estimate of μ-r in terms of the volatility.It is perhaps surprising that there were no significances especially in the case of C in Table1 and AAPL in Table 2, the former because of the long time span and the approximation assuming a fixed risk-free rate in a world in which it is constantly changing, and the latter because on June 29, 2007 Apple introduced its first iPhone a major milestone in the company's rising fortunes. The former is in line with the sample size and observed effect size while the latter is consistent with a fundamental change.


## 5.2. Likelihood Ratio Tests

Likelihood ratio tests present an alternative possibility. They are known to be optimal (see, e.g., [12]) in large samples. According to Wilks’ theorem, the null hypothesis should be rejected when -2lnλ(x) is too large. Setting(5.5)w*=2[1+∑j=1nxj2n-1],
one has the following.Lemma 5.2.
The likelihood ratio test of (5.2) of size α rejects H0 if the test statistic
(5.6)τ=n(ln⁡⁡w*-1)+1w*∑j=1n(xj+w*2)2-nln⁡(∑j=1n(xj-x¯)2n)
exceeds χα2. See proof in the Appendix.P values are found in Tables 1 and 2 both for the UMPU test and for the LRT and one observes that they are quite close and, again, there are no significances at α=0.05.



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